Average Error: 37.8 → 12.9
Time: 19.2s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2.0} \cdot 0.5\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2.0} \cdot 0.5
double f(double re, double im) {
        double r1448597 = 0.5;
        double r1448598 = 2.0;
        double r1448599 = re;
        double r1448600 = r1448599 * r1448599;
        double r1448601 = im;
        double r1448602 = r1448601 * r1448601;
        double r1448603 = r1448600 + r1448602;
        double r1448604 = sqrt(r1448603);
        double r1448605 = r1448604 + r1448599;
        double r1448606 = r1448598 * r1448605;
        double r1448607 = sqrt(r1448606);
        double r1448608 = r1448597 * r1448607;
        return r1448608;
}
↓
double f(double re, double im) {
        double r1448609 = re;
        double r1448610 = im;
        double r1448611 = hypot(r1448609, r1448610);
        double r1448612 = r1448609 + r1448611;
        double r1448613 = 2.0;
        double r1448614 = r1448612 * r1448613;
        double r1448615 = sqrt(r1448614);
        double r1448616 = 0.5;
        double r1448617 = r1448615 * r1448616;
        return r1448617;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.8
Target32.8
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.8

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

  2. Simplified12.9

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) + re\right) \cdot 2.0}}\]

  3. Final simplification12.9

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

Reproduce

herbie shell --seed 2019142 +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)))))