Average Error: 37.5 → 12.9
Time: 13.2s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\sqrt{\left(re + \sqrt{re^2 + im^2}^*\right) \cdot 2.0} \cdot 0.5\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\sqrt{\left(re + \sqrt{re^2 + im^2}^*\right) \cdot 2.0} \cdot 0.5
double f(double re, double im) {
        double r16332009 = 0.5;
        double r16332010 = 2.0;
        double r16332011 = re;
        double r16332012 = r16332011 * r16332011;
        double r16332013 = im;
        double r16332014 = r16332013 * r16332013;
        double r16332015 = r16332012 + r16332014;
        double r16332016 = sqrt(r16332015);
        double r16332017 = r16332016 + r16332011;
        double r16332018 = r16332010 * r16332017;
        double r16332019 = sqrt(r16332018);
        double r16332020 = r16332009 * r16332019;
        return r16332020;
}
↓
double f(double re, double im) {
        double r16332021 = re;
        double r16332022 = im;
        double r16332023 = hypot(r16332021, r16332022);
        double r16332024 = r16332021 + r16332023;
        double r16332025 = 2.0;
        double r16332026 = r16332024 * r16332025;
        double r16332027 = sqrt(r16332026);
        double r16332028 = 0.5;
        double r16332029 = r16332027 * r16332028;
        return r16332029;
}

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
Target32.8
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.5

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

  2. Simplified12.9

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\sqrt{re^2 + im^2}^* + re\right) \cdot 2.0}}\]

  3. Final simplification12.9

    \[\leadsto \sqrt{\left(re + \sqrt{re^2 + im^2}^*\right) \cdot 2.0} \cdot 0.5\]

Reproduce

herbie shell --seed 2019112 +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)))))