Average Error: 38.1 → 13.0
Time: 17.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) + re\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) + re\right) \cdot 2}
double f(double re, double im) {
        double r11158924 = 0.5;
        double r11158925 = 2.0;
        double r11158926 = re;
        double r11158927 = r11158926 * r11158926;
        double r11158928 = im;
        double r11158929 = r11158928 * r11158928;
        double r11158930 = r11158927 + r11158929;
        double r11158931 = sqrt(r11158930);
        double r11158932 = r11158931 + r11158926;
        double r11158933 = r11158925 * r11158932;
        double r11158934 = sqrt(r11158933);
        double r11158935 = r11158924 * r11158934;
        return r11158935;
}
↓
double f(double re, double im) {
        double r11158936 = 0.5;
        double r11158937 = re;
        double r11158938 = im;
        double r11158939 = hypot(r11158937, r11158938);
        double r11158940 = r11158939 + r11158937;
        double r11158941 = 2.0;
        double r11158942 = r11158940 * r11158941;
        double r11158943 = sqrt(r11158942);
        double r11158944 = r11158936 * r11158943;
        return r11158944;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.1
Target33.2
Herbie13.0
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.1

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

  2. Simplified13.0

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

  3. Final simplification13.0

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

Reproduce

herbie shell --seed 2019174 +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)))))