2cos (problem 3.3.5)

Average Error: 39.5 → 15.7

Time: 6.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.9500988603657079 \cdot 10^{-11}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \log \left(e^{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)}\right)\\ \mathbf{elif}\;\varepsilon \le 4.8007364015852064 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{1}{6} \cdot {x}^{3} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double VAR;
	if ((eps <= -2.950098860365708e-11)) {
		VAR = ((cos(x) * cos(eps)) - log(exp(fma(sin(x), sin(eps), cos(x)))));
	} else {
		double VAR_1;
		if ((eps <= 4.8007364015852064e-11)) {
			VAR_1 = (eps * ((0.16666666666666666 * pow(x, 3.0)) - fma(0.5, eps, x)));
		} else {
			VAR_1 = fma(cos(x), cos(eps), -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 < -2.950098860365708e-11

   1. Initial program 30.6

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

   3. Applied cos-sum 1.5

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

   4. Applied associate--l- 1.6

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

   5. Simplified 1.6

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

   7. Applied add-log-exp 1.6

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

   if -2.950098860365708e-11 < eps < 4.8007364015852064e-11

   1. Initial program 49.6

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

   3. Applied cos-sum 49.4

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

   4. Applied associate--l- 49.4

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

   5. Simplified 49.4

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

   7. Applied fma-neg 49.4

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

   8. Taylor expanded around 0 31.1

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({x}^{3} \cdot \varepsilon\right) - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]

   9. Simplified 31.1

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{1}{6} \cdot {x}^{3} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)}\]

   if 4.8007364015852064e-11 < eps

   1. Initial program 29.9

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

   3. Applied cos-sum 1.5

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

   4. Applied associate--l- 1.5

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

   5. Simplified 1.5

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

   7. Applied fma-neg 1.5

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

4. Final simplification 15.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.9500988603657079 \cdot 10^{-11}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \log \left(e^{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)}\right)\\ \mathbf{elif}\;\varepsilon \le 4.8007364015852064 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{1}{6} \cdot {x}^{3} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))