Average Error: 32.2 → 0.3
Time: 7.1s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[-1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(-2\right) \cdot \log \left({\left(\frac{1}{base}\right)}^{\frac{-1}{3}}\right) + \log \left(\frac{1}{\sqrt[3]{base}}\right)}\]
\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}
↓
-1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\left(-2\right) \cdot \log \left({\left(\frac{1}{base}\right)}^{\frac{-1}{3}}\right) + \log \left(\frac{1}{\sqrt[3]{base}}\right)}
double f(double re, double im, double base) {
        double r48162 = im;
        double r48163 = re;
        double r48164 = atan2(r48162, r48163);
        double r48165 = base;
        double r48166 = log(r48165);
        double r48167 = r48164 * r48166;
        double r48168 = r48163 * r48163;
        double r48169 = r48162 * r48162;
        double r48170 = r48168 + r48169;
        double r48171 = sqrt(r48170);
        double r48172 = log(r48171);
        double r48173 = 0.0;
        double r48174 = r48172 * r48173;
        double r48175 = r48167 - r48174;
        double r48176 = r48166 * r48166;
        double r48177 = r48173 * r48173;
        double r48178 = r48176 + r48177;
        double r48179 = r48175 / r48178;
        return r48179;
}
↓
double f(double re, double im, double base) {
        double r48180 = -1.0;
        double r48181 = im;
        double r48182 = re;
        double r48183 = atan2(r48181, r48182);
        double r48184 = 2.0;
        double r48185 = -r48184;
        double r48186 = 1.0;
        double r48187 = base;
        double r48188 = r48186 / r48187;
        double r48189 = -0.3333333333333333;
        double r48190 = pow(r48188, r48189);
        double r48191 = log(r48190);
        double r48192 = r48185 * r48191;
        double r48193 = cbrt(r48187);
        double r48194 = r48186 / r48193;
        double r48195 = log(r48194);
        double r48196 = r48192 + r48195;
        double r48197 = r48183 / r48196;
        double r48198 = r48180 * r48197;
        return r48198;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.2

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

  2. Taylor expanded around inf 0.3

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

  4. Applied add-cube-cbrt0.3

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\frac{1}{\color{blue}{\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}}}\right)}\]

  5. Applied add-cube-cbrt0.3

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}}\right)}\]

  6. Applied times-frac0.3

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{base} \cdot \sqrt[3]{base}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{base}}\right)}}\]

  7. Applied log-prod0.4

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{base} \cdot \sqrt[3]{base}}\right) + \log \left(\frac{\sqrt[3]{1}}{\sqrt[3]{base}}\right)}}\]

  8. Simplified0.4

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

  9. Simplified0.4

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

  10. Taylor expanded around inf 0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  :precision binary64
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))