Average Error: 38.3 → 14.2
Time: 15.2s
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 r2125390 = 0.5;
        double r2125391 = 2.0;
        double r2125392 = re;
        double r2125393 = r2125392 * r2125392;
        double r2125394 = im;
        double r2125395 = r2125394 * r2125394;
        double r2125396 = r2125393 + r2125395;
        double r2125397 = sqrt(r2125396);
        double r2125398 = r2125397 + r2125392;
        double r2125399 = r2125391 * r2125398;
        double r2125400 = sqrt(r2125399);
        double r2125401 = r2125390 * r2125400;
        return r2125401;
}
↓
double f(double re, double im) {
        double r2125402 = re;
        double r2125403 = im;
        double r2125404 = hypot(r2125402, r2125403);
        double r2125405 = r2125402 + r2125404;
        double r2125406 = 2.0;
        double r2125407 = r2125405 * r2125406;
        double r2125408 = sqrt(r2125407);
        double r2125409 = 0.5;
        double r2125410 = r2125408 * r2125409;
        return r2125410;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.3
Target33.2
Herbie14.2
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.3

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

  2. Simplified14.2

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

  3. Final simplification14.2

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

Reproduce

herbie shell --seed 2019153 +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)))))