Average Error: 36.8 → 13.1
Time: 18.8s
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 r745954 = 0.5;
        double r745955 = 2.0;
        double r745956 = re;
        double r745957 = r745956 * r745956;
        double r745958 = im;
        double r745959 = r745958 * r745958;
        double r745960 = r745957 + r745959;
        double r745961 = sqrt(r745960);
        double r745962 = r745961 - r745956;
        double r745963 = r745955 * r745962;
        double r745964 = sqrt(r745963);
        double r745965 = r745954 * r745964;
        return r745965;
}
↓
double f(double re, double im) {
        double r745966 = re;
        double r745967 = im;
        double r745968 = hypot(r745966, r745967);
        double r745969 = r745968 - r745966;
        double r745970 = 2.0;
        double r745971 = r745969 * r745970;
        double r745972 = sqrt(r745971);
        double r745973 = 0.5;
        double r745974 = r745972 * r745973;
        return r745974;
}

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 36.8

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

  2. Simplified13.1

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

  3. Final simplification13.1

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

Reproduce

herbie shell --seed 2019158 +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)))))