Average Error: 37.6 → 13.1
Time: 18.5s
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 r16652087 = 0.5;
        double r16652088 = 2.0;
        double r16652089 = re;
        double r16652090 = r16652089 * r16652089;
        double r16652091 = im;
        double r16652092 = r16652091 * r16652091;
        double r16652093 = r16652090 + r16652092;
        double r16652094 = sqrt(r16652093);
        double r16652095 = r16652094 + r16652089;
        double r16652096 = r16652088 * r16652095;
        double r16652097 = sqrt(r16652096);
        double r16652098 = r16652087 * r16652097;
        return r16652098;
}
↓
double f(double re, double im) {
        double r16652099 = re;
        double r16652100 = im;
        double r16652101 = hypot(r16652099, r16652100);
        double r16652102 = r16652099 + r16652101;
        double r16652103 = 2.0;
        double r16652104 = r16652102 * r16652103;
        double r16652105 = sqrt(r16652104);
        double r16652106 = 0.5;
        double r16652107 = r16652105 * r16652106;
        return r16652107;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target32.8
Herbie13.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.6

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

  2. Simplified13.1

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

  3. Final simplification13.1

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

Reproduce

herbie shell --seed 2019120 +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)))))