Average Error: 32.3 → 0.5
Time: 8.0s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{1}{\mathsf{hypot}\left(\log base, 0.0\right)} \cdot \frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right)}\]
\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 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)}
double f(double re, double im, double base) {
        double r110196 = re;
        double r110197 = r110196 * r110196;
        double r110198 = im;
        double r110199 = r110198 * r110198;
        double r110200 = r110197 + r110199;
        double r110201 = sqrt(r110200);
        double r110202 = log(r110201);
        double r110203 = base;
        double r110204 = log(r110203);
        double r110205 = r110202 * r110204;
        double r110206 = atan2(r110198, r110196);
        double r110207 = 0.0;
        double r110208 = r110206 * r110207;
        double r110209 = r110205 + r110208;
        double r110210 = r110204 * r110204;
        double r110211 = r110207 * r110207;
        double r110212 = r110210 + r110211;
        double r110213 = r110209 / r110212;
        return r110213;
}
↓
double f(double re, double im, double base) {
        double r110214 = 1.0;
        double r110215 = base;
        double r110216 = log(r110215);
        double r110217 = 0.0;
        double r110218 = hypot(r110216, r110217);
        double r110219 = r110214 / r110218;
        double r110220 = re;
        double r110221 = im;
        double r110222 = hypot(r110220, r110221);
        double r110223 = log(r110222);
        double r110224 = atan2(r110221, r110220);
        double r110225 = r110224 * r110217;
        double r110226 = fma(r110216, r110223, r110225);
        double r110227 = r110226 / r110218;
        double r110228 = r110219 * r110227;
        return r110228;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Initial program 32.3

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

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

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

  4. Applied sqrt-prod32.3

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

  5. Simplified32.3

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

  6. Simplified0.5

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

  8. Applied add-sqr-sqrt0.5

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

  9. Applied associate-/r*0.4

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

  10. Simplified0.4

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

  12. Applied *-un-lft-identity0.4

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

  13. Applied sqrt-prod0.4

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

  14. Applied *-un-lft-identity0.4

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

  15. Applied times-frac0.5

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

  16. Applied times-frac0.5

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

  17. Simplified0.5

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

  18. Simplified0.5

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

  19. Final simplification0.5

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

Reproduce

herbie shell --seed 2020002 +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))))