Average Error: 37.5 → 13.6
Time: 15.8s
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 r2589377 = 0.5;
        double r2589378 = 2.0;
        double r2589379 = re;
        double r2589380 = r2589379 * r2589379;
        double r2589381 = im;
        double r2589382 = r2589381 * r2589381;
        double r2589383 = r2589380 + r2589382;
        double r2589384 = sqrt(r2589383);
        double r2589385 = r2589384 + r2589379;
        double r2589386 = r2589378 * r2589385;
        double r2589387 = sqrt(r2589386);
        double r2589388 = r2589377 * r2589387;
        return r2589388;
}
↓
double f(double re, double im) {
        double r2589389 = re;
        double r2589390 = im;
        double r2589391 = hypot(r2589389, r2589390);
        double r2589392 = r2589389 + r2589391;
        double r2589393 = 2.0;
        double r2589394 = r2589392 * r2589393;
        double r2589395 = sqrt(r2589394);
        double r2589396 = 0.5;
        double r2589397 = r2589395 * r2589396;
        return r2589397;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.5
Target32.4
Herbie13.6
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.5

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

  2. Simplified13.6

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

  3. Final simplification13.6

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

Reproduce

herbie shell --seed 2019152 +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)))))