Average Error: 12.9 → 13.2
Time: 36.2s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt[3]{e^{\log \left(\left(\sqrt[3]{0.5 + x \cdot \frac{0.5}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}} \cdot \sqrt[3]{0.5 + x \cdot \frac{0.5}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}\right) \cdot \sqrt[3]{0.5 + x \cdot \frac{0.5}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}\right)} \cdot \sqrt{e^{\log \left(0.5 + x \cdot \frac{0.5}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}\right)}}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt[3]{e^{\log \left(\left(\sqrt[3]{0.5 + x \cdot \frac{0.5}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}} \cdot \sqrt[3]{0.5 + x \cdot \frac{0.5}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}\right) \cdot \sqrt[3]{0.5 + x \cdot \frac{0.5}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}}\right)} \cdot \sqrt{e^{\log \left(0.5 + x \cdot \frac{0.5}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4\right)}}\right)}}}
double f(double p, double x) {
        double r115219459 = 0.5;
        double r115219460 = 1.0;
        double r115219461 = x;
        double r115219462 = 4.0;
        double r115219463 = p;
        double r115219464 = r115219462 * r115219463;
        double r115219465 = r115219464 * r115219463;
        double r115219466 = r115219461 * r115219461;
        double r115219467 = r115219465 + r115219466;
        double r115219468 = sqrt(r115219467);
        double r115219469 = r115219461 / r115219468;
        double r115219470 = r115219460 + r115219469;
        double r115219471 = r115219459 * r115219470;
        double r115219472 = sqrt(r115219471);
        return r115219472;
}
↓
double f(double p, double x) {
        double r115219473 = 0.5;
        double r115219474 = x;
        double r115219475 = r115219474 * r115219474;
        double r115219476 = p;
        double r115219477 = 4.0;
        double r115219478 = r115219476 * r115219477;
        double r115219479 = r115219476 * r115219478;
        double r115219480 = r115219475 + r115219479;
        double r115219481 = sqrt(r115219480);
        double r115219482 = r115219473 / r115219481;
        double r115219483 = r115219474 * r115219482;
        double r115219484 = r115219473 + r115219483;
        double r115219485 = cbrt(r115219484);
        double r115219486 = r115219485 * r115219485;
        double r115219487 = r115219486 * r115219485;
        double r115219488 = log(r115219487);
        double r115219489 = exp(r115219488);
        double r115219490 = log(r115219484);
        double r115219491 = exp(r115219490);
        double r115219492 = sqrt(r115219491);
        double r115219493 = r115219489 * r115219492;
        double r115219494 = cbrt(r115219493);
        return r115219494;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.9
Target12.9
Herbie13.2
\[\sqrt{\frac{1}{2} + \frac{\mathsf{copysign}\left(\frac{1}{2}, x\right)}{\mathsf{hypot}\left(1, \left(\frac{2 \cdot p}{x}\right)\right)}}\]

Derivation

  1. Initial program 12.9

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

  2. Simplified12.9

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

  4. Applied add-cbrt-cube12.9

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

  5. Simplified13.1

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

  7. Applied add-exp-log13.1

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

  9. Applied add-exp-log13.1

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

  11. Applied add-cube-cbrt13.2

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

  12. Final simplification13.2

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

Reproduce

herbie shell --seed 2019128 
(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))))))))