Average Error: 32.1 → 0.4
Time: 4.6s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{\frac{\mathsf{fma}\left(-0.0, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot \log base\right)}{\mathsf{hypot}\left(\log base, 0.0\right)}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]
\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}
↓
\frac{\frac{\mathsf{fma}\left(-0.0, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot \log base\right)}{\mathsf{hypot}\left(\log base, 0.0\right)}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}
double f(double re, double im, double base) {
        double r41599 = im;
        double r41600 = re;
        double r41601 = atan2(r41599, r41600);
        double r41602 = base;
        double r41603 = log(r41602);
        double r41604 = r41601 * r41603;
        double r41605 = r41600 * r41600;
        double r41606 = r41599 * r41599;
        double r41607 = r41605 + r41606;
        double r41608 = sqrt(r41607);
        double r41609 = log(r41608);
        double r41610 = 0.0;
        double r41611 = r41609 * r41610;
        double r41612 = r41604 - r41611;
        double r41613 = r41603 * r41603;
        double r41614 = r41610 * r41610;
        double r41615 = r41613 + r41614;
        double r41616 = r41612 / r41615;
        return r41616;
}
↓
double f(double re, double im, double base) {
        double r41617 = 0.0;
        double r41618 = -r41617;
        double r41619 = re;
        double r41620 = im;
        double r41621 = hypot(r41619, r41620);
        double r41622 = log(r41621);
        double r41623 = atan2(r41620, r41619);
        double r41624 = base;
        double r41625 = log(r41624);
        double r41626 = r41623 * r41625;
        double r41627 = fma(r41618, r41622, r41626);
        double r41628 = hypot(r41625, r41617);
        double r41629 = r41627 / r41628;
        double r41630 = r41625 * r41625;
        double r41631 = r41617 * r41617;
        double r41632 = r41630 + r41631;
        double r41633 = sqrt(r41632);
        double r41634 = r41629 / r41633;
        return r41634;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Initial program 32.1

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

  3. Applied *-un-lft-identity32.1

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

  4. Applied sqrt-prod32.1

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

  5. Simplified32.1

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

  6. Simplified0.4

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

  8. Applied add-sqr-sqrt0.4

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

  9. Applied associate-/r*0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  :precision binary64
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))