Average Error: 37.9 → 13.1
Time: 20.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2} \cdot 0.5\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2} \cdot 0.5
double f(double re, double im) {
        double r5371802 = 0.5;
        double r5371803 = 2.0;
        double r5371804 = re;
        double r5371805 = r5371804 * r5371804;
        double r5371806 = im;
        double r5371807 = r5371806 * r5371806;
        double r5371808 = r5371805 + r5371807;
        double r5371809 = sqrt(r5371808);
        double r5371810 = r5371809 + r5371804;
        double r5371811 = r5371803 * r5371810;
        double r5371812 = sqrt(r5371811);
        double r5371813 = r5371802 * r5371812;
        return r5371813;
}
↓
double f(double re, double im) {
        double r5371814 = re;
        double r5371815 = im;
        double r5371816 = hypot(r5371814, r5371815);
        double r5371817 = r5371814 + r5371816;
        double r5371818 = 2.0;
        double r5371819 = r5371817 * r5371818;
        double r5371820 = sqrt(r5371819);
        double r5371821 = 0.5;
        double r5371822 = r5371820 * r5371821;
        return r5371822;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.9
Target32.7
Herbie13.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.9

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

  2. Simplified13.1

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

  3. Final simplification13.1

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (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)))))