2cos (problem 3.3.5)

Average Error: 40.0 → 15.7

Time: 6.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.5330496627903165 \cdot 10^{-7} \lor \neg \left(\varepsilon \le 4.3765411525387599 \cdot 10^{-4}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right) + \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right) \cdot \left(\left(-1\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) + \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right) \cdot \left(\left(-1\right) + 1\right)\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double temp;
	if (((eps <= -1.5330496627903165e-07) || !(eps <= 0.000437654115253876))) {
		temp = (fma(cos(eps), cos(x), -fma(sin(x), sin(eps), cos(x))) + (fma(sin(x), sin(eps), cos(x)) * (-1.0 + 1.0)));
	} else {
		temp = ((eps * ((pow(eps, 3.0) * 0.041666666666666664) - fma(0.5, eps, x))) + (fma(sin(x), sin(eps), cos(x)) * (-1.0 + 1.0)));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if eps < -1.5330496627903165e-07 or 0.000437654115253876 < eps

   1. Initial program 30.5

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

   3. Applied cos-sum 1.1

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

   4. Applied associate--l- 1.1

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

   5. Simplified 1.1

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

   7. Applied add-cube-cbrt 1.5

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

   8. Applied prod-diff 1.5

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

   9. Simplified 1.0

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

   10. Simplified 1.0

       \[\leadsto \mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right) + \color{blue}{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right) \cdot \left(\left(-1\right) + 1\right)}\]

   if -1.5330496627903165e-07 < eps < 0.000437654115253876

   1. Initial program 49.8

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

   3. Applied cos-sum 49.2

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

   4. Applied associate--l- 49.2

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

   5. Simplified 49.2

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

   7. Applied add-cube-cbrt 49.8

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

   8. Applied prod-diff 49.8

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

   9. Simplified 49.8

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

   10. Simplified 49.2

       \[\leadsto \mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right) + \color{blue}{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right) \cdot \left(\left(-1\right) + 1\right)}\]

   11. Taylor expanded around 0 30.8

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

   12. Simplified 30.8

       \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)} + \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right) \cdot \left(\left(-1\right) + 1\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 15.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.5330496627903165 \cdot 10^{-7} \lor \neg \left(\varepsilon \le 4.3765411525387599 \cdot 10^{-4}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right) + \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right) \cdot \left(\left(-1\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) + \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right) \cdot \left(\left(-1\right) + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))