Average Error: 31.2 → 0
Time: 967.0ms
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\log \left(1 \cdot \mathsf{hypot}\left(re, im\right)\right)\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\log \left(1 \cdot \mathsf{hypot}\left(re, im\right)\right)
double f(double re, double im) {
        double r25807 = re;
        double r25808 = r25807 * r25807;
        double r25809 = im;
        double r25810 = r25809 * r25809;
        double r25811 = r25808 + r25810;
        double r25812 = sqrt(r25811);
        double r25813 = log(r25812);
        return r25813;
}
↓
double f(double re, double im) {
        double r25814 = 1.0;
        double r25815 = re;
        double r25816 = im;
        double r25817 = hypot(r25815, r25816);
        double r25818 = r25814 * r25817;
        double r25819 = log(r25818);
        return r25819;
}

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 31.2

    \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity31.2

    \[\leadsto \log \left(\sqrt{\color{blue}{1 \cdot \left(re \cdot re + im \cdot im\right)}}\right)\]

  4. Applied sqrt-prod31.2

    \[\leadsto \log \color{blue}{\left(\sqrt{1} \cdot \sqrt{re \cdot re + im \cdot im}\right)}\]

  5. Simplified31.2

    \[\leadsto \log \left(\color{blue}{1} \cdot \sqrt{re \cdot re + im \cdot im}\right)\]

  6. Simplified0

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

  7. Final simplification0

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))