Average Error: 13.5 → 13.8
Time: 15.0s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \left(1 + x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
double f(double p, double x) {
        double r271434 = 0.5;
        double r271435 = 1.0;
        double r271436 = x;
        double r271437 = 4.0;
        double r271438 = p;
        double r271439 = r271437 * r271438;
        double r271440 = r271439 * r271438;
        double r271441 = r271436 * r271436;
        double r271442 = r271440 + r271441;
        double r271443 = sqrt(r271442);
        double r271444 = r271436 / r271443;
        double r271445 = r271435 + r271444;
        double r271446 = r271434 * r271445;
        double r271447 = sqrt(r271446);
        return r271447;
}
↓
double f(double p, double x) {
        double r271448 = 0.5;
        double r271449 = 1.0;
        double r271450 = x;
        double r271451 = 1.0;
        double r271452 = 4.0;
        double r271453 = p;
        double r271454 = r271452 * r271453;
        double r271455 = r271454 * r271453;
        double r271456 = r271450 * r271450;
        double r271457 = r271455 + r271456;
        double r271458 = sqrt(r271457);
        double r271459 = r271451 / r271458;
        double r271460 = r271450 * r271459;
        double r271461 = r271449 + r271460;
        double r271462 = r271448 * r271461;
        double r271463 = sqrt(r271462);
        return r271463;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.5
Target13.5
Herbie13.8
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.5

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

  3. Applied div-inv13.8

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

  4. Final simplification13.8

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

Reproduce

herbie shell --seed 2019305 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (< 1.00000000000000001e-150 (fabs x) 9.99999999999999981e149)

  :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))))))))