Average Error: 39.5 → 13.3
Time: 11.6s
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 r244264 = 0.5;
        double r244265 = 2.0;
        double r244266 = re;
        double r244267 = r244266 * r244266;
        double r244268 = im;
        double r244269 = r244268 * r244268;
        double r244270 = r244267 + r244269;
        double r244271 = sqrt(r244270);
        double r244272 = r244271 + r244266;
        double r244273 = r244265 * r244272;
        double r244274 = sqrt(r244273);
        double r244275 = r244264 * r244274;
        return r244275;
}
↓
double f(double re, double im) {
        double r244276 = 0.5;
        double r244277 = re;
        double r244278 = im;
        double r244279 = hypot(r244277, r244278);
        double r244280 = r244277 + r244279;
        double r244281 = 2.0;
        double r244282 = r244280 * r244281;
        double r244283 = sqrt(r244282);
        double r244284 = r244276 * r244283;
        return r244284;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.5
Target34.4
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 39.5

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

  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 2020047 +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)))))