powComplex, real part

Average Error: 33.3 → 9.5

Time: 7.5s

Precision: 64

Math

\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;x.re \le -1.45038257289459698 \cdot 10^{43}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le -1.22786918581455784 \cdot 10^{-111}:\\ \;\;\;\;e^{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}\right) \cdot \left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le -3.36793833987596067 \cdot 10^{-217}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le 1.3585741357606672 \cdot 10^{-276}:\\ \;\;\;\;e^{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}\right) \cdot \left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le 5.6833266414007387 \cdot 10^{-174}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le 1.6471946110688965 \cdot 10^{-33}:\\ \;\;\;\;e^{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}\right) \cdot \left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \end{array}\]

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re))));
}

↓

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double temp;
	if ((x_46_re <= -1.450382572894597e+43)) {
		temp = (exp(((-1.0 * (y_46_re * log((-1.0 / x_46_re)))) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0);
	} else {
		double temp_1;
		if ((x_46_re <= -1.2278691858145578e-111)) {
			temp_1 = (exp((((cbrt(log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) * cbrt(log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))))) * (cbrt(log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) * y_46_re)) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0);
		} else {
			double temp_2;
			if ((x_46_re <= -3.3679383398759607e-217)) {
				temp_2 = (exp(((-1.0 * (y_46_re * log((-1.0 / x_46_re)))) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0);
			} else {
				double temp_3;
				if ((x_46_re <= 1.3585741357606672e-276)) {
					temp_3 = (exp((((cbrt(log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) * cbrt(log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))))) * (cbrt(log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) * y_46_re)) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0);
				} else {
					double temp_4;
					if ((x_46_re <= 5.683326641400739e-174)) {
						temp_4 = (exp(((log(x_46_re) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0);
					} else {
						double temp_5;
						if ((x_46_re <= 1.6471946110688965e-33)) {
							temp_5 = (exp((((cbrt(log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) * cbrt(log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))))) * (cbrt(log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) * y_46_re)) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0);
						} else {
							temp_5 = (exp(((log(x_46_re) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0);
						}
						temp_4 = temp_5;
					}
					temp_3 = temp_4;
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

1. Split input into 3 regimes

2. if x.re < -1.450382572894597e+43 or -1.2278691858145578e-111 < x.re < -3.3679383398759607e-217

   1. Initial program 38.7

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

   2. Taylor expanded around 0 21.9

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1}\]

   3. Taylor expanded around -inf 3.8

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right)} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]

   if -1.450382572894597e+43 < x.re < -1.2278691858145578e-111 or -3.3679383398759607e-217 < x.re < 1.3585741357606672e-276 or 5.683326641400739e-174 < x.re < 1.6471946110688965e-33

   1. Initial program 21.9

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

   2. Taylor expanded around 0 12.4

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 12.4

      \[\leadsto e^{\color{blue}{\left(\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]

   5. Applied associate-*l* 12.4

      \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}\right) \cdot \left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.re\right)} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]

   if 1.3585741357606672e-276 < x.re < 5.683326641400739e-174 or 1.6471946110688965e-33 < x.re

   1. Initial program 39.0

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

   2. Taylor expanded around 0 25.3

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1}\]

   3. Taylor expanded around inf 11.4

      \[\leadsto e^{\log \color{blue}{x.re} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]
3. Recombined 3 regimes into one program.

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -1.45038257289459698 \cdot 10^{43}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le -1.22786918581455784 \cdot 10^{-111}:\\ \;\;\;\;e^{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}\right) \cdot \left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le -3.36793833987596067 \cdot 10^{-217}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le 1.3585741357606672 \cdot 10^{-276}:\\ \;\;\;\;e^{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}\right) \cdot \left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le 5.6833266414007387 \cdot 10^{-174}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le 1.6471946110688965 \cdot 10^{-33}:\\ \;\;\;\;e^{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}\right) \cdot \left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))