Average Error: 37.6 → 13.3
Time: 13.7s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\sqrt{\left(re + \sqrt{re^2 + im^2}^*\right) \cdot 2.0} \cdot 0.5\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\sqrt{\left(re + \sqrt{re^2 + im^2}^*\right) \cdot 2.0} \cdot 0.5
double f(double re, double im) {
        double r27720364 = 0.5;
        double r27720365 = 2.0;
        double r27720366 = re;
        double r27720367 = r27720366 * r27720366;
        double r27720368 = im;
        double r27720369 = r27720368 * r27720368;
        double r27720370 = r27720367 + r27720369;
        double r27720371 = sqrt(r27720370);
        double r27720372 = r27720371 + r27720366;
        double r27720373 = r27720365 * r27720372;
        double r27720374 = sqrt(r27720373);
        double r27720375 = r27720364 * r27720374;
        return r27720375;
}
↓
double f(double re, double im) {
        double r27720376 = re;
        double r27720377 = im;
        double r27720378 = hypot(r27720376, r27720377);
        double r27720379 = r27720376 + r27720378;
        double r27720380 = 2.0;
        double r27720381 = r27720379 * r27720380;
        double r27720382 = sqrt(r27720381);
        double r27720383 = 0.5;
        double r27720384 = r27720382 * r27720383;
        return r27720384;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target32.6
Herbie13.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.6

    \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

  2. Simplified13.3

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\sqrt{re^2 + im^2}^* + re\right) \cdot 2.0}}\]

  3. Final simplification13.3

    \[\leadsto \sqrt{\left(re + \sqrt{re^2 + im^2}^*\right) \cdot 2.0} \cdot 0.5\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))