2cos (problem 3.3.5)

Average Error: 39.7 → 0.5

Time: 1.2min

Precision: binary64

Cost: 33218

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.003293150329827447:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0024034558787377884:\\ \;\;\;\;\sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right) + \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

FPCore

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.003293150329827447)
   (- (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x))
   (if (<= eps 0.0024034558787377884)
     (+
      (* (sin x) (- (* (pow eps 3.0) 0.16666666666666666) eps))
      (*
       (cos x)
       (+ (* (* eps eps) -0.5) (* 0.041666666666666664 (pow eps 4.0)))))
     (- (* (cos x) (cos eps)) (+ (cos x) (* (sin x) (sin eps)))))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.003293150329827447) {
		tmp = ((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x);
	} else if (eps <= 0.0024034558787377884) {
		tmp = (sin(x) * ((pow(eps, 3.0) * 0.16666666666666666) - eps)) + (cos(x) * (((eps * eps) * -0.5) + (0.041666666666666664 * pow(eps, 4.0))));
	} else {
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Alternatives

Alternative 1
Error	0.5
Cost	32904

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.002533648414460582 \lor \neg \left(\varepsilon \leq 0.003030661213187838\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right) + \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)\\ \end{array}\]

Alternative 2
Error	15.1
Cost	13632

\[\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Alternative 3
Error	15.2
Cost	13632

\[-2 \cdot \left(\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Alternative 4
Error	15.1
Cost	13504

\[\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(x + \varepsilon \cdot 0.5\right)\]

Alternative 5
Error	15.0
Cost	14018

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1115726505891959 \cdot 10^{-07}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{elif}\;\varepsilon \leq 0.008384642448738832:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.5\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array}\]

Alternative 6
Error	21.1
Cost	13512

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.816593790645222 \cdot 10^{-48} \lor \neg \left(\varepsilon \leq 4.5984648178483713 \cdot 10^{-51}\right):\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array}\]

Alternative 7
Error	20.8
Cost	13955

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -112837442.24806044:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq -1.2971198777042918 \cdot 10^{-47}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;\varepsilon \leq 3.7642385778515147 \cdot 10^{-07}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array}\]

Alternative 8
Error	21.0
Cost	7682

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.005528182842280636:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq -8.046750886431058 \cdot 10^{-48}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + 0.041666666666666664 \cdot {\varepsilon}^{4}\\ \mathbf{elif}\;\varepsilon \leq 4.7601751627301173 \cdot 10^{-07}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array}\]

Alternative 9
Error	21.0
Cost	7619

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00014227753523047773:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq -8.276907982216895 \cdot 10^{-48}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;\varepsilon \leq 2.921523006031159 \cdot 10^{-07}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array}\]

Alternative 10
Error	34.0
Cost	6920

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00014227753523047773 \lor \neg \left(\varepsilon \leq 0.0001400930034384832\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array}\]

Alternative 11
Error	46.4
Cost	962

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.3869469113144555:\\ \;\;\;\;-1\\ \mathbf{elif}\;\varepsilon \leq 1.3738129382540136:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Alternative 12
Error	51.8
Cost	706

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.143728817397363 \cdot 10^{-77}:\\ \;\;\;\;-1\\ \mathbf{elif}\;\varepsilon \leq 1.462416093196927 \cdot 10^{-105}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Alternative 13
Error	55.7
Cost	64

\[0\]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00329315032982744694

   1. Initial program 30.8

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

   3. Applied cos-sum_binary64_212 0.8

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

   4. Simplified 0.8

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

   if -0.00329315032982744694 < eps < 0.00240345587873778835

   1. Initial program 49.0

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

   3. Applied diff-cos_binary64_229 37.4

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

   4. Simplified 0.5

      \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]
   5. Using strategy rm

   6. Applied associate-*r*_binary64_18 0.5

      \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)}\]

   7. Simplified 0.5

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)} \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\]
   8. Using strategy rm

   9. Applied pow1_binary64_139 0.5

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

   10. Taylor expanded around 0 0.1

       \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left(\cos x \cdot {\varepsilon}^{4}\right) + 0.16666666666666666 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right)\right) - \left(\sin x \cdot \varepsilon + 0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\right)}\]

   11. Simplified 0.1

       \[\leadsto \color{blue}{\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)}\]

   12. Simplified 0.1

       \[\leadsto \color{blue}{\sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right) + \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)}\]

   if 0.00240345587873778835 < eps

   1. Initial program 30.1

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

   3. Applied cos-sum_binary64_212 0.8

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

   4. Applied associate--l-_binary64_16 0.9

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

   5. Simplified 0.9

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

   6. Simplified 0.9

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.003293150329827447:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0024034558787377884:\\ \;\;\;\;\sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right) + \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021040 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))