2cos (problem 3.3.5)

Average Error: 39.6 → 0.5

Time: 1.1min

Precision: binary64

Cost: 33992

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0024335825693093255 \lor \neg \left(\varepsilon \leq 0.0036073965070522497\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \left(\varepsilon \cdot 0.5 + {\varepsilon}^{3} \cdot -0.020833333333333332\right) - \left(0.125 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x\right)\right)\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0024335825693093255) (not (<= eps 0.0036073965070522497)))
   (- (- (* (cos eps) (cos x)) (* (sin eps) (sin x))) (cos x))
   (*
    -2.0
    (*
     (sin (/ eps 2.0))
     (-
      (* (cos x) (+ (* eps 0.5) (* (pow eps 3.0) -0.020833333333333332)))
      (- (* 0.125 (* (sin x) (* eps eps))) (sin x)))))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0024335825693093255) || !(eps <= 0.0036073965070522497)) {
		tmp = ((cos(eps) * cos(x)) - (sin(eps) * sin(x))) - cos(x);
	} else {
		tmp = -2.0 * (sin(eps / 2.0) * ((cos(x) * ((eps * 0.5) + (pow(eps, 3.0) * -0.020833333333333332))) - ((0.125 * (sin(x) * (eps * eps))) - sin(x))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Alternatives

Alternative 1
Error	0.5
Cost	32904

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0024335825693093255 \lor \neg \left(\varepsilon \leq 0.003493861581670858\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right) + \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right)\\ \end{array}\]

Alternative 2
Error	15.0
Cost	13632

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

Alternative 3
Error	15.0
Cost	13504

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

Alternative 4
Error	15.0
Cost	14018

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -408962436006.8145:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0007460669567238816:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.5\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array}\]

Alternative 5
Error	18.9
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -5.530951086961855 \cdot 10^{-09} \lor \neg \left(x \leq 0.00039483389046432855\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array}\]

Alternative 6
Error	20.1
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -1.76112650866227 \cdot 10^{-06} \lor \neg \left(x \leq 0.00039483389046432855\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array}\]

Alternative 7
Error	21.8
Cost	14276

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -408962436006.8145:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq -1.5790679630181446 \cdot 10^{-33}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\\ \mathbf{elif}\;\varepsilon \leq 9.677252196218986 \cdot 10^{-136}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.1473734955250018 \cdot 10^{-07}:\\ \;\;\;\;\varepsilon \cdot \left(0.16666666666666666 \cdot {x}^{3} + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)\right) - \varepsilon \cdot \left(x + \varepsilon \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array}\]

Alternative 8
Error	22.2
Cost	9092

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -408962436006.8145:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq -1.4992305999204856 \cdot 10^{-33}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\\ \mathbf{elif}\;\varepsilon \leq 5.630040587748545 \cdot 10^{-136}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.1473734955250018 \cdot 10^{-07}:\\ \;\;\;\;\varepsilon \cdot \left(0.16666666666666666 \cdot {x}^{3} + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)\right) - \varepsilon \cdot \left(x + \varepsilon \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array}\]

Alternative 9
Error	21.3
Cost	7682

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -408962436006.8145:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq -1.8808535976661016 \cdot 10^{-34}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\\ \mathbf{elif}\;\varepsilon \leq 0.00010016414844791195:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array}\]

Alternative 10
Error	21.0
Cost	7619

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0001756687504501214:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq -3.917817851588561 \cdot 10^{-38}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 9.302691634614032 \cdot 10^{-07}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array}\]

Alternative 11
Error	33.9
Cost	6920

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0001756687504501214 \lor \neg \left(\varepsilon \leq 2.1473734955250018 \cdot 10^{-07}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array}\]

Alternative 12
Error	46.5
Cost	962

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.4215077523960264:\\ \;\;\;\;-1\\ \mathbf{elif}\;\varepsilon \leq 1.5083393991063354:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Alternative 13
Error	52.0
Cost	706

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.0077704307918553 \cdot 10^{-87}:\\ \;\;\;\;-1\\ \mathbf{elif}\;\varepsilon \leq 5.0249605220960307 \cdot 10^{-88}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Alternative 14
Error	58.0
Cost	64

\[-1\]

Derivation

1. Split input into 2 regimes

2. if eps < -0.002433582569309325 or 0.0036073965070522497 < eps

   1. Initial program 30.1

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

   3. Applied cos-sum_binary64_553 0.8

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

   4. Simplified 0.8

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

   if -0.002433582569309325 < eps < 0.0036073965070522497

   1. Initial program 49.4

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

   3. Applied diff-cos_binary64_570 37.3

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

   4. Simplified 0.6

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

   5. Taylor expanded around 0 0.2

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\left(\sin x + 0.5 \cdot \left(\cos x \cdot \varepsilon\right)\right) - \left(0.125 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + 0.020833333333333332 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right)\right)\right)}\right)\]

   6. Simplified 0.2

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\cos x \cdot \left(\varepsilon \cdot 0.5 + {\varepsilon}^{3} \cdot -0.020833333333333332\right) - \left(0.125 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x\right)\right)}\right)\]

   7. Simplified 0.2

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \left(0.5 \cdot \varepsilon + {\varepsilon}^{3} \cdot -0.020833333333333332\right) - \left(0.125 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x\right)\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0024335825693093255 \lor \neg \left(\varepsilon \leq 0.0036073965070522497\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \left(\varepsilon \cdot 0.5 + {\varepsilon}^{3} \cdot -0.020833333333333332\right) - \left(0.125 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021014 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))