Average Error: 38.6 → 13.5
Time: 38.2s
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 r5087249 = 0.5;
        double r5087250 = 2.0;
        double r5087251 = re;
        double r5087252 = r5087251 * r5087251;
        double r5087253 = im;
        double r5087254 = r5087253 * r5087253;
        double r5087255 = r5087252 + r5087254;
        double r5087256 = sqrt(r5087255);
        double r5087257 = r5087256 + r5087251;
        double r5087258 = r5087250 * r5087257;
        double r5087259 = sqrt(r5087258);
        double r5087260 = r5087249 * r5087259;
        return r5087260;
}
↓
double f(double re, double im) {
        double r5087261 = re;
        double r5087262 = im;
        double r5087263 = hypot(r5087261, r5087262);
        double r5087264 = r5087261 + r5087263;
        double r5087265 = 2.0;
        double r5087266 = r5087264 * r5087265;
        double r5087267 = sqrt(r5087266);
        double r5087268 = 0.5;
        double r5087269 = r5087267 * r5087268;
        return r5087269;
}

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
Target33.8
Herbie13.5
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.6

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

  2. Simplified13.5

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

  3. Final simplification13.5

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

Reproduce

herbie shell --seed 2019168 +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)))))