Average Error: 38.2 → 13.8
Time: 19.2s
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 r549213 = 0.5;
        double r549214 = 2.0;
        double r549215 = re;
        double r549216 = r549215 * r549215;
        double r549217 = im;
        double r549218 = r549217 * r549217;
        double r549219 = r549216 + r549218;
        double r549220 = sqrt(r549219);
        double r549221 = r549220 - r549215;
        double r549222 = r549214 * r549221;
        double r549223 = sqrt(r549222);
        double r549224 = r549213 * r549223;
        return r549224;
}
↓
double f(double re, double im) {
        double r549225 = re;
        double r549226 = im;
        double r549227 = hypot(r549225, r549226);
        double r549228 = r549227 - r549225;
        double r549229 = 2.0;
        double r549230 = r549228 * r549229;
        double r549231 = sqrt(r549230);
        double r549232 = 0.5;
        double r549233 = r549231 * r549232;
        return r549233;
}

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 38.2

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

  2. Simplified13.8

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

  3. Final simplification13.8

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

Reproduce

herbie shell --seed 2019142 +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)))))