Average Error: 37.5 → 13.5
Time: 14.6s
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 r1280802 = 0.5;
        double r1280803 = 2.0;
        double r1280804 = re;
        double r1280805 = r1280804 * r1280804;
        double r1280806 = im;
        double r1280807 = r1280806 * r1280806;
        double r1280808 = r1280805 + r1280807;
        double r1280809 = sqrt(r1280808);
        double r1280810 = r1280809 + r1280804;
        double r1280811 = r1280803 * r1280810;
        double r1280812 = sqrt(r1280811);
        double r1280813 = r1280802 * r1280812;
        return r1280813;
}
↓
double f(double re, double im) {
        double r1280814 = re;
        double r1280815 = im;
        double r1280816 = hypot(r1280814, r1280815);
        double r1280817 = r1280814 + r1280816;
        double r1280818 = 2.0;
        double r1280819 = r1280817 * r1280818;
        double r1280820 = sqrt(r1280819);
        double r1280821 = 0.5;
        double r1280822 = r1280820 * r1280821;
        return r1280822;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.5
Target32.7
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 37.5

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

  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 2019139 +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)))))