Average Error: 39.2 → 14.0
Time: 19.5s
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 r25323 = 0.5;
        double r25324 = 2.0;
        double r25325 = re;
        double r25326 = r25325 * r25325;
        double r25327 = im;
        double r25328 = r25327 * r25327;
        double r25329 = r25326 + r25328;
        double r25330 = sqrt(r25329);
        double r25331 = r25330 - r25325;
        double r25332 = r25324 * r25331;
        double r25333 = sqrt(r25332);
        double r25334 = r25323 * r25333;
        return r25334;
}
↓
double f(double re, double im) {
        double r25335 = 0.5;
        double r25336 = re;
        double r25337 = im;
        double r25338 = hypot(r25336, r25337);
        double r25339 = r25338 - r25336;
        double r25340 = 2.0;
        double r25341 = r25339 * r25340;
        double r25342 = sqrt(r25341);
        double r25343 = r25335 * r25342;
        return r25343;
}

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 39.2

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

  2. Simplified14.0

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

  3. Final simplification14.0

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))