Average Error: 38.0 → 13.1
Time: 19.6s
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 r5244809 = 0.5;
        double r5244810 = 2.0;
        double r5244811 = re;
        double r5244812 = r5244811 * r5244811;
        double r5244813 = im;
        double r5244814 = r5244813 * r5244813;
        double r5244815 = r5244812 + r5244814;
        double r5244816 = sqrt(r5244815);
        double r5244817 = r5244816 + r5244811;
        double r5244818 = r5244810 * r5244817;
        double r5244819 = sqrt(r5244818);
        double r5244820 = r5244809 * r5244819;
        return r5244820;
}
↓
double f(double re, double im) {
        double r5244821 = re;
        double r5244822 = im;
        double r5244823 = hypot(r5244821, r5244822);
        double r5244824 = r5244821 + r5244823;
        double r5244825 = 2.0;
        double r5244826 = r5244824 * r5244825;
        double r5244827 = sqrt(r5244826);
        double r5244828 = 0.5;
        double r5244829 = r5244827 * r5244828;
        return r5244829;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.0
Target33.2
Herbie13.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.0

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

  2. Simplified13.1

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

  3. Final simplification13.1

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

Reproduce

herbie shell --seed 2019133 +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)))))