Average Error: 13.0 → 13.0
Time: 16.5s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}
double f(double p, double x) {
        double r267474 = 0.5;
        double r267475 = 1.0;
        double r267476 = x;
        double r267477 = 4.0;
        double r267478 = p;
        double r267479 = r267477 * r267478;
        double r267480 = r267479 * r267478;
        double r267481 = r267476 * r267476;
        double r267482 = r267480 + r267481;
        double r267483 = sqrt(r267482);
        double r267484 = r267476 / r267483;
        double r267485 = r267475 + r267484;
        double r267486 = r267474 * r267485;
        double r267487 = sqrt(r267486);
        return r267487;
}
↓
double f(double p, double x) {
        double r267488 = 0.5;
        double r267489 = 1.0;
        double r267490 = x;
        double r267491 = 4.0;
        double r267492 = p;
        double r267493 = r267491 * r267492;
        double r267494 = r267493 * r267492;
        double r267495 = r267490 * r267490;
        double r267496 = r267494 + r267495;
        double r267497 = sqrt(r267496);
        double r267498 = r267490 / r267497;
        double r267499 = r267489 + r267498;
        double r267500 = exp(r267499);
        double r267501 = log(r267500);
        double r267502 = r267488 * r267501;
        double r267503 = sqrt(r267502);
        return r267503;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.0
Target13.0
Herbie13.0
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.0

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

  3. Applied add-log-exp13.0

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

  4. Applied add-log-exp13.0

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

  5. Applied sum-log13.0

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

  6. Simplified13.0

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

  7. Final simplification13.0

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

Reproduce

herbie shell --seed 2019323 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1 (/ (* 2 p) x)))))

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