return (x1[1] * x2[0] - x1[0] * x2[1]) / (np.square(x1[0]) + np.square(x2[0])) register_diff(np.add, lambda x1, x2: x1[1] + x2[1]) register_diff(np.subtract, lambda x1, x2: x1[1] - x2[1]) register_diff(np.multiply, multiply_diff) register_diff(np.matmul, matmul_diff) register_diff(np.divide, divide_diff) register_diff(np.true_divide, divide_diff) register_diff(np.power, pow_diff) register_diff(np.positive, lambda x: +x[1]) register_diff(np.negative, lambda x: -x[1]) register_diff(np.conj, lambda x: np.conj(x[1])) register_diff(np.conj, lambda x: np.conj(x[1])) register_diff(np.exp, lambda x: x[1] * np.exp(x[0])) register_diff(np.exp2, lambda x: x[1] * np.log(2) * np.exp(x[0])) register_diff(np.log, lambda x: x[1] / x[0]) register_diff(np.log2, lambda x: x[1] / (np.log(2) * x[0])) register_diff(np.log10, lambda x: x[1] / (np.log(10) * x[0])) register_diff(np.sqrt, lambda x: x[1] / (2 * np.sqrt(x[0]))) register_diff(np.square, lambda x: 2 * x[1] * x[0]) register_diff(np.cbrt, lambda x: x[1] / (3 * (x[0]**(2 / 3)))) register_diff(np.reciprocal, lambda x: -x[1] / np.square(x[0])) register_diff(np.broadcast_to, lambda x, shape: np.broadcast_to(x[1], shape)) register_diff(np.sin, lambda x: x[1] * np.cos(x[0])) register_diff(np.cos, lambda x: -x[1] * np.sin(x[0])) register_diff(np.tan, lambda x: x[1] / np.square(np.cos(x[0]))) register_diff(np.arcsin, lambda x: x[1] / np.sqrt(1 - np.square(x[0]))) register_diff(np.arccos, lambda x: -x[1] / np.sqrt(1 - np.square(x[0])))
lambda ans, x, y: lambda g: g * balanced_eq(x, ans, y), lambda ans, x, y: lambda g: g * balanced_eq(y, ans, x), ) defjvp( np.fmax, lambda ans, x, y: lambda g: g * balanced_eq(x, ans, y), lambda ans, x, y: lambda g: g * balanced_eq(y, ans, x), ) defjvp( np.fmin, lambda ans, x, y: lambda g: g * balanced_eq(x, ans, y), lambda ans, x, y: lambda g: g * balanced_eq(y, ans, x), ) defjvp( np.logaddexp, lambda ans, x, y: lambda g: g * np.exp(x - ans), lambda ans, x, y: lambda g: g * np.exp(y - ans), ) defjvp( np.logaddexp2, lambda ans, x, y: lambda g: g * 2**(x - ans), lambda ans, x, y: lambda g: g * 2**(y - ans), ) defjvp(np.true_divide, "same", lambda ans, x, y: lambda g: -g * x / y**2) defjvp( np.mod, lambda ans, x, y: lambda g: np.broadcast_to(g, np.shape(ans)), lambda ans, x, y: lambda g: -g * np.floor(x / y), ) defjvp( np.remainder,
lambda ans, x, y: unbroadcast_f(x, lambda g: g * balanced_eq(x, ans, y)), lambda ans, x, y: unbroadcast_f(y, lambda g: g * balanced_eq(y, ans, x)), ) defvjp( np.fmax, lambda ans, x, y: unbroadcast_f(x, lambda g: g * balanced_eq(x, ans, y)), lambda ans, x, y: unbroadcast_f(y, lambda g: g * balanced_eq(y, ans, x)), ) defvjp( np.fmin, lambda ans, x, y: unbroadcast_f(x, lambda g: g * balanced_eq(x, ans, y)), lambda ans, x, y: unbroadcast_f(y, lambda g: g * balanced_eq(y, ans, x)), ) defvjp( np.logaddexp, lambda ans, x, y: unbroadcast_f(x, lambda g: g * np.exp(x - ans)), lambda ans, x, y: unbroadcast_f(y, lambda g: g * np.exp(y - ans)), ) defvjp( np.logaddexp2, lambda ans, x, y: unbroadcast_f(x, lambda g: g * 2 ** (x - ans)), lambda ans, x, y: unbroadcast_f(y, lambda g: g * 2 ** (y - ans)), ) defvjp( np.true_divide, lambda ans, x, y: unbroadcast_f(x, lambda g: g / y), lambda ans, x, y: unbroadcast_f(y, lambda g: -g * x / y ** 2), ) defvjp( np.mod, lambda ans, x, y: unbroadcast_f(x, lambda g: g),
) register_diff(np.add, lambda x1, x2: x1[1] + x2[1]) register_diff(np.subtract, lambda x1, x2: x1[1] - x2[1]) register_diff(np.multiply, multiply_diff) register_diff(np.matmul, matmul_diff) register_diff(np.divide, divide_diff) register_diff(np.true_divide, divide_diff) register_diff(np.power, pow_diff) register_diff( np.absolute, lambda x: x[1] * np.where(np.sign(x[0]) == 0, float("nan"), np.sign(x[0])), ) register_diff(np.positive, lambda x: +x[1]) register_diff(np.negative, lambda x: -x[1]) register_diff(np.conj, lambda x: np.conj(x[1])) register_diff(np.exp, lambda x: x[1] * np.exp(x[0])) register_diff(np.exp2, lambda x: x[1] * np.log(2) * np.exp2(x[0])) register_diff(np.log, lambda x: x[1] / x[0]) register_diff(np.log2, lambda x: x[1] / (np.log(2) * x[0])) register_diff(np.log10, lambda x: x[1] / (np.log(10) * x[0])) register_diff(np.sqrt, lambda x: x[1] / (2 * np.sqrt(x[0]))) register_diff(np.square, lambda x: 2 * x[1] * x[0]) register_diff(np.cbrt, lambda x: x[1] / (3 * (x[0]**(2 / 3)))) register_diff(np.reciprocal, lambda x: -x[1] / np.square(x[0])) register_diff(np.broadcast_to, lambda x, shape: np.broadcast_to(x[1], shape)) register_diff(np.sin, lambda x: x[1] * np.cos(x[0])) register_diff(np.cos, lambda x: -x[1] * np.sin(x[0])) register_diff(np.tan, lambda x: x[1] / np.square(np.cos(x[0]))) register_diff(np.arcsin, lambda x: x[1] / np.sqrt(1 - np.square(x[0]))) register_diff(np.arccos, lambda x: -x[1] / np.sqrt(1 - np.square(x[0])))