Average Error: 37.7 → 13.2
Time: 21.0s
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 r463267 = 0.5;
        double r463268 = 2.0;
        double r463269 = re;
        double r463270 = r463269 * r463269;
        double r463271 = im;
        double r463272 = r463271 * r463271;
        double r463273 = r463270 + r463272;
        double r463274 = sqrt(r463273);
        double r463275 = r463274 - r463269;
        double r463276 = r463268 * r463275;
        double r463277 = sqrt(r463276);
        double r463278 = r463267 * r463277;
        return r463278;
}
↓
double f(double re, double im) {
        double r463279 = re;
        double r463280 = im;
        double r463281 = hypot(r463279, r463280);
        double r463282 = r463281 - r463279;
        double r463283 = 2.0;
        double r463284 = r463282 * r463283;
        double r463285 = sqrt(r463284);
        double r463286 = 0.5;
        double r463287 = r463285 * r463286;
        return r463287;
}

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 37.7

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

  2. Simplified13.2

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

  3. Final simplification13.2

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

Reproduce

herbie shell --seed 2019112 +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)))))