Average Error: 39.0 → 13.3
Time: 8.8s
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 r24435 = 0.5;
        double r24436 = 2.0;
        double r24437 = re;
        double r24438 = r24437 * r24437;
        double r24439 = im;
        double r24440 = r24439 * r24439;
        double r24441 = r24438 + r24440;
        double r24442 = sqrt(r24441);
        double r24443 = r24442 - r24437;
        double r24444 = r24436 * r24443;
        double r24445 = sqrt(r24444);
        double r24446 = r24435 * r24445;
        return r24446;
}
↓
double f(double re, double im) {
        double r24447 = 0.5;
        double r24448 = re;
        double r24449 = im;
        double r24450 = hypot(r24448, r24449);
        double r24451 = r24450 - r24448;
        double r24452 = 2.0;
        double r24453 = r24451 * r24452;
        double r24454 = sqrt(r24453);
        double r24455 = r24447 * r24454;
        return r24455;
}

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 39.0

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

  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 2020047 +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)))))