Average Error: 37.9 → 13.4
Time: 25.9s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\sqrt{\left(re + \sqrt{re^2 + im^2}^*\right) \cdot 2.0} \cdot 0.5\]
double f(double re, double im) {
        double r18774071 = 0.5;
        double r18774072 = 2.0;
        double r18774073 = re;
        double r18774074 = r18774073 * r18774073;
        double r18774075 = im;
        double r18774076 = r18774075 * r18774075;
        double r18774077 = r18774074 + r18774076;
        double r18774078 = sqrt(r18774077);
        double r18774079 = r18774078 + r18774073;
        double r18774080 = r18774072 * r18774079;
        double r18774081 = sqrt(r18774080);
        double r18774082 = r18774071 * r18774081;
        return r18774082;
}
↓
double f(double re, double im) {
        double r18774083 = re;
        double r18774084 = im;
        double r18774085 = hypot(r18774083, r18774084);
        double r18774086 = r18774083 + r18774085;
        double r18774087 = 2.0;
        double r18774088 = r18774086 * r18774087;
        double r18774089 = sqrt(r18774088);
        double r18774090 = 0.5;
        double r18774091 = r18774089 * r18774090;
        return r18774091;
}


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

Error

Bits error versus re

Bits error versus im

Target

Original37.9
Target33.0
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 37.9

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

  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 2019102 +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)))))