Average Error: 32.1 → 0.4
Time: 17.0s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(re, im\right)\right), \log base, \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(0.0, \log base\right)}}{\sqrt{\mathsf{fma}\left(\log base, \log base, 0.0 \cdot 0.0\right)}}\]
\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{\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(0.0, \log base\right)}}{\sqrt{\mathsf{fma}\left(\log base, \log base, 0.0 \cdot 0.0\right)}}
double f(double re, double im, double base) {
        double r49943 = re;
        double r49944 = r49943 * r49943;
        double r49945 = im;
        double r49946 = r49945 * r49945;
        double r49947 = r49944 + r49946;
        double r49948 = sqrt(r49947);
        double r49949 = log(r49948);
        double r49950 = base;
        double r49951 = log(r49950);
        double r49952 = r49949 * r49951;
        double r49953 = atan2(r49945, r49943);
        double r49954 = 0.0;
        double r49955 = r49953 * r49954;
        double r49956 = r49952 + r49955;
        double r49957 = r49951 * r49951;
        double r49958 = r49954 * r49954;
        double r49959 = r49957 + r49958;
        double r49960 = r49956 / r49959;
        return r49960;
}
↓
double f(double re, double im, double base) {
        double r49961 = re;
        double r49962 = im;
        double r49963 = hypot(r49961, r49962);
        double r49964 = log(r49963);
        double r49965 = base;
        double r49966 = log(r49965);
        double r49967 = atan2(r49962, r49961);
        double r49968 = 0.0;
        double r49969 = r49967 * r49968;
        double r49970 = fma(r49964, r49966, r49969);
        double r49971 = hypot(r49968, r49966);
        double r49972 = r49970 / r49971;
        double r49973 = r49968 * r49968;
        double r49974 = fma(r49966, r49966, r49973);
        double r49975 = sqrt(r49974);
        double r49976 = r49972 / r49975;
        return r49976;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Initial program 32.1

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

  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 *-un-lft-identity0.4

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

  9. Applied sqrt-prod0.4

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

  10. Applied associate-/r*0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019351 +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))))