Average Error: 37.3 → 12.8
Time: 20.3s
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 r6208181 = 0.5;
        double r6208182 = 2.0;
        double r6208183 = re;
        double r6208184 = r6208183 * r6208183;
        double r6208185 = im;
        double r6208186 = r6208185 * r6208185;
        double r6208187 = r6208184 + r6208186;
        double r6208188 = sqrt(r6208187);
        double r6208189 = r6208188 + r6208183;
        double r6208190 = r6208182 * r6208189;
        double r6208191 = sqrt(r6208190);
        double r6208192 = r6208181 * r6208191;
        return r6208192;
}
↓
double f(double re, double im) {
        double r6208193 = re;
        double r6208194 = im;
        double r6208195 = hypot(r6208193, r6208194);
        double r6208196 = r6208193 + r6208195;
        double r6208197 = 2.0;
        double r6208198 = r6208196 * r6208197;
        double r6208199 = sqrt(r6208198);
        double r6208200 = 0.5;
        double r6208201 = r6208199 * r6208200;
        return r6208201;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target32.5
Herbie12.8
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.3

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

  2. Simplified12.8

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) + re\right) \cdot 2.0}}\]

  3. Final simplification12.8

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

Reproduce

herbie shell --seed 2019163 +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)))))