math.log/2 on complex, real part

Average Error: 30.6 → 0.6

Time: 1.3m

Precision: 64

• could not determine a ground truth for program body

  1. re = 4.674895467543714e+201
  2. im = 1.0016805978280525e+269
  3. base = 1.5180684099834806e-149

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

↓

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

C

double f(double re, double im, double base) {
        double r1122688 = re;
        double r1122689 = r1122688 * r1122688;
        double r1122690 = im;
        double r1122691 = r1122690 * r1122690;
        double r1122692 = r1122689 + r1122691;
        double r1122693 = sqrt(r1122692);
        double r1122694 = log(r1122693);
        double r1122695 = base;
        double r1122696 = log(r1122695);
        double r1122697 = r1122694 * r1122696;
        double r1122698 = atan2(r1122690, r1122688);
        double r1122699 = 0.0;
        double r1122700 = r1122698 * r1122699;
        double r1122701 = r1122697 + r1122700;
        double r1122702 = r1122696 * r1122696;
        double r1122703 = r1122699 * r1122699;
        double r1122704 = r1122702 + r1122703;
        double r1122705 = r1122701 / r1122704;
        return r1122705;
}

↓

double f(double re, double im, double base) {
        double r1122706 = re;
        double r1122707 = im;
        double r1122708 = hypot(r1122706, r1122707);
        double r1122709 = log(r1122708);
        double r1122710 = base;
        double r1122711 = log(r1122710);
        double r1122712 = r1122709 / r1122711;
        double r1122713 = r1122712 * r1122712;
        double r1122714 = r1122713 * r1122712;
        double r1122715 = cbrt(r1122714);
        double r1122716 = r1122715 * r1122712;
        double r1122717 = r1122716 * r1122712;
        double r1122718 = cbrt(r1122717);
        return r1122718;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Initial program 30.6

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

2. Simplified 0.4

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

4. Applied add-cbrt-cube 0.6

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

5. Applied add-cbrt-cube 0.8

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

6. Applied cbrt-undiv 0.7

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

7. Simplified 0.7

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

9. Applied add-cbrt-cube 0.7

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

10. Applied add-cbrt-cube 0.7

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

11. Applied cbrt-undiv 0.7

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

12. Simplified 0.6

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

13. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0)) (+ (* (log base) (log base)) (* 0 0))))