Average Error: 38.3 → 13.0
Time: 11.2s
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 r15109 = 0.5;
        double r15110 = 2.0;
        double r15111 = re;
        double r15112 = r15111 * r15111;
        double r15113 = im;
        double r15114 = r15113 * r15113;
        double r15115 = r15112 + r15114;
        double r15116 = sqrt(r15115);
        double r15117 = r15116 - r15111;
        double r15118 = r15110 * r15117;
        double r15119 = sqrt(r15118);
        double r15120 = r15109 * r15119;
        return r15120;
}
↓
double f(double re, double im) {
        double r15121 = 0.5;
        double r15122 = re;
        double r15123 = im;
        double r15124 = hypot(r15122, r15123);
        double r15125 = r15124 - r15122;
        double r15126 = 2.0;
        double r15127 = r15125 * r15126;
        double r15128 = sqrt(r15127);
        double r15129 = r15121 * r15128;
        return r15129;
}

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 38.3

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

  2. Simplified13.0

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

  3. Final simplification13.0

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

Reproduce

herbie shell --seed 2019351 +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)))))