Average Error: 38.1 → 13.0
Time: 16.0s
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 r6445227 = 0.5;
        double r6445228 = 2.0;
        double r6445229 = re;
        double r6445230 = r6445229 * r6445229;
        double r6445231 = im;
        double r6445232 = r6445231 * r6445231;
        double r6445233 = r6445230 + r6445232;
        double r6445234 = sqrt(r6445233);
        double r6445235 = r6445234 + r6445229;
        double r6445236 = r6445228 * r6445235;
        double r6445237 = sqrt(r6445236);
        double r6445238 = r6445227 * r6445237;
        return r6445238;
}
↓
double f(double re, double im) {
        double r6445239 = re;
        double r6445240 = im;
        double r6445241 = hypot(r6445239, r6445240);
        double r6445242 = r6445239 + r6445241;
        double r6445243 = 2.0;
        double r6445244 = r6445242 * r6445243;
        double r6445245 = sqrt(r6445244);
        double r6445246 = 0.5;
        double r6445247 = r6445245 * r6445246;
        return r6445247;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.1
Target33.2
Herbie13.0
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.1

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

  2. Simplified13.0

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

  3. Final simplification13.0

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

Reproduce

herbie shell --seed 2019174 +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)))))