Average Error: 32.2 → 0.5
Time: 19.5s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(re, im\right)\right), \log base, \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right)} \cdot \frac{1}{\mathsf{hypot}\left(\log base, 0.0\right)}\]
\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{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(re, im\right)\right), \log base, \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right)} \cdot \frac{1}{\mathsf{hypot}\left(\log base, 0.0\right)}
double f(double re, double im, double base) {
        double r53383 = re;
        double r53384 = r53383 * r53383;
        double r53385 = im;
        double r53386 = r53385 * r53385;
        double r53387 = r53384 + r53386;
        double r53388 = sqrt(r53387);
        double r53389 = log(r53388);
        double r53390 = base;
        double r53391 = log(r53390);
        double r53392 = r53389 * r53391;
        double r53393 = atan2(r53385, r53383);
        double r53394 = 0.0;
        double r53395 = r53393 * r53394;
        double r53396 = r53392 + r53395;
        double r53397 = r53391 * r53391;
        double r53398 = r53394 * r53394;
        double r53399 = r53397 + r53398;
        double r53400 = r53396 / r53399;
        return r53400;
}
↓
double f(double re, double im, double base) {
        double r53401 = re;
        double r53402 = im;
        double r53403 = hypot(r53401, r53402);
        double r53404 = log(r53403);
        double r53405 = base;
        double r53406 = log(r53405);
        double r53407 = atan2(r53402, r53401);
        double r53408 = 0.0;
        double r53409 = r53407 * r53408;
        double r53410 = fma(r53404, r53406, r53409);
        double r53411 = hypot(r53406, r53408);
        double r53412 = r53410 / r53411;
        double r53413 = 1.0;
        double r53414 = r53413 / r53411;
        double r53415 = r53412 * r53414;
        return r53415;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Initial program 32.2

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

  2. Simplified0.5

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied associate-/r*0.4

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

  6. Simplified0.4

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

  8. Applied div-inv0.5

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

  9. Simplified0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2020043 +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))))