Average Error: 38.4 → 13.1
Time: 20.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2} \cdot 0.5\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2} \cdot 0.5
double f(double re, double im) {
        double r6024423 = 0.5;
        double r6024424 = 2.0;
        double r6024425 = re;
        double r6024426 = r6024425 * r6024425;
        double r6024427 = im;
        double r6024428 = r6024427 * r6024427;
        double r6024429 = r6024426 + r6024428;
        double r6024430 = sqrt(r6024429);
        double r6024431 = r6024430 + r6024425;
        double r6024432 = r6024424 * r6024431;
        double r6024433 = sqrt(r6024432);
        double r6024434 = r6024423 * r6024433;
        return r6024434;
}
↓
double f(double re, double im) {
        double r6024435 = re;
        double r6024436 = im;
        double r6024437 = hypot(r6024435, r6024436);
        double r6024438 = r6024435 + r6024437;
        double r6024439 = 2.0;
        double r6024440 = r6024438 * r6024439;
        double r6024441 = sqrt(r6024440);
        double r6024442 = 0.5;
        double r6024443 = r6024441 * r6024442;
        return r6024443;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.4
Target33.3
Herbie13.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.4

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

  2. Simplified13.1

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

  3. Final simplification13.1

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

Reproduce

herbie shell --seed 2019179 +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)))))