Average Error: 38.0 → 13.7
Time: 20.8s
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 r5228197 = 0.5;
        double r5228198 = 2.0;
        double r5228199 = re;
        double r5228200 = r5228199 * r5228199;
        double r5228201 = im;
        double r5228202 = r5228201 * r5228201;
        double r5228203 = r5228200 + r5228202;
        double r5228204 = sqrt(r5228203);
        double r5228205 = r5228204 + r5228199;
        double r5228206 = r5228198 * r5228205;
        double r5228207 = sqrt(r5228206);
        double r5228208 = r5228197 * r5228207;
        return r5228208;
}
↓
double f(double re, double im) {
        double r5228209 = re;
        double r5228210 = im;
        double r5228211 = hypot(r5228209, r5228210);
        double r5228212 = r5228209 + r5228211;
        double r5228213 = 2.0;
        double r5228214 = r5228212 * r5228213;
        double r5228215 = sqrt(r5228214);
        double r5228216 = 0.5;
        double r5228217 = r5228215 * r5228216;
        return r5228217;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.0
Target33.1
Herbie13.7
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.0

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

  2. Simplified13.7

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

  3. Final simplification13.7

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

Reproduce

herbie shell --seed 2019144 +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)))))