Average Error: 39.0 → 13.3
Time: 13.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
0.5 \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}
double f(double re, double im) {
        double r112140 = 0.5;
        double r112141 = 2.0;
        double r112142 = re;
        double r112143 = r112142 * r112142;
        double r112144 = im;
        double r112145 = r112144 * r112144;
        double r112146 = r112143 + r112145;
        double r112147 = sqrt(r112146);
        double r112148 = r112147 + r112142;
        double r112149 = r112141 * r112148;
        double r112150 = sqrt(r112149);
        double r112151 = r112140 * r112150;
        return r112151;
}
↓
double f(double re, double im) {
        double r112152 = 0.5;
        double r112153 = re;
        double r112154 = im;
        double r112155 = hypot(r112153, r112154);
        double r112156 = r112153 + r112155;
        double r112157 = 2.0;
        double r112158 = r112156 * r112157;
        double r112159 = sqrt(r112158);
        double r112160 = r112152 * r112159;
        return r112160;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.0
Target33.9
Herbie13.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 39.0

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

  2. Simplified13.3

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

  3. Final simplification13.3

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))