Given's Rotation SVD example, simplified

Average Error: 15.7 → 0.2

Time: 19.4s

Precision: 64

Math

\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.00287753907949529894:\\ \;\;\;\;\frac{{1}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)} - \frac{{\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{2}}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\\ \mathbf{elif}\;x \le 0.00142444280501321705:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.5, \frac{1}{\sqrt{1}}, 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({1}^{\frac{3}{2}} + {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{4}}\right) \cdot \left({1}^{\frac{3}{2}} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{4}}\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\\ \end{array}\]

C

double code(double x) {
	return (1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x))))));
}

↓

double code(double x) {
	double VAR;
	if ((x <= -0.002877539079495299)) {
		VAR = ((pow(1.0, 3.0) / fma(1.0, 1.0, fma(0.5, (1.0 + (1.0 / hypot(1.0, x))), (1.0 * sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x))))))))) - (pow((0.5 * (1.0 + (1.0 / hypot(1.0, x)))), 1.5) / fma(1.0, 1.0, fma(0.5, (1.0 + (1.0 / hypot(1.0, x))), (1.0 * sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x))))))))));
	} else {
		double VAR_1;
		if ((x <= 0.001424442805013217)) {
			VAR_1 = (fma(0.25, (pow(x, 2.0) / pow(sqrt(1.0), 3.0)), (0.5 - fma(0.5, (1.0 / sqrt(1.0)), (0.1875 * (pow(x, 4.0) / pow(sqrt(1.0), 5.0)))))) / (1.0 + sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))))));
		} else {
			VAR_1 = (((pow(1.0, 1.5) + pow((0.5 * (1.0 + (1.0 / hypot(1.0, x)))), 0.75)) * (pow(1.0, 1.5) - pow((0.5 * (1.0 + (1.0 / hypot(1.0, x)))), 0.75))) / fma(1.0, 1.0, fma(0.5, (1.0 + (1.0 / hypot(1.0, x))), (1.0 * sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.002877539079495299

   1. Initial program 1.1

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
   2. Using strategy rm

   3. Applied flip3-- 1.6

      \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}}{1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}}\]

   4. Simplified 1.1

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

   6. Applied pow1/2 1.1

      \[\leadsto \frac{{1}^{3} - {\color{blue}{\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{1}{2}}\right)}}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\]

   7. Applied pow-pow 0.1

      \[\leadsto \frac{{1}^{3} - \color{blue}{{\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\left(\frac{1}{2} \cdot 3\right)}}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\]

   8. Simplified 0.1

      \[\leadsto \frac{{1}^{3} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\color{blue}{\frac{3}{2}}}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\]
   9. Using strategy rm

   10. Applied div-sub 0.1

       \[\leadsto \color{blue}{\frac{{1}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)} - \frac{{\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{2}}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}}\]

   if -0.002877539079495299 < x < 0.001424442805013217

   1. Initial program 30.4

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
   2. Using strategy rm

   3. Applied flip-- 30.4

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

   4. Simplified 30.4

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

   5. Taylor expanded around 0 30.4

      \[\leadsto \frac{\color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}} + 0.5\right) - \left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

   6. Simplified 0.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.5, \frac{1}{\sqrt{1}}, 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

   if 0.001424442805013217 < x

   1. Initial program 1.0

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
   2. Using strategy rm

   3. Applied flip3-- 1.6

      \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}}{1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}}\]

   4. Simplified 1.0

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

   6. Applied pow1/2 1.0

      \[\leadsto \frac{{1}^{3} - {\color{blue}{\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{1}{2}}\right)}}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\]

   7. Applied pow-pow 0.1

      \[\leadsto \frac{{1}^{3} - \color{blue}{{\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\left(\frac{1}{2} \cdot 3\right)}}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\]

   8. Simplified 0.1

      \[\leadsto \frac{{1}^{3} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\color{blue}{\frac{3}{2}}}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\]
   9. Using strategy rm

   10. Applied sqr-pow 0.1

       \[\leadsto \frac{{1}^{3} - \color{blue}{{\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\]

   11. Applied sqr-pow 0.1

       \[\leadsto \frac{\color{blue}{{1}^{\left(\frac{3}{2}\right)} \cdot {1}^{\left(\frac{3}{2}\right)}} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\]

   12. Applied difference-of-squares 0.1

       \[\leadsto \frac{\color{blue}{\left({1}^{\left(\frac{3}{2}\right)} + {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}\right) \cdot \left({1}^{\left(\frac{3}{2}\right)} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}\right)}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\]

   13. Simplified 0.1

       \[\leadsto \frac{\color{blue}{\left({1}^{\frac{3}{2}} + {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{4}}\right)} \cdot \left({1}^{\left(\frac{3}{2}\right)} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\]

   14. Simplified 0.1

       \[\leadsto \frac{\left({1}^{\frac{3}{2}} + {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{4}}\right) \cdot \color{blue}{\left({1}^{\frac{3}{2}} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{4}}\right)}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00287753907949529894:\\ \;\;\;\;\frac{{1}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)} - \frac{{\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{2}}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\\ \mathbf{elif}\;x \le 0.00142444280501321705:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.5, \frac{1}{\sqrt{1}}, 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({1}^{\frac{3}{2}} + {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{4}}\right) \cdot \left({1}^{\frac{3}{2}} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{4}}\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))