Average Error: 38.5 → 13.1
Time: 16.3s
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 r6081096 = 0.5;
        double r6081097 = 2.0;
        double r6081098 = re;
        double r6081099 = r6081098 * r6081098;
        double r6081100 = im;
        double r6081101 = r6081100 * r6081100;
        double r6081102 = r6081099 + r6081101;
        double r6081103 = sqrt(r6081102);
        double r6081104 = r6081103 + r6081098;
        double r6081105 = r6081097 * r6081104;
        double r6081106 = sqrt(r6081105);
        double r6081107 = r6081096 * r6081106;
        return r6081107;
}
↓
double f(double re, double im) {
        double r6081108 = re;
        double r6081109 = im;
        double r6081110 = hypot(r6081108, r6081109);
        double r6081111 = r6081108 + r6081110;
        double r6081112 = 2.0;
        double r6081113 = r6081111 * r6081112;
        double r6081114 = sqrt(r6081113);
        double r6081115 = 0.5;
        double r6081116 = r6081114 * r6081115;
        return r6081116;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.5
Target33.6
Herbie13.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.5

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

  2. Simplified13.1

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

  3. Final simplification13.1

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

Reproduce

herbie shell --seed 2019192 +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)))))