Average Error: 37.4 → 13.4
Time: 18.0s
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 r7012138 = 0.5;
        double r7012139 = 2.0;
        double r7012140 = re;
        double r7012141 = r7012140 * r7012140;
        double r7012142 = im;
        double r7012143 = r7012142 * r7012142;
        double r7012144 = r7012141 + r7012143;
        double r7012145 = sqrt(r7012144);
        double r7012146 = r7012145 + r7012140;
        double r7012147 = r7012139 * r7012146;
        double r7012148 = sqrt(r7012147);
        double r7012149 = r7012138 * r7012148;
        return r7012149;
}
↓
double f(double re, double im) {
        double r7012150 = re;
        double r7012151 = im;
        double r7012152 = hypot(r7012150, r7012151);
        double r7012153 = r7012150 + r7012152;
        double r7012154 = 2.0;
        double r7012155 = r7012153 * r7012154;
        double r7012156 = sqrt(r7012155);
        double r7012157 = 0.5;
        double r7012158 = r7012156 * r7012157;
        return r7012158;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.4
Target32.3
Herbie13.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.4

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

  2. Simplified13.4

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

  3. Final simplification13.4

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

Reproduce

herbie shell --seed 2019164 +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)))))