2cos (problem 3.3.5)

Average Error: 39.8 → 0.5

Time: 15.6s

Precision: binary64

Cost: 39177

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0055 \lor \neg \left(\varepsilon \leq 0.0055\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0055) (not (<= eps 0.0055)))
   (- (fma (cos x) (cos eps) (* (sin eps) (- (sin x)))) (cos x))
   (-
    (* (cos x) (+ (* (* eps eps) -0.5) (* (pow eps 4.0) 0.041666666666666664)))
    (* (sin 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 <= -0.0055) || !(eps <= 0.0055)) {
		tmp = fma(cos(x), cos(eps), (sin(eps) * -sin(x))) - cos(x);
	} else {
		tmp = (cos(x) * (((eps * eps) * -0.5) + (pow(eps, 4.0) * 0.041666666666666664))) - (sin(eps) * sin(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end

↓

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0055) || !(eps <= 0.0055))
		tmp = Float64(fma(cos(x), cos(eps), Float64(sin(eps) * Float64(-sin(x)))) - cos(x));
	else
		tmp = Float64(Float64(cos(x) * Float64(Float64(Float64(eps * eps) * -0.5) + Float64((eps ^ 4.0) * 0.041666666666666664))) - Float64(sin(eps) * sin(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := If[Or[LessEqual[eps, -0.0055], N[Not[LessEqual[eps, 0.0055]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0054999999999999997 or 0.0054999999999999997 < eps

   1. Initial program 30.4

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

   2. Applied egg-rr 0.8

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

   if -0.0054999999999999997 < eps < 0.0054999999999999997

   1. Initial program 49.1

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

   2. Applied egg-rr 49.4

      \[\leadsto \color{blue}{\sqrt{{\cos \left(x + \varepsilon\right)}^{2}}} - \cos x \]

   3. Simplified 49.4

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

      [Start] 49.4	\[ \sqrt{{\cos \left(x + \varepsilon\right)}^{2}} - \cos x \]
      unpow2 [=>] 49.4	\[ \sqrt{\color{blue}{\cos \left(x + \varepsilon\right) \cdot \cos \left(x + \varepsilon\right)}} - \cos x \]
      rem-sqrt-square [=>] 49.4	\[ \color{blue}{\left|\cos \left(x + \varepsilon\right)\right|} - \cos x \]
      +-commutative [<=] 49.4	\[ \left|\cos \color{blue}{\left(\varepsilon + x\right)}\right| - \cos x \]

   4. Applied egg-rr 48.4

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

   5. Simplified 11.8

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

      [Start] 48.4	\[ \cos x \cdot \cos \varepsilon + \left(\sin x \cdot \left(-\sin \varepsilon\right) + \left(-\cos x\right)\right) \]
      +-commutative [=>] 48.4	\[ \cos x \cdot \cos \varepsilon + \color{blue}{\left(\left(-\cos x\right) + \sin x \cdot \left(-\sin \varepsilon\right)\right)} \]
      distribute-rgt-neg-out [=>] 48.4	\[ \cos x \cdot \cos \varepsilon + \left(\left(-\cos x\right) + \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)}\right) \]
      *-commutative [<=] 48.4	\[ \cos x \cdot \cos \varepsilon + \left(\left(-\cos x\right) + \left(-\color{blue}{\sin \varepsilon \cdot \sin x}\right)\right) \]
      unsub-neg [=>] 48.4	\[ \cos x \cdot \cos \varepsilon + \color{blue}{\left(\left(-\cos x\right) - \sin \varepsilon \cdot \sin x\right)} \]
      associate-+r- [=>] 11.8	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right) - \sin \varepsilon \cdot \sin x} \]
      *-commutative [=>] 11.8	\[ \left(\color{blue}{\cos \varepsilon \cdot \cos x} + \left(-\cos x\right)\right) - \sin \varepsilon \cdot \sin x \]
      neg-mul-1 [=>] 11.8	\[ \left(\cos \varepsilon \cdot \cos x + \color{blue}{-1 \cdot \cos x}\right) - \sin \varepsilon \cdot \sin x \]
      distribute-rgt-out [=>] 11.8	\[ \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)} - \sin \varepsilon \cdot \sin x \]
      *-commutative [=>] 11.8	\[ \cos x \cdot \left(\cos \varepsilon + -1\right) - \color{blue}{\sin x \cdot \sin \varepsilon} \]

   6. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\frac{{\sin \varepsilon}^{2} \cdot \cos x}{-1 - \cos \varepsilon}} - \sin x \cdot \sin \varepsilon \]

   7. Taylor expanded in eps around 0 0.1

      \[\leadsto \color{blue}{\left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)} - \sin x \cdot \sin \varepsilon \]

   8. Simplified 0.1

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

      [Start] 0.1	\[ \left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) - \sin x \cdot \sin \varepsilon \]
      +-commutative [=>] 0.1	\[ \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left({\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right)\right)} - \sin x \cdot \sin \varepsilon \]
      mul-1-neg [=>] 0.1	\[ \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \color{blue}{\left(-{\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right)}\right) - \sin x \cdot \sin \varepsilon \]
      unsub-neg [=>] 0.1	\[ \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) - {\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right)} - \sin x \cdot \sin \varepsilon \]
      associate-*r* [=>] 0.1	\[ \left(\color{blue}{\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x} - {\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right) - \sin x \cdot \sin \varepsilon \]
      *-commutative [=>] 0.1	\[ \left(\color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)} - {\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right) - \sin x \cdot \sin \varepsilon \]
      *-commutative [=>] 0.1	\[ \left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \color{blue}{\left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right) \cdot {\varepsilon}^{4}}\right) - \sin x \cdot \sin \varepsilon \]
      distribute-rgt-out-- [=>] 0.1	\[ \left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \color{blue}{\left(\cos x \cdot \left(-0.16666666666666666 - -0.125\right)\right)} \cdot {\varepsilon}^{4}\right) - \sin x \cdot \sin \varepsilon \]
      metadata-eval [=>] 0.1	\[ \left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \left(\cos x \cdot \color{blue}{-0.041666666666666664}\right) \cdot {\varepsilon}^{4}\right) - \sin x \cdot \sin \varepsilon \]
      associate-*l* [=>] 0.1	\[ \left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \color{blue}{\cos x \cdot \left(-0.041666666666666664 \cdot {\varepsilon}^{4}\right)}\right) - \sin x \cdot \sin \varepsilon \]
      distribute-lft-out-- [=>] 0.1	\[ \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} - -0.041666666666666664 \cdot {\varepsilon}^{4}\right)} - \sin x \cdot \sin \varepsilon \]
      *-commutative [=>] 0.1	\[ \cos x \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot -0.5} - -0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin x \cdot \sin \varepsilon \]
      unpow2 [=>] 0.1	\[ \cos x \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot -0.5 - -0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin x \cdot \sin \varepsilon \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0055 \lor \neg \left(\varepsilon \leq 0.0055\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	39168

\[\frac{{\sin \varepsilon}^{2} \cdot \cos x}{-1 - \cos \varepsilon} - \sin \varepsilon \cdot \sin x \]

Alternative 2
Error	0.5
Cost	32644

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0045:\\ \;\;\;\;\mathsf{fma}\left(t_0, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0044:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot t_0 - t_1\\ \end{array} \]

Alternative 3
Error	0.5
Cost	26889

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0045 \lor \neg \left(\varepsilon \leq 0.0044\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right) - t_0\\ \end{array} \]

Alternative 4
Error	0.5
Cost	26441

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00014 \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) - t_0\\ \end{array} \]

Alternative 5
Error	14.9
Cost	13768

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.013:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00105:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 6
Error	15.1
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0064:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0002:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 7
Error	15.1
Cost	13632

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

Alternative 8
Error	15.1
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0085:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.000185:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 9
Error	15.1
Cost	13124

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0028:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0018:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]

Alternative 10
Error	15.3
Cost	7241

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0012 \lor \neg \left(\varepsilon \leq 0.008\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 11
Error	32.5
Cost	7120

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{if}\;\varepsilon \leq -0.00044:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -3.6 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 1.52 \cdot 10^{-162}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00015:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	21.1
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.5 \lor \neg \left(\varepsilon \leq 1.7 \cdot 10^{-5}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 13
Error	48.5
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{-104}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+50}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \cos x\\ \end{array} \]

Alternative 14
Error	48.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1 \cdot 10^{-121} \lor \neg \left(\varepsilon \leq 1.12 \cdot 10^{-159}\right):\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \end{array} \]

Alternative 15
Error	52.2
Cost	256

\[\varepsilon \cdot \left(-x\right) \]

Reproduce

herbie shell --seed 2023028 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))