Average Error: 31.7 → 0.4
Time: 27.5s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\right)
double f(double re, double im) {
        double r34297 = re;
        double r34298 = r34297 * r34297;
        double r34299 = im;
        double r34300 = r34299 * r34299;
        double r34301 = r34298 + r34300;
        double r34302 = sqrt(r34301);
        double r34303 = log(r34302);
        double r34304 = 10.0;
        double r34305 = log(r34304);
        double r34306 = r34303 / r34305;
        return r34306;
}
↓
double f(double re, double im) {
        double r34307 = 1.0;
        double r34308 = 10.0;
        double r34309 = log(r34308);
        double r34310 = sqrt(r34309);
        double r34311 = r34307 / r34310;
        double r34312 = re;
        double r34313 = im;
        double r34314 = hypot(r34312, r34313);
        double r34315 = log(r34314);
        double r34316 = r34315 * r34311;
        double r34317 = r34311 * r34316;
        return r34317;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.7

    \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

  2. Simplified0.6

    \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.6

    \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

  5. Applied pow10.6

    \[\leadsto \frac{\log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

  6. Applied log-pow0.6

    \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

  7. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}}\]
  8. Using strategy rm

  9. Applied div-inv0.4

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\right)}\]

  10. Final simplification0.4

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))