Average Error: 37.9 → 13.1
Time: 21.0s
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 r6241031 = 0.5;
        double r6241032 = 2.0;
        double r6241033 = re;
        double r6241034 = r6241033 * r6241033;
        double r6241035 = im;
        double r6241036 = r6241035 * r6241035;
        double r6241037 = r6241034 + r6241036;
        double r6241038 = sqrt(r6241037);
        double r6241039 = r6241038 + r6241033;
        double r6241040 = r6241032 * r6241039;
        double r6241041 = sqrt(r6241040);
        double r6241042 = r6241031 * r6241041;
        return r6241042;
}
↓
double f(double re, double im) {
        double r6241043 = re;
        double r6241044 = im;
        double r6241045 = hypot(r6241043, r6241044);
        double r6241046 = r6241043 + r6241045;
        double r6241047 = 2.0;
        double r6241048 = r6241046 * r6241047;
        double r6241049 = sqrt(r6241048);
        double r6241050 = 0.5;
        double r6241051 = r6241049 * r6241050;
        return r6241051;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.9
Target32.7
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 37.9

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

  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 2019172 +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)))))