Average Error: 37.5 → 13.6
Time: 21.7s
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 r5139965 = 0.5;
        double r5139966 = 2.0;
        double r5139967 = re;
        double r5139968 = r5139967 * r5139967;
        double r5139969 = im;
        double r5139970 = r5139969 * r5139969;
        double r5139971 = r5139968 + r5139970;
        double r5139972 = sqrt(r5139971);
        double r5139973 = r5139972 + r5139967;
        double r5139974 = r5139966 * r5139973;
        double r5139975 = sqrt(r5139974);
        double r5139976 = r5139965 * r5139975;
        return r5139976;
}
↓
double f(double re, double im) {
        double r5139977 = re;
        double r5139978 = im;
        double r5139979 = hypot(r5139977, r5139978);
        double r5139980 = r5139977 + r5139979;
        double r5139981 = 2.0;
        double r5139982 = r5139980 * r5139981;
        double r5139983 = sqrt(r5139982);
        double r5139984 = 0.5;
        double r5139985 = r5139983 * r5139984;
        return r5139985;
}

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
Target32.7
Herbie13.6
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.5

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

  2. Simplified13.6

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

  3. Final simplification13.6

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

Reproduce

herbie shell --seed 2019162 +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)))))