Average Error: 37.6 → 12.9
Time: 15.0s
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 r107859 = 0.5;
        double r107860 = 2.0;
        double r107861 = re;
        double r107862 = r107861 * r107861;
        double r107863 = im;
        double r107864 = r107863 * r107863;
        double r107865 = r107862 + r107864;
        double r107866 = sqrt(r107865);
        double r107867 = r107866 + r107861;
        double r107868 = r107860 * r107867;
        double r107869 = sqrt(r107868);
        double r107870 = r107859 * r107869;
        return r107870;
}
↓
double f(double re, double im) {
        double r107871 = re;
        double r107872 = im;
        double r107873 = hypot(r107871, r107872);
        double r107874 = r107871 + r107873;
        double r107875 = 2.0;
        double r107876 = r107874 * r107875;
        double r107877 = sqrt(r107876);
        double r107878 = 0.5;
        double r107879 = r107877 * r107878;
        return r107879;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target32.7
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.6

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

  2. Simplified12.9

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

  3. Final simplification12.9

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

Reproduce

herbie shell --seed 2019194 +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)))))