Average Error: 38.9 → 13.5
Time: 4.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}
double f(double re, double im) {
        double r339849 = 0.5;
        double r339850 = 2.0;
        double r339851 = re;
        double r339852 = r339851 * r339851;
        double r339853 = im;
        double r339854 = r339853 * r339853;
        double r339855 = r339852 + r339854;
        double r339856 = sqrt(r339855);
        double r339857 = r339856 + r339851;
        double r339858 = r339850 * r339857;
        double r339859 = sqrt(r339858);
        double r339860 = r339849 * r339859;
        return r339860;
}
↓
double f(double re, double im) {
        double r339861 = 0.5;
        double r339862 = 2.0;
        double r339863 = 1.0;
        double r339864 = re;
        double r339865 = im;
        double r339866 = hypot(r339864, r339865);
        double r339867 = r339864 + r339866;
        double r339868 = r339863 * r339867;
        double r339869 = r339862 * r339868;
        double r339870 = sqrt(r339869);
        double r339871 = r339861 * r339870;
        return r339871;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.9
Target34.1
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 38.9

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

  3. Applied *-un-lft-identity38.9

    \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + \color{blue}{1 \cdot re}\right)}\]

  4. Applied *-un-lft-identity38.9

    \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1 \cdot \sqrt{re \cdot re + im \cdot im}} + 1 \cdot re\right)}\]

  5. Applied distribute-lft-out38.9

    \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)}}\]

  6. Simplified13.5

    \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)}\]

  7. Final simplification13.5

    \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))