Average Error: 37.8 → 13.5
Time: 12.5s
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 r2776876 = 0.5;
        double r2776877 = 2.0;
        double r2776878 = re;
        double r2776879 = r2776878 * r2776878;
        double r2776880 = im;
        double r2776881 = r2776880 * r2776880;
        double r2776882 = r2776879 + r2776881;
        double r2776883 = sqrt(r2776882);
        double r2776884 = r2776883 + r2776878;
        double r2776885 = r2776877 * r2776884;
        double r2776886 = sqrt(r2776885);
        double r2776887 = r2776876 * r2776886;
        return r2776887;
}
↓
double f(double re, double im) {
        double r2776888 = re;
        double r2776889 = im;
        double r2776890 = hypot(r2776888, r2776889);
        double r2776891 = r2776888 + r2776890;
        double r2776892 = 2.0;
        double r2776893 = r2776891 * r2776892;
        double r2776894 = sqrt(r2776893);
        double r2776895 = 0.5;
        double r2776896 = r2776894 * r2776895;
        return r2776896;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.8
Target32.9
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 37.8

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

  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 2019156 +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)))))