Average Error: 37.3 → 12.9
Time: 19.4s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5
double f(double re, double im) {
        double r835186 = 0.5;
        double r835187 = 2.0;
        double r835188 = re;
        double r835189 = r835188 * r835188;
        double r835190 = im;
        double r835191 = r835190 * r835190;
        double r835192 = r835189 + r835191;
        double r835193 = sqrt(r835192);
        double r835194 = r835193 - r835188;
        double r835195 = r835187 * r835194;
        double r835196 = sqrt(r835195);
        double r835197 = r835186 * r835196;
        return r835197;
}
↓
double f(double re, double im) {
        double r835198 = re;
        double r835199 = im;
        double r835200 = hypot(r835198, r835199);
        double r835201 = r835200 - r835198;
        double r835202 = 2.0;
        double r835203 = r835201 * r835202;
        double r835204 = sqrt(r835203);
        double r835205 = 0.5;
        double r835206 = r835204 * r835205;
        return r835206;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 37.3

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

  2. Simplified12.9

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

  3. Final simplification12.9

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))