Average Error: 38.1 → 12.7
Time: 16.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}
double f(double re, double im) {
        double r37268 = 0.5;
        double r37269 = 2.0;
        double r37270 = re;
        double r37271 = r37270 * r37270;
        double r37272 = im;
        double r37273 = r37272 * r37272;
        double r37274 = r37271 + r37273;
        double r37275 = sqrt(r37274);
        double r37276 = r37275 - r37270;
        double r37277 = r37269 * r37276;
        double r37278 = sqrt(r37277);
        double r37279 = r37268 * r37278;
        return r37279;
}
↓
double f(double re, double im) {
        double r37280 = 0.5;
        double r37281 = re;
        double r37282 = im;
        double r37283 = hypot(r37281, r37282);
        double r37284 = r37283 - r37281;
        double r37285 = 2.0;
        double r37286 = r37284 * r37285;
        double r37287 = sqrt(r37286);
        double r37288 = r37280 * r37287;
        return r37288;
}

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 38.1

    \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

  2. Simplified12.7

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}}\]

  3. Final simplification12.7

    \[\leadsto 0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))