2cos (problem 3.3.5)

Average Error: 39.5 → 15.6

Time: 7.0s

Precision: 64

Math

\[\cos \left(x + \varepsilon\right) - \cos x\]

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.0852323357979476 \cdot 10^{-10}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right)\right)}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon, \mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right) + \cos x, \cos x \cdot \cos x\right)}\\ \mathbf{elif}\;\varepsilon \le 6.27540161566563378 \cdot 10^{-24}:\\ \;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

C

double code(double x, double eps) {
	return (cos((x + eps)) - cos(x));
}

↓

double code(double x, double eps) {
	double VAR;
	if ((eps <= -3.0852323357979476e-10)) {
		VAR = ((pow(fma(cos(x), cos(eps), -(sin(eps) * sin(x))), 3.0) - pow(cos(x), 3.0)) / fma(((cos(eps) * cos(x)) - (sin(x) * sin(eps))), (fma(cos(x), cos(eps), -(sin(eps) * sin(x))) + cos(x)), (cos(x) * cos(x))));
	} else {
		double VAR_1;
		if ((eps <= 6.275401615665634e-24)) {
			VAR_1 = (eps * ((pow(eps, 3.0) * 0.041666666666666664) - fma(0.5, eps, x)));
		} else {
			VAR_1 = ((log1p(expm1((cos(x) * cos(eps)))) - (sin(x) * sin(eps))) - cos(x));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if eps < -3.0852323357979476e-10

   1. Initial program 31.1

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   2. Using strategy rm

   3. Applied cos-sum 1.4

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   4. Using strategy rm

   5. Applied fma-neg 1.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin x \cdot \sin \varepsilon\right)} - \cos x\]

   6. Simplified 1.4

      \[\leadsto \mathsf{fma}\left(\cos x, \cos \varepsilon, \color{blue}{-\sin \varepsilon \cdot \sin x}\right) - \cos x\]
   7. Using strategy rm

   8. Applied flip3-- 1.5

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right)\right)}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right) \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x + \mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right) \cdot \cos x\right)}}\]

   9. Simplified 1.5

      \[\leadsto \frac{{\left(\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right)\right)}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon, \mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right) + \cos x, \cos x \cdot \cos x\right)}}\]

   if -3.0852323357979476e-10 < eps < 6.275401615665634e-24

   1. Initial program 48.7

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   2. Using strategy rm

   3. Applied cos-sum 48.5

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   4. Using strategy rm

   5. Applied add-log-exp 48.8

      \[\leadsto \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \color{blue}{\log \left(e^{\cos x}\right)}\]

   6. Applied add-log-exp 48.8

      \[\leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\log \left(e^{\sin x \cdot \sin \varepsilon}\right)}\right) - \log \left(e^{\cos x}\right)\]

   7. Applied add-log-exp 48.5

      \[\leadsto \left(\color{blue}{\log \left(e^{\cos x \cdot \cos \varepsilon}\right)} - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right) - \log \left(e^{\cos x}\right)\]

   8. Applied diff-log 48.5

      \[\leadsto \color{blue}{\log \left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}\right)} - \log \left(e^{\cos x}\right)\]

   9. Applied diff-log 48.5

      \[\leadsto \color{blue}{\log \left(\frac{\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}}{e^{\cos x}}\right)}\]

   10. Simplified 48.5

       \[\leadsto \log \color{blue}{\left(e^{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)}\right)}\]

   11. Taylor expanded around 0 30.8

       \[\leadsto \color{blue}{\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]

   12. Simplified 30.8

       \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)}\]

   if 6.275401615665634e-24 < eps

   1. Initial program 32.5

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   2. Using strategy rm

   3. Applied cos-sum 4.2

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   4. Using strategy rm

   5. Applied log1p-expm1-u 4.3

      \[\leadsto \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right)} - \sin x \cdot \sin \varepsilon\right) - \cos x\]
3. Recombined 3 regimes into one program.

4. Final simplification 15.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.0852323357979476 \cdot 10^{-10}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right)\right)}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon, \mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right) + \cos x, \cos x \cdot \cos x\right)}\\ \mathbf{elif}\;\varepsilon \le 6.27540161566563378 \cdot 10^{-24}:\\ \;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))