Average Error: 38.6 → 13.5
Time: 30.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) + re}\right)\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) + re}\right)
double f(double re, double im) {
        double r6440306 = 0.5;
        double r6440307 = 2.0;
        double r6440308 = re;
        double r6440309 = r6440308 * r6440308;
        double r6440310 = im;
        double r6440311 = r6440310 * r6440310;
        double r6440312 = r6440309 + r6440311;
        double r6440313 = sqrt(r6440312);
        double r6440314 = r6440313 + r6440308;
        double r6440315 = r6440307 * r6440314;
        double r6440316 = sqrt(r6440315);
        double r6440317 = r6440306 * r6440316;
        return r6440317;
}
↓
double f(double re, double im) {
        double r6440318 = 0.5;
        double r6440319 = 2.0;
        double r6440320 = sqrt(r6440319);
        double r6440321 = re;
        double r6440322 = im;
        double r6440323 = hypot(r6440321, r6440322);
        double r6440324 = r6440323 + r6440321;
        double r6440325 = sqrt(r6440324);
        double r6440326 = r6440320 * r6440325;
        double r6440327 = r6440318 * r6440326;
        return r6440327;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.6
Target33.5
Herbie13.5
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.6

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

  2. Simplified13.2

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}}\]
  3. Using strategy rm

  4. Applied sqrt-prod13.5

    \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{re + \mathsf{hypot}\left(re, im\right)} \cdot \sqrt{2}\right)}\]

  5. Final simplification13.5

    \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) + re}\right)\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))