powComplex, imaginary part

Average Error: 33.3 → 5.8

Time: 15.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \le -0.0:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\sqrt[3]{y.im} \cdot \left(\left(\sqrt[3]{\sqrt[3]{y.im}} \cdot \sqrt[3]{\sqrt[3]{y.im}}\right) \cdot \sqrt[3]{\sqrt[3]{y.im}}\right)\right)\right) \cdot \sqrt[3]{y.im}}}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r23514 = x_re;
        double r23515 = r23514 * r23514;
        double r23516 = x_im;
        double r23517 = r23516 * r23516;
        double r23518 = r23515 + r23517;
        double r23519 = sqrt(r23518);
        double r23520 = log(r23519);
        double r23521 = y_re;
        double r23522 = r23520 * r23521;
        double r23523 = atan2(r23516, r23514);
        double r23524 = y_im;
        double r23525 = r23523 * r23524;
        double r23526 = r23522 - r23525;
        double r23527 = exp(r23526);
        double r23528 = r23520 * r23524;
        double r23529 = r23523 * r23521;
        double r23530 = r23528 + r23529;
        double r23531 = sin(r23530);
        double r23532 = r23527 * r23531;
        return r23532;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r23533 = x_re;
        double r23534 = r23533 * r23533;
        double r23535 = x_im;
        double r23536 = r23535 * r23535;
        double r23537 = r23534 + r23536;
        double r23538 = sqrt(r23537);
        double r23539 = log(r23538);
        double r23540 = y_re;
        double r23541 = r23539 * r23540;
        double r23542 = atan2(r23535, r23533);
        double r23543 = y_im;
        double r23544 = r23542 * r23543;
        double r23545 = r23541 - r23544;
        double r23546 = exp(r23545);
        double r23547 = r23539 * r23543;
        double r23548 = r23542 * r23540;
        double r23549 = r23547 + r23548;
        double r23550 = sin(r23549);
        double r23551 = r23546 * r23550;
        double r23552 = -0.0;
        bool r23553 = r23551 <= r23552;
        double r23554 = hypot(r23533, r23535);
        double r23555 = pow(r23554, r23540);
        double r23556 = log(r23554);
        double r23557 = r23556 * r23543;
        double r23558 = sin(r23557);
        double r23559 = log1p(r23558);
        double r23560 = expm1(r23559);
        double r23561 = cos(r23548);
        double r23562 = r23560 * r23561;
        double r23563 = cos(r23557);
        double r23564 = sin(r23548);
        double r23565 = r23563 * r23564;
        double r23566 = r23562 + r23565;
        double r23567 = r23555 * r23566;
        double r23568 = cbrt(r23543);
        double r23569 = cbrt(r23568);
        double r23570 = r23569 * r23569;
        double r23571 = r23570 * r23569;
        double r23572 = r23568 * r23571;
        double r23573 = r23542 * r23572;
        double r23574 = r23573 * r23568;
        double r23575 = exp(r23574);
        double r23576 = r23567 / r23575;
        double r23577 = r23553 ? r23551 : r23576;
        return r23577;
}

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 2 regimes

2. if (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))) < -0.0

   1. Initial program 1.7

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

   if -0.0 < (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re))))

   1. Initial program 53.6

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

   2. Simplified 8.4

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

   4. Applied add-cube-cbrt 8.4

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

   5. Applied associate-*r* 8.4

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

   7. Applied add-cube-cbrt 8.4

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

   9. Applied fma-udef 8.4

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

   10. Applied sin-sum 8.4

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

   12. Applied expm1-log1p-u 8.4

       \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\right)} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\sqrt[3]{y.im} \cdot \left(\left(\sqrt[3]{\sqrt[3]{y.im}} \cdot \sqrt[3]{\sqrt[3]{y.im}}\right) \cdot \sqrt[3]{\sqrt[3]{y.im}}\right)\right)\right) \cdot \sqrt[3]{y.im}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \le -0.0:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\sqrt[3]{y.im} \cdot \left(\left(\sqrt[3]{\sqrt[3]{y.im}} \cdot \sqrt[3]{\sqrt[3]{y.im}}\right) \cdot \sqrt[3]{\sqrt[3]{y.im}}\right)\right)\right) \cdot \sqrt[3]{y.im}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))