math.log/2 on complex, real part

Average Error: 32.5 → 0.5

Time: 8.7s

Precision: 64

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

↓

\[\frac{1}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{1}} \cdot \frac{1}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}\]

C

double f(double re, double im, double base) {
        double r52558 = re;
        double r52559 = r52558 * r52558;
        double r52560 = im;
        double r52561 = r52560 * r52560;
        double r52562 = r52559 + r52561;
        double r52563 = sqrt(r52562);
        double r52564 = log(r52563);
        double r52565 = base;
        double r52566 = log(r52565);
        double r52567 = r52564 * r52566;
        double r52568 = atan2(r52560, r52558);
        double r52569 = 0.0;
        double r52570 = r52568 * r52569;
        double r52571 = r52567 + r52570;
        double r52572 = r52566 * r52566;
        double r52573 = r52569 * r52569;
        double r52574 = r52572 + r52573;
        double r52575 = r52571 / r52574;
        return r52575;
}

↓

double f(double re, double im, double base) {
        double r52576 = 1.0;
        double r52577 = base;
        double r52578 = log(r52577);
        double r52579 = 0.0;
        double r52580 = hypot(r52578, r52579);
        double r52581 = r52580 / r52576;
        double r52582 = r52576 / r52581;
        double r52583 = re;
        double r52584 = im;
        double r52585 = hypot(r52583, r52584);
        double r52586 = log(r52585);
        double r52587 = atan2(r52584, r52583);
        double r52588 = r52587 * r52579;
        double r52589 = fma(r52578, r52586, r52588);
        double r52590 = r52580 / r52589;
        double r52591 = r52576 / r52590;
        double r52592 = r52582 * r52591;
        return r52592;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Initial program 32.5

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
2. Using strategy rm

3. Applied *-un-lft-identity 32.5

   \[\leadsto \frac{\log \left(\sqrt{\color{blue}{1 \cdot \left(re \cdot re + im \cdot im\right)}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

4. Applied sqrt-prod 32.5

   \[\leadsto \frac{\log \color{blue}{\left(\sqrt{1} \cdot \sqrt{re \cdot re + im \cdot im}\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

5. Simplified 32.5

   \[\leadsto \frac{\log \left(\color{blue}{1} \cdot \sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

6. Simplified 0.5

   \[\leadsto \frac{\log \left(1 \cdot \color{blue}{\mathsf{hypot}\left(re, im\right)}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
7. Using strategy rm

8. Applied add-sqr-sqrt 0.5

   \[\leadsto \frac{\log \left(1 \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

9. Applied *-un-lft-identity 0.5

   \[\leadsto \frac{\color{blue}{1 \cdot \left(\log \left(1 \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

10. Applied times-frac 0.5

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(1 \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

11. Simplified 0.5

    \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{1}}} \cdot \frac{\log \left(1 \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

12. Simplified 0.5

    \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{1}} \cdot \color{blue}{\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1}}\]
13. Using strategy rm

14. Applied clear-num 0.5

    \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{1}} \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1}{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}}\]

15. Simplified 0.5

    \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{1}} \cdot \frac{1}{\color{blue}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}}\]

16. Final simplification 0.5

    \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{1}} \cdot \frac{1}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))