Average Error: 31.9 → 0
Time: 768.0ms
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 r33227 = re;
        double r33228 = r33227 * r33227;
        double r33229 = im;
        double r33230 = r33229 * r33229;
        double r33231 = r33228 + r33230;
        double r33232 = sqrt(r33231);
        double r33233 = log(r33232);
        return r33233;
}
↓
double f(double re, double im) {
        double r33234 = 1.0;
        double r33235 = re;
        double r33236 = im;
        double r33237 = hypot(r33235, r33236);
        double r33238 = r33234 * r33237;
        double r33239 = log(r33238);
        return r33239;
}

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 31.9

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

  3. Applied *-un-lft-identity31.9

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

  4. Applied sqrt-prod31.9

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

  5. Simplified31.9

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

  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 2020062 +o rules:numerics
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))