Average Error: 37.7 → 13.4
Time: 18.2s
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 r1820447 = 0.5;
        double r1820448 = 2.0;
        double r1820449 = re;
        double r1820450 = r1820449 * r1820449;
        double r1820451 = im;
        double r1820452 = r1820451 * r1820451;
        double r1820453 = r1820450 + r1820452;
        double r1820454 = sqrt(r1820453);
        double r1820455 = r1820454 + r1820449;
        double r1820456 = r1820448 * r1820455;
        double r1820457 = sqrt(r1820456);
        double r1820458 = r1820447 * r1820457;
        return r1820458;
}
↓
double f(double re, double im) {
        double r1820459 = re;
        double r1820460 = im;
        double r1820461 = hypot(r1820459, r1820460);
        double r1820462 = r1820459 + r1820461;
        double r1820463 = 2.0;
        double r1820464 = r1820462 * r1820463;
        double r1820465 = sqrt(r1820464);
        double r1820466 = 0.5;
        double r1820467 = r1820465 * r1820466;
        return r1820467;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.7
Target32.8
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 37.7

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

  2. Simplified13.4

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

  3. Final simplification13.4

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

Reproduce

herbie shell --seed 2019135 +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)))))