Average Error: 12.7 → 12.7
Time: 13.0s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\left(\sqrt{\frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}} + 1} \cdot \sqrt{\frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}} + 1}\right) \cdot 0.5}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\left(\sqrt{\frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}} + 1} \cdot \sqrt{\frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}} + 1}\right) \cdot 0.5}
double f(double p, double x) {
        double r239359 = 0.5;
        double r239360 = 1.0;
        double r239361 = x;
        double r239362 = 4.0;
        double r239363 = p;
        double r239364 = r239362 * r239363;
        double r239365 = r239364 * r239363;
        double r239366 = r239361 * r239361;
        double r239367 = r239365 + r239366;
        double r239368 = sqrt(r239367);
        double r239369 = r239361 / r239368;
        double r239370 = r239360 + r239369;
        double r239371 = r239359 * r239370;
        double r239372 = sqrt(r239371);
        return r239372;
}
↓
double f(double p, double x) {
        double r239373 = x;
        double r239374 = 4.0;
        double r239375 = p;
        double r239376 = r239374 * r239375;
        double r239377 = r239373 * r239373;
        double r239378 = fma(r239376, r239375, r239377);
        double r239379 = sqrt(r239378);
        double r239380 = r239373 / r239379;
        double r239381 = 1.0;
        double r239382 = r239380 + r239381;
        double r239383 = sqrt(r239382);
        double r239384 = r239383 * r239383;
        double r239385 = 0.5;
        double r239386 = r239384 * r239385;
        double r239387 = sqrt(r239386);
        return r239387;
}

Error

Bits error versus p

Bits error versus x

Target

Original12.7
Target12.7
Herbie12.7
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 12.7

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

  2. Simplified12.7

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

  4. Applied add-sqr-sqrt12.7

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

  5. Final simplification12.7

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(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))))))))