Average Error: 38.2 → 13.5
Time: 18.3s
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 r7056886 = 0.5;
        double r7056887 = 2.0;
        double r7056888 = re;
        double r7056889 = r7056888 * r7056888;
        double r7056890 = im;
        double r7056891 = r7056890 * r7056890;
        double r7056892 = r7056889 + r7056891;
        double r7056893 = sqrt(r7056892);
        double r7056894 = r7056893 + r7056888;
        double r7056895 = r7056887 * r7056894;
        double r7056896 = sqrt(r7056895);
        double r7056897 = r7056886 * r7056896;
        return r7056897;
}
↓
double f(double re, double im) {
        double r7056898 = re;
        double r7056899 = im;
        double r7056900 = hypot(r7056898, r7056899);
        double r7056901 = r7056898 + r7056900;
        double r7056902 = 2.0;
        double r7056903 = r7056901 * r7056902;
        double r7056904 = sqrt(r7056903);
        double r7056905 = 0.5;
        double r7056906 = r7056904 * r7056905;
        return r7056906;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.2
Target33.0
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 38.2

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

  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 2019143 +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)))))