Average Error: 37.6 → 13.4
Time: 22.3s
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 r3535472 = 0.5;
        double r3535473 = 2.0;
        double r3535474 = re;
        double r3535475 = r3535474 * r3535474;
        double r3535476 = im;
        double r3535477 = r3535476 * r3535476;
        double r3535478 = r3535475 + r3535477;
        double r3535479 = sqrt(r3535478);
        double r3535480 = r3535479 + r3535474;
        double r3535481 = r3535473 * r3535480;
        double r3535482 = sqrt(r3535481);
        double r3535483 = r3535472 * r3535482;
        return r3535483;
}
↓
double f(double re, double im) {
        double r3535484 = re;
        double r3535485 = im;
        double r3535486 = hypot(r3535484, r3535485);
        double r3535487 = r3535484 + r3535486;
        double r3535488 = 2.0;
        double r3535489 = r3535487 * r3535488;
        double r3535490 = sqrt(r3535489);
        double r3535491 = 0.5;
        double r3535492 = r3535490 * r3535491;
        return r3535492;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target32.6
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 37.6

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

  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 2019151 +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)))))