Average Error: 31.0 → 0.5
Time: 24.2s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{1}{\mathsf{hypot}\left(\log base, 0.0\right)} \cdot \frac{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(re, im\right)\right), \log base, \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(0.0, \log base\right)}\]
\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}{\mathsf{hypot}\left(\log base, 0.0\right)} \cdot \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)}
double f(double re, double im, double base) {
        double r111930 = re;
        double r111931 = r111930 * r111930;
        double r111932 = im;
        double r111933 = r111932 * r111932;
        double r111934 = r111931 + r111933;
        double r111935 = sqrt(r111934);
        double r111936 = log(r111935);
        double r111937 = base;
        double r111938 = log(r111937);
        double r111939 = r111936 * r111938;
        double r111940 = atan2(r111932, r111930);
        double r111941 = 0.0;
        double r111942 = r111940 * r111941;
        double r111943 = r111939 + r111942;
        double r111944 = r111938 * r111938;
        double r111945 = r111941 * r111941;
        double r111946 = r111944 + r111945;
        double r111947 = r111943 / r111946;
        return r111947;
}
↓
double f(double re, double im, double base) {
        double r111948 = 1.0;
        double r111949 = base;
        double r111950 = log(r111949);
        double r111951 = 0.0;
        double r111952 = hypot(r111950, r111951);
        double r111953 = r111948 / r111952;
        double r111954 = re;
        double r111955 = im;
        double r111956 = hypot(r111954, r111955);
        double r111957 = log(r111956);
        double r111958 = atan2(r111955, r111954);
        double r111959 = r111958 * r111951;
        double r111960 = fma(r111957, r111950, r111959);
        double r111961 = hypot(r111951, r111950);
        double r111962 = r111960 / r111961;
        double r111963 = r111953 * r111962;
        return r111963;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Initial program 31.0

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

  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 add-sqr-sqrt0.4

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

  9. Applied sqrt-prod0.7

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

  10. Applied add-sqr-sqrt1.0

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

  11. Applied *-un-lft-identity1.0

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

  12. Applied times-frac1.0

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

  13. Applied times-frac1.0

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

  14. Simplified0.7

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

  15. Simplified0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2019303 +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))))