Given's Rotation SVD example, simplified

Average Error: 15.4 → 0.2

Time: 15.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.0023638476354078493:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, 1 + \sqrt[3]{{\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}, 1 \cdot 1\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{elif}\;x \le 0.001099911291066277:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.5, \frac{1}{\sqrt{1}}, 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot 1\right)}{\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)\right)\right)}\\ \end{array}\]

C

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

↓

double code(double x) {
	double temp;
	if ((x <= -0.0023638476354078493)) {
		temp = (fma(-0.5, (1.0 + cbrt(pow((1.0 / hypot(1.0, x)), 3.0))), (1.0 * 1.0)) / (1.0 + sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))))));
	} else {
		double temp_1;
		if ((x <= 0.0010999112910662773)) {
			temp_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 {
			temp_1 = (fma(-0.5, (1.0 + (1.0 / hypot(1.0, x))), (1.0 * 1.0)) / expm1(expm1(log1p(log1p((1.0 + sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))))))))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.0023638476354078493

   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 flip-- 1.0

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

   4. Simplified 0.1

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

   6. Applied add-cbrt-cube 0.1

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

   7. Applied add-cbrt-cube 0.1

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

   8. Applied cbrt-undiv 0.1

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

   9. Simplified 0.1

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

   if -0.0023638476354078493 < x < 0.0010999112910662773

   1. Initial program 30.1

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

   3. Applied flip-- 30.1

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

   4. Simplified 30.1

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

   5. Taylor expanded around 0 30.1

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

   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 0.0010999112910662773 < x

   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 flip-- 1.1

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

   4. Simplified 0.1

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

   6. Applied expm1-log1p-u 0.1

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot 1\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}}\]
   7. Using strategy rm

   8. Applied expm1-log1p-u 0.1

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot 1\right)}{\mathsf{expm1}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)\right)}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0023638476354078493:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, 1 + \sqrt[3]{{\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}, 1 \cdot 1\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{elif}\;x \le 0.001099911291066277:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.5, \frac{1}{\sqrt{1}}, 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot 1\right)}{\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))