Average Error: 38.6 → 13.3
Time: 17.5s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2} \cdot 0.5\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2} \cdot 0.5
double f(double re, double im) {
        double r6654333 = 0.5;
        double r6654334 = 2.0;
        double r6654335 = re;
        double r6654336 = r6654335 * r6654335;
        double r6654337 = im;
        double r6654338 = r6654337 * r6654337;
        double r6654339 = r6654336 + r6654338;
        double r6654340 = sqrt(r6654339);
        double r6654341 = r6654340 + r6654335;
        double r6654342 = r6654334 * r6654341;
        double r6654343 = sqrt(r6654342);
        double r6654344 = r6654333 * r6654343;
        return r6654344;
}
↓
double f(double re, double im) {
        double r6654345 = re;
        double r6654346 = im;
        double r6654347 = hypot(r6654345, r6654346);
        double r6654348 = r6654345 + r6654347;
        double r6654349 = 2.0;
        double r6654350 = r6654348 * r6654349;
        double r6654351 = sqrt(r6654350);
        double r6654352 = 0.5;
        double r6654353 = r6654351 * r6654352;
        return r6654353;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.6
Target33.8
Herbie13.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.6

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

  2. Simplified13.3

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) + re\right) \cdot 2}}\]

  3. Final simplification13.3

    \[\leadsto \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2} \cdot 0.5\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (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)))))