Average Error: 37.4 → 12.9
Time: 23.0s
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 r6453340 = 0.5;
        double r6453341 = 2.0;
        double r6453342 = re;
        double r6453343 = r6453342 * r6453342;
        double r6453344 = im;
        double r6453345 = r6453344 * r6453344;
        double r6453346 = r6453343 + r6453345;
        double r6453347 = sqrt(r6453346);
        double r6453348 = r6453347 + r6453342;
        double r6453349 = r6453341 * r6453348;
        double r6453350 = sqrt(r6453349);
        double r6453351 = r6453340 * r6453350;
        return r6453351;
}
↓
double f(double re, double im) {
        double r6453352 = re;
        double r6453353 = im;
        double r6453354 = hypot(r6453352, r6453353);
        double r6453355 = r6453352 + r6453354;
        double r6453356 = 2.0;
        double r6453357 = r6453355 * r6453356;
        double r6453358 = sqrt(r6453357);
        double r6453359 = 0.5;
        double r6453360 = r6453358 * r6453359;
        return r6453360;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.4
Target32.6
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.4

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

  2. Simplified12.9

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

  3. Final simplification12.9

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

Reproduce

herbie shell --seed 2019149 +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)))))