Average Error: 38.6 → 13.6
Time: 14.3s
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 r86007 = 0.5;
        double r86008 = 2.0;
        double r86009 = re;
        double r86010 = r86009 * r86009;
        double r86011 = im;
        double r86012 = r86011 * r86011;
        double r86013 = r86010 + r86012;
        double r86014 = sqrt(r86013);
        double r86015 = r86014 + r86009;
        double r86016 = r86008 * r86015;
        double r86017 = sqrt(r86016);
        double r86018 = r86007 * r86017;
        return r86018;
}
↓
double f(double re, double im) {
        double r86019 = re;
        double r86020 = im;
        double r86021 = hypot(r86019, r86020);
        double r86022 = r86019 + r86021;
        double r86023 = 2.0;
        double r86024 = r86022 * r86023;
        double r86025 = sqrt(r86024);
        double r86026 = 0.5;
        double r86027 = r86025 * r86026;
        return r86027;
}

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

Derivation

  1. Initial program 38.6

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

  2. Simplified13.6

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

  3. Final simplification13.6

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

Reproduce

herbie shell --seed 2019195 +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)))))