2cos (problem 3.3.5)

Average Error: 39.6 → 16.0

Time: 5.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.9500525747086803 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\sin \varepsilon \cdot \sin x\right) + \mathsf{fma}\left(\sin x, \left(-\sin \varepsilon\right) + \sin \varepsilon, -\cos x\right)\\ \mathbf{elif}\;\varepsilon \le 4.69434538371788253 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double VAR;
	if ((eps <= -2.9500525747086803e-33)) {
		VAR = (fma(cos(eps), cos(x), -(sin(eps) * sin(x))) + fma(sin(x), (-sin(eps) + sin(eps)), -cos(x)));
	} else {
		double VAR_1;
		if ((eps <= 4.6943453837178825e-06)) {
			VAR_1 = fma(0.041666666666666664, pow(eps, 4.0), -fma(x, eps, (0.5 * pow(eps, 2.0))));
		} else {
			VAR_1 = fma(cos(eps), cos(x), -fma(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 < -2.9500525747086803e-33

   1. Initial program 33.0

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

   3. Applied cos-sum 4.9

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

   5. Applied *-commutative 4.9

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

   6. Applied prod-diff 4.9

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

   7. Applied associate--l+ 4.9

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

   8. Simplified 4.9

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

   if -2.9500525747086803e-33 < eps < 4.6943453837178825e-06

   1. Initial program 48.6

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

   3. Applied cos-sum 48.3

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

   5. Applied *-commutative 48.3

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

   6. Applied prod-diff 48.3

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

   7. Applied associate--l+ 48.3

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

   8. Simplified 48.3

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

   9. Taylor expanded around 0 30.9

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

   10. Simplified 30.9

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

   if 4.6943453837178825e-06 < eps

   1. Initial program 30.4

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

   3. Applied cos-sum 0.9

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

   4. Taylor expanded around inf 1.0

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

   5. Simplified 0.9

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

4. Final simplification 16.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.9500525747086803 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\sin \varepsilon \cdot \sin x\right) + \mathsf{fma}\left(\sin x, \left(-\sin \varepsilon\right) + \sin \varepsilon, -\cos x\right)\\ \mathbf{elif}\;\varepsilon \le 4.69434538371788253 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))