Average Error: 37.1 → 13.1
Time: 23.9s
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 r22377829 = 0.5;
        double r22377830 = 2.0;
        double r22377831 = re;
        double r22377832 = r22377831 * r22377831;
        double r22377833 = im;
        double r22377834 = r22377833 * r22377833;
        double r22377835 = r22377832 + r22377834;
        double r22377836 = sqrt(r22377835);
        double r22377837 = r22377836 + r22377831;
        double r22377838 = r22377830 * r22377837;
        double r22377839 = sqrt(r22377838);
        double r22377840 = r22377829 * r22377839;
        return r22377840;
}
↓
double f(double re, double im) {
        double r22377841 = re;
        double r22377842 = im;
        double r22377843 = hypot(r22377841, r22377842);
        double r22377844 = r22377841 + r22377843;
        double r22377845 = 2.0;
        double r22377846 = r22377844 * r22377845;
        double r22377847 = sqrt(r22377846);
        double r22377848 = 0.5;
        double r22377849 = r22377847 * r22377848;
        return r22377849;
}


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

Original37.1
Target32.0
Herbie13.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.1

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

  2. Simplified13.1

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

  3. Final simplification13.1

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

Reproduce

herbie shell --seed 2019101 +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)))))