Average Error: 37.6 → 13.4
Time: 19.2s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\sqrt{\left(re + \sqrt{re^2 + im^2}^*\right) \cdot 2.0} \cdot 0.5\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\sqrt{\left(re + \sqrt{re^2 + im^2}^*\right) \cdot 2.0} \cdot 0.5
double f(double re, double im) {
        double r39454130 = 0.5;
        double r39454131 = 2.0;
        double r39454132 = re;
        double r39454133 = r39454132 * r39454132;
        double r39454134 = im;
        double r39454135 = r39454134 * r39454134;
        double r39454136 = r39454133 + r39454135;
        double r39454137 = sqrt(r39454136);
        double r39454138 = r39454137 + r39454132;
        double r39454139 = r39454131 * r39454138;
        double r39454140 = sqrt(r39454139);
        double r39454141 = r39454130 * r39454140;
        return r39454141;
}
↓
double f(double re, double im) {
        double r39454142 = re;
        double r39454143 = im;
        double r39454144 = hypot(r39454142, r39454143);
        double r39454145 = r39454142 + r39454144;
        double r39454146 = 2.0;
        double r39454147 = r39454145 * r39454146;
        double r39454148 = sqrt(r39454147);
        double r39454149 = 0.5;
        double r39454150 = r39454148 * r39454149;
        return r39454150;
}

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
Target32.9
Herbie13.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.6

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

  2. Simplified13.4

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\sqrt{re^2 + im^2}^* + re\right) \cdot 2.0}}\]

  3. Final simplification13.4

    \[\leadsto \sqrt{\left(re + \sqrt{re^2 + im^2}^*\right) \cdot 2.0} \cdot 0.5\]

Reproduce

herbie shell --seed 2019104 +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)))))