Average Error: 39.3 → 13.3
Time: 10.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}
double f(double re, double im) {
        double r24228 = 0.5;
        double r24229 = 2.0;
        double r24230 = re;
        double r24231 = r24230 * r24230;
        double r24232 = im;
        double r24233 = r24232 * r24232;
        double r24234 = r24231 + r24233;
        double r24235 = sqrt(r24234);
        double r24236 = r24235 - r24230;
        double r24237 = r24229 * r24236;
        double r24238 = sqrt(r24237);
        double r24239 = r24228 * r24238;
        return r24239;
}
↓
double f(double re, double im) {
        double r24240 = 0.5;
        double r24241 = re;
        double r24242 = im;
        double r24243 = hypot(r24241, r24242);
        double r24244 = r24243 - r24241;
        double r24245 = 2.0;
        double r24246 = r24244 * r24245;
        double r24247 = sqrt(r24246);
        double r24248 = r24240 * r24247;
        return r24248;
}

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 39.3

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

  2. Simplified13.3

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

  3. Final simplification13.3

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))