Average Error: 31.3 → 0.5
Time: 32.5s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\frac{1}{\frac{1}{3} \cdot \frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\frac{1}{\frac{1}{3} \cdot \frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}}
double f(double re, double im) {
        double r1220018 = re;
        double r1220019 = r1220018 * r1220018;
        double r1220020 = im;
        double r1220021 = r1220020 * r1220020;
        double r1220022 = r1220019 + r1220021;
        double r1220023 = sqrt(r1220022);
        double r1220024 = log(r1220023);
        double r1220025 = 10.0;
        double r1220026 = log(r1220025);
        double r1220027 = r1220024 / r1220026;
        return r1220027;
}
↓
double f(double re, double im) {
        double r1220028 = 1.0;
        double r1220029 = 0.3333333333333333;
        double r1220030 = 10.0;
        double r1220031 = log(r1220030);
        double r1220032 = re;
        double r1220033 = im;
        double r1220034 = hypot(r1220032, r1220033);
        double r1220035 = cbrt(r1220034);
        double r1220036 = log(r1220035);
        double r1220037 = r1220031 / r1220036;
        double r1220038 = r1220029 * r1220037;
        double r1220039 = r1220028 / r1220038;
        return r1220039;
}

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 31.3

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

  2. Simplified0.6

    \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log 10}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.6

    \[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}}{\log 10}\]
  5. Using strategy rm

  6. Applied pow10.6

    \[\leadsto \frac{\log \left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}^{1}}\right)}{\log 10}\]

  7. Applied pow10.6

    \[\leadsto \frac{\log \left(\color{blue}{{\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}^{1}} \cdot {\left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}^{1}\right)}{\log 10}\]

  8. Applied pow-prod-down0.6

    \[\leadsto \frac{\log \color{blue}{\left({\left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}^{1}\right)}}{\log 10}\]

  9. Applied log-pow0.6

    \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}}{\log 10}\]

  10. Applied associate-/l*0.6

    \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}}}\]
  11. Using strategy rm

  12. Applied pow10.6

    \[\leadsto \frac{1}{\frac{\log 10}{\log \left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}^{1}}\right)}}\]

  13. Applied pow20.6

    \[\leadsto \frac{1}{\frac{\log 10}{\log \left(\color{blue}{{\left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}^{1}\right)}}\]

  14. Applied pow-prod-up0.6

    \[\leadsto \frac{1}{\frac{\log 10}{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}^{\left(2 + 1\right)}\right)}}}\]

  15. Applied log-pow0.7

    \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}}}\]

  16. Applied *-un-lft-identity0.7

    \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot \log 10}}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}}\]

  17. Applied times-frac0.5

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{2 + 1} \cdot \frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}}}\]

  18. Simplified0.5

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{3}} \cdot \frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}}\]

  19. Final simplification0.5

    \[\leadsto \frac{1}{\frac{1}{3} \cdot \frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))