Average Error: 37.6 → 13.5
Time: 20.8s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\sqrt{\left(\sqrt{re^2 + im^2}^* - re\right) \cdot 2.0} \cdot 0.5\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\sqrt{\left(\sqrt{re^2 + im^2}^* - re\right) \cdot 2.0} \cdot 0.5
double f(double re, double im) {
        double r463203 = 0.5;
        double r463204 = 2.0;
        double r463205 = re;
        double r463206 = r463205 * r463205;
        double r463207 = im;
        double r463208 = r463207 * r463207;
        double r463209 = r463206 + r463208;
        double r463210 = sqrt(r463209);
        double r463211 = r463210 - r463205;
        double r463212 = r463204 * r463211;
        double r463213 = sqrt(r463212);
        double r463214 = r463203 * r463213;
        return r463214;
}
↓
double f(double re, double im) {
        double r463215 = re;
        double r463216 = im;
        double r463217 = hypot(r463215, r463216);
        double r463218 = r463217 - r463215;
        double r463219 = 2.0;
        double r463220 = r463218 * r463219;
        double r463221 = sqrt(r463220);
        double r463222 = 0.5;
        double r463223 = r463221 * r463222;
        return r463223;
}

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 37.6

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

  2. Simplified13.5

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\sqrt{re^2 + im^2}^* - re\right) \cdot 2.0}}\]

  3. Final simplification13.5

    \[\leadsto \sqrt{\left(\sqrt{re^2 + im^2}^* - re\right) \cdot 2.0} \cdot 0.5\]

Reproduce

herbie shell --seed 2019119 +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)))))