Average Error: 37.4 → 12.7
Time: 16.9s
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 r310154 = 0.5;
        double r310155 = 2.0;
        double r310156 = re;
        double r310157 = r310156 * r310156;
        double r310158 = im;
        double r310159 = r310158 * r310158;
        double r310160 = r310157 + r310159;
        double r310161 = sqrt(r310160);
        double r310162 = r310161 - r310156;
        double r310163 = r310155 * r310162;
        double r310164 = sqrt(r310163);
        double r310165 = r310154 * r310164;
        return r310165;
}
↓
double f(double re, double im) {
        double r310166 = re;
        double r310167 = im;
        double r310168 = hypot(r310166, r310167);
        double r310169 = r310168 - r310166;
        double r310170 = 2.0;
        double r310171 = r310169 * r310170;
        double r310172 = sqrt(r310171);
        double r310173 = 0.5;
        double r310174 = r310172 * r310173;
        return r310174;
}

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 37.4

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

  2. Simplified12.7

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

  3. Final simplification12.7

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

Reproduce

herbie shell --seed 2019153 +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)))))