Average Error: 37.4 → 13.5
Time: 26.1s
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 r38365133 = 0.5;
        double r38365134 = 2.0;
        double r38365135 = re;
        double r38365136 = r38365135 * r38365135;
        double r38365137 = im;
        double r38365138 = r38365137 * r38365137;
        double r38365139 = r38365136 + r38365138;
        double r38365140 = sqrt(r38365139);
        double r38365141 = r38365140 + r38365135;
        double r38365142 = r38365134 * r38365141;
        double r38365143 = sqrt(r38365142);
        double r38365144 = r38365133 * r38365143;
        return r38365144;
}
↓
double f(double re, double im) {
        double r38365145 = re;
        double r38365146 = im;
        double r38365147 = hypot(r38365145, r38365146);
        double r38365148 = r38365145 + r38365147;
        double r38365149 = 2.0;
        double r38365150 = r38365148 * r38365149;
        double r38365151 = sqrt(r38365150);
        double r38365152 = 0.5;
        double r38365153 = r38365151 * r38365152;
        return r38365153;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.4
Target32.4
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 37.4

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

  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 2019128 +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)))))