Average Error: 37.7 → 13.5
Time: 22.3s
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 r23532588 = 0.5;
        double r23532589 = 2.0;
        double r23532590 = re;
        double r23532591 = r23532590 * r23532590;
        double r23532592 = im;
        double r23532593 = r23532592 * r23532592;
        double r23532594 = r23532591 + r23532593;
        double r23532595 = sqrt(r23532594);
        double r23532596 = r23532595 + r23532590;
        double r23532597 = r23532589 * r23532596;
        double r23532598 = sqrt(r23532597);
        double r23532599 = r23532588 * r23532598;
        return r23532599;
}
↓
double f(double re, double im) {
        double r23532600 = re;
        double r23532601 = im;
        double r23532602 = hypot(r23532600, r23532601);
        double r23532603 = r23532600 + r23532602;
        double r23532604 = 2.0;
        double r23532605 = r23532603 * r23532604;
        double r23532606 = sqrt(r23532605);
        double r23532607 = 0.5;
        double r23532608 = r23532606 * r23532607;
        return r23532608;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.7
Target32.8
Herbie13.5
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.7

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

  2. Simplified13.5

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

  3. Final simplification13.5

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

Reproduce

herbie shell --seed 2019124 +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)))))