Average Error: 38.3 → 14.2
Time: 16.6s
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 r2358232 = 0.5;
        double r2358233 = 2.0;
        double r2358234 = re;
        double r2358235 = r2358234 * r2358234;
        double r2358236 = im;
        double r2358237 = r2358236 * r2358236;
        double r2358238 = r2358235 + r2358237;
        double r2358239 = sqrt(r2358238);
        double r2358240 = r2358239 + r2358234;
        double r2358241 = r2358233 * r2358240;
        double r2358242 = sqrt(r2358241);
        double r2358243 = r2358232 * r2358242;
        return r2358243;
}
↓
double f(double re, double im) {
        double r2358244 = re;
        double r2358245 = im;
        double r2358246 = hypot(r2358244, r2358245);
        double r2358247 = r2358244 + r2358246;
        double r2358248 = 2.0;
        double r2358249 = r2358247 * r2358248;
        double r2358250 = sqrt(r2358249);
        double r2358251 = 0.5;
        double r2358252 = r2358250 * r2358251;
        return r2358252;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.3
Target33.2
Herbie14.2
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.3

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

  2. Simplified14.2

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

  3. Final simplification14.2

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

Reproduce

herbie shell --seed 2019153 +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)))))