Average Error: 13.2 → 14.1
Time: 16.1s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\frac{\left(\frac{\left(0.5 \cdot x\right) \cdot \left(\sqrt[3]{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \left(\sqrt[3]{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \sqrt[3]{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p}}\right)\right)}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} + 0.5\right) \cdot \left(0.5 \cdot 0.5\right)}{0.5 \cdot 0.5 + \frac{0.5 \cdot x}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \left(\frac{0.5 \cdot x}{e^{\log \left(\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}\right)}} - 0.5\right)}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\frac{\left(\frac{\left(0.5 \cdot x\right) \cdot \left(\sqrt[3]{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \left(\sqrt[3]{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \sqrt[3]{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p}}\right)\right)}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} + 0.5\right) \cdot \left(0.5 \cdot 0.5\right)}{0.5 \cdot 0.5 + \frac{0.5 \cdot x}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \left(\frac{0.5 \cdot x}{e^{\log \left(\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}\right)}} - 0.5\right)}}
double f(double p, double x) {
        double r8181493 = 0.5;
        double r8181494 = 1.0;
        double r8181495 = x;
        double r8181496 = 4.0;
        double r8181497 = p;
        double r8181498 = r8181496 * r8181497;
        double r8181499 = r8181498 * r8181497;
        double r8181500 = r8181495 * r8181495;
        double r8181501 = r8181499 + r8181500;
        double r8181502 = sqrt(r8181501);
        double r8181503 = r8181495 / r8181502;
        double r8181504 = r8181494 + r8181503;
        double r8181505 = r8181493 * r8181504;
        double r8181506 = sqrt(r8181505);
        return r8181506;
}
↓
double f(double p, double x) {
        double r8181507 = 0.5;
        double r8181508 = x;
        double r8181509 = r8181507 * r8181508;
        double r8181510 = r8181508 * r8181508;
        double r8181511 = p;
        double r8181512 = 4.0;
        double r8181513 = r8181511 * r8181512;
        double r8181514 = r8181513 * r8181511;
        double r8181515 = r8181510 + r8181514;
        double r8181516 = r8181510 / r8181515;
        double r8181517 = cbrt(r8181516);
        double r8181518 = r8181517 * r8181517;
        double r8181519 = r8181517 * r8181518;
        double r8181520 = r8181509 * r8181519;
        double r8181521 = sqrt(r8181515);
        double r8181522 = r8181520 / r8181521;
        double r8181523 = r8181522 + r8181507;
        double r8181524 = r8181507 * r8181507;
        double r8181525 = r8181523 * r8181524;
        double r8181526 = r8181509 / r8181521;
        double r8181527 = log(r8181521);
        double r8181528 = exp(r8181527);
        double r8181529 = r8181509 / r8181528;
        double r8181530 = r8181529 - r8181507;
        double r8181531 = r8181526 * r8181530;
        double r8181532 = r8181524 + r8181531;
        double r8181533 = r8181525 / r8181532;
        double r8181534 = sqrt(r8181533);
        return r8181534;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.2
Target13.2
Herbie14.1
\[\sqrt{\frac{1}{2} + \frac{\mathsf{copysign}\left(\frac{1}{2}, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.2

    \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]

  2. Simplified13.2

    \[\leadsto \color{blue}{\sqrt{0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}} + 0.5}}\]
  3. Using strategy rm

  4. Applied flip3-+13.2

    \[\leadsto \sqrt{\color{blue}{\frac{{\left(0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right)}^{3} + {0.5}^{3}}{\left(0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) \cdot \left(0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) + \left(0.5 \cdot 0.5 - \left(0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) \cdot 0.5\right)}}}\]

  5. Simplified13.2

    \[\leadsto \sqrt{\frac{\color{blue}{\left(0.5 \cdot 0.5\right) \cdot \left(\frac{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p} \cdot \left(x \cdot 0.5\right)}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} + 0.5\right)}}{\left(0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) \cdot \left(0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) + \left(0.5 \cdot 0.5 - \left(0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) \cdot 0.5\right)}}\]

  6. Simplified13.2

    \[\leadsto \sqrt{\frac{\left(0.5 \cdot 0.5\right) \cdot \left(\frac{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p} \cdot \left(x \cdot 0.5\right)}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} + 0.5\right)}{\color{blue}{\frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \left(\frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} - 0.5\right) + 0.5 \cdot 0.5}}}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt13.2

    \[\leadsto \sqrt{\frac{\left(0.5 \cdot 0.5\right) \cdot \left(\frac{\color{blue}{\left(\left(\sqrt[3]{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \sqrt[3]{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p}}\right) \cdot \sqrt[3]{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p}}\right)} \cdot \left(x \cdot 0.5\right)}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} + 0.5\right)}{\frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \left(\frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} - 0.5\right) + 0.5 \cdot 0.5}}\]
  9. Using strategy rm

  10. Applied add-exp-log14.1

    \[\leadsto \sqrt{\frac{\left(0.5 \cdot 0.5\right) \cdot \left(\frac{\left(\left(\sqrt[3]{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \sqrt[3]{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p}}\right) \cdot \sqrt[3]{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p}}\right) \cdot \left(x \cdot 0.5\right)}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} + 0.5\right)}{\frac{x \cdot 0.5}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \left(\frac{x \cdot 0.5}{\color{blue}{e^{\log \left(\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}\right)}}} - 0.5\right) + 0.5 \cdot 0.5}}\]

  11. Final simplification14.1

    \[\leadsto \sqrt{\frac{\left(\frac{\left(0.5 \cdot x\right) \cdot \left(\sqrt[3]{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \left(\sqrt[3]{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \sqrt[3]{\frac{x \cdot x}{x \cdot x + \left(p \cdot 4\right) \cdot p}}\right)\right)}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} + 0.5\right) \cdot \left(0.5 \cdot 0.5\right)}{0.5 \cdot 0.5 + \frac{0.5 \cdot x}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \cdot \left(\frac{0.5 \cdot x}{e^{\log \left(\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}\right)}} - 0.5\right)}}\]

Reproduce

herbie shell --seed 2019135 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 1/2 (/ (copysign 1/2 x) (hypot 1 (/ (* 2 p) x)))))

  (sqrt (* 0.5 (+ 1 (/ x (sqrt (+ (* (* 4 p) p) (* x x))))))))