Average Error: 38.1 → 13.3
Time: 11.8s
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 r14941 = 0.5;
        double r14942 = 2.0;
        double r14943 = re;
        double r14944 = r14943 * r14943;
        double r14945 = im;
        double r14946 = r14945 * r14945;
        double r14947 = r14944 + r14946;
        double r14948 = sqrt(r14947);
        double r14949 = r14948 - r14943;
        double r14950 = r14942 * r14949;
        double r14951 = sqrt(r14950);
        double r14952 = r14941 * r14951;
        return r14952;
}
↓
double f(double re, double im) {
        double r14953 = 0.5;
        double r14954 = re;
        double r14955 = im;
        double r14956 = hypot(r14954, r14955);
        double r14957 = r14956 - r14954;
        double r14958 = 2.0;
        double r14959 = r14957 * r14958;
        double r14960 = sqrt(r14959);
        double r14961 = r14953 * r14960;
        return r14961;
}

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 38.1

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

  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 2019350 +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)))))