Average Error: 13.3 → 13.3
Time: 14.2s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt[3]{{\left(\sqrt{\left(\frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}} + 1\right) \cdot 0.5}\right)}^{3}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt[3]{{\left(\sqrt{\left(\frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}} + 1\right) \cdot 0.5}\right)}^{3}}
double f(double p, double x) {
        double r236567 = 0.5;
        double r236568 = 1.0;
        double r236569 = x;
        double r236570 = 4.0;
        double r236571 = p;
        double r236572 = r236570 * r236571;
        double r236573 = r236572 * r236571;
        double r236574 = r236569 * r236569;
        double r236575 = r236573 + r236574;
        double r236576 = sqrt(r236575);
        double r236577 = r236569 / r236576;
        double r236578 = r236568 + r236577;
        double r236579 = r236567 * r236578;
        double r236580 = sqrt(r236579);
        return r236580;
}
↓
double f(double p, double x) {
        double r236581 = x;
        double r236582 = r236581 * r236581;
        double r236583 = p;
        double r236584 = 4.0;
        double r236585 = r236583 * r236584;
        double r236586 = r236583 * r236585;
        double r236587 = r236582 + r236586;
        double r236588 = sqrt(r236587);
        double r236589 = r236581 / r236588;
        double r236590 = 1.0;
        double r236591 = r236589 + r236590;
        double r236592 = 0.5;
        double r236593 = r236591 * r236592;
        double r236594 = sqrt(r236593);
        double r236595 = 3.0;
        double r236596 = pow(r236594, r236595);
        double r236597 = cbrt(r236596);
        return r236597;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.3
Target13.3
Herbie13.3
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.3

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

  2. Simplified13.3

    \[\leadsto \color{blue}{\sqrt{\left(1 + \frac{x}{\sqrt{x \cdot x + \left(p \cdot p\right) \cdot 4}}\right) \cdot 0.5}}\]
  3. Using strategy rm

  4. Applied add-log-exp13.3

    \[\leadsto \color{blue}{\log \left(e^{\sqrt{\left(1 + \frac{x}{\sqrt{x \cdot x + \left(p \cdot p\right) \cdot 4}}\right) \cdot 0.5}}\right)}\]

  5. Simplified13.3

    \[\leadsto \log \color{blue}{\left(e^{\sqrt{0.5 \cdot \left(\frac{x}{\sqrt{x \cdot x + 4 \cdot {p}^{2}}} + 1\right)}}\right)}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt13.3

    \[\leadsto \log \left(e^{\sqrt{0.5 \cdot \left(\frac{x}{\sqrt{\color{blue}{\sqrt{x \cdot x + 4 \cdot {p}^{2}} \cdot \sqrt{x \cdot x + 4 \cdot {p}^{2}}}}} + 1\right)}}\right)\]

  8. Applied sqrt-prod14.3

    \[\leadsto \log \left(e^{\sqrt{0.5 \cdot \left(\frac{x}{\color{blue}{\sqrt{\sqrt{x \cdot x + 4 \cdot {p}^{2}}} \cdot \sqrt{\sqrt{x \cdot x + 4 \cdot {p}^{2}}}}} + 1\right)}}\right)\]

  9. Simplified14.3

    \[\leadsto \log \left(e^{\sqrt{0.5 \cdot \left(\frac{x}{\color{blue}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt{\sqrt{x \cdot x + 4 \cdot {p}^{2}}}} + 1\right)}}\right)\]

  10. Simplified14.3

    \[\leadsto \log \left(e^{\sqrt{0.5 \cdot \left(\frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \color{blue}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} + 1\right)}}\right)\]
  11. Using strategy rm

  12. Applied add-cbrt-cube14.3

    \[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(e^{\sqrt{0.5 \cdot \left(\frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} + 1\right)}}\right) \cdot \log \left(e^{\sqrt{0.5 \cdot \left(\frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} + 1\right)}}\right)\right) \cdot \log \left(e^{\sqrt{0.5 \cdot \left(\frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} + 1\right)}}\right)}}\]

  13. Simplified13.3

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right) \cdot 0.5}\right)}^{3}}}\]

  14. Final simplification13.3

    \[\leadsto \sqrt[3]{{\left(\sqrt{\left(\frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}} + 1\right) \cdot 0.5}\right)}^{3}}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))