2cos (problem 3.3.5)

Percentage Accurate: 38.4% → 99.2%

Time: 16.3s

Precision: binary64

Cost: 40136

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00325:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0014:\\ \;\;\;\;\left(-2 \cdot \left(\left(\sin x \cdot {\varepsilon}^{3}\right) \cdot -0.08333333333333333\right) - \varepsilon \cdot \sin x\right) + \cos x \cdot \mathsf{fma}\left({\varepsilon}^{4}, 0.041666666666666664, \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + -1\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00325)
   (fma (cos x) (cos eps) (- (- (cos x)) (* (sin x) (sin eps))))
   (if (<= eps 0.0014)
     (+
      (-
       (* -2.0 (* (* (sin x) (pow eps 3.0)) -0.08333333333333333))
       (* eps (sin x)))
      (*
       (cos x)
       (fma (pow eps 4.0) 0.041666666666666664 (* eps (* eps -0.5)))))
     (fma (sin x) (- (sin eps)) (* (cos x) (+ (cos eps) -1.0))))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00325) {
		tmp = fma(cos(x), cos(eps), (-cos(x) - (sin(x) * sin(eps))));
	} else if (eps <= 0.0014) {
		tmp = ((-2.0 * ((sin(x) * pow(eps, 3.0)) * -0.08333333333333333)) - (eps * sin(x))) + (cos(x) * fma(pow(eps, 4.0), 0.041666666666666664, (eps * (eps * -0.5))));
	} else {
		tmp = fma(sin(x), -sin(eps), (cos(x) * (cos(eps) + -1.0)));
	}
	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 <= -0.00325)
		tmp = fma(cos(x), cos(eps), Float64(Float64(-cos(x)) - Float64(sin(x) * sin(eps))));
	elseif (eps <= 0.0014)
		tmp = Float64(Float64(Float64(-2.0 * Float64(Float64(sin(x) * (eps ^ 3.0)) * -0.08333333333333333)) - Float64(eps * sin(x))) + Float64(cos(x) * fma((eps ^ 4.0), 0.041666666666666664, Float64(eps * Float64(eps * -0.5)))));
	else
		tmp = fma(sin(x), Float64(-sin(eps)), Float64(cos(x) * Float64(cos(eps) + -1.0)));
	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[LessEqual[eps, -0.00325], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[((-N[Cos[x], $MachinePrecision]) - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0014], N[(N[(N[(-2.0 * N[(N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Power[eps, 4.0], $MachinePrecision] * 0.041666666666666664 + N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00324999999999999985

   1. Initial program 58.5%

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

   2. Applied egg-rr 98.8%

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

      [Start] 58.5%	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      cos-sum [=>] 98.7%	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]
      associate--l- [=>] 98.7%	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \]
      fma-neg [=>] 98.8%	\[ \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \]
      fma-def [=>] 98.8%	\[ \mathsf{fma}\left(\cos x, \cos \varepsilon, -\color{blue}{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)}\right) \]

   3. Applied egg-rr 98.8%

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

      [Start] 98.8%	\[ \mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right) \]
      fma-udef [=>] 98.8%	\[ \mathsf{fma}\left(\cos x, \cos \varepsilon, -\color{blue}{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}\right) \]

   if -0.00324999999999999985 < eps < 0.00139999999999999999

   1. Initial program 19.4%

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

   2. Applied egg-rr 44.1%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
      Step-by-step derivation

      [Start] 19.4%	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      diff-cos [=>] 44.1%	\[ \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]
      div-inv [=>] 44.1%	\[ -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
      metadata-eval [=>] 44.1%	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
      div-inv [=>] 44.1%	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]
      +-commutative [=>] 44.1%	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]
      metadata-eval [=>] 44.1%	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right)} \]
      Step-by-step derivation

      [Start] 44.1%	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
      *-commutative [=>] 44.1%	\[ -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
      +-commutative [<=] 44.1%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
      associate--l+ [=>] 99.2%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
      +-inverses [=>] 99.2%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \color{blue}{0}\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
      distribute-lft-in [=>] 99.2%	\[ -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \varepsilon + 0.5 \cdot 0\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
      metadata-eval [=>] 99.2%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + \color{blue}{0}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
      *-commutative [=>] 99.2%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]
      +-commutative [<=] 99.2%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \color{blue}{\left(\varepsilon + x\right)}\right)\right)\right) \]

   4. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + -2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right)\right)\right)} \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\left(-2 \cdot \left(\left(\sin x \cdot {\varepsilon}^{3}\right) \cdot -0.08333333333333333\right) - \varepsilon \cdot \sin x\right) + \cos x \cdot \mathsf{fma}\left({\varepsilon}^{4}, 0.041666666666666664, \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right)} \]
      Step-by-step derivation

      [Start] 99.8%	\[ 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + -2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right)\right)\right) \]
      associate-+r+ [=>] 99.8%	\[ \color{blue}{\left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + -2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right)\right)} \]
      +-commutative [=>] 99.8%	\[ \color{blue}{\left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + -2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right)\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)} \]

   if 0.00139999999999999999 < eps

   1. Initial program 54.9%

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

   2. Applied egg-rr 98.8%

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

      [Start] 54.9%	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      cos-sum [=>] 98.8%	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]
      sub-neg [=>] 98.8%	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]

   3. Simplified 98.9%

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

      [Start] 98.8%	\[ \left(\cos x \cdot \cos \varepsilon + \left(-\sin x \cdot \sin \varepsilon\right)\right) - \cos x \]
      +-commutative [=>] 98.8%	\[ \color{blue}{\left(\left(-\sin x \cdot \sin \varepsilon\right) + \cos x \cdot \cos \varepsilon\right)} - \cos x \]
      distribute-lft-neg-in [=>] 98.8%	\[ \left(\color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon} + \cos x \cdot \cos \varepsilon\right) - \cos x \]
      *-commutative [=>] 98.8%	\[ \left(\color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} + \cos x \cdot \cos \varepsilon\right) - \cos x \]
      fma-def [=>] 98.9%	\[ \color{blue}{\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \cos \varepsilon\right)} - \cos x \]
      *-commutative [=>] 98.9%	\[ \mathsf{fma}\left(\sin \varepsilon, -\sin x, \color{blue}{\cos \varepsilon \cdot \cos x}\right) - \cos x \]

   4. Taylor expanded in eps around inf 98.8%

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

   5. Simplified 99.1%

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

      [Start] 98.8%	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos x \cdot \cos \varepsilon\right) - \cos x \]
      neg-mul-1 [<=] 98.8%	\[ \left(\color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)} + \cos x \cdot \cos \varepsilon\right) - \cos x \]
      associate--l+ [=>] 98.9%	\[ \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right) + \left(\cos x \cdot \cos \varepsilon - \cos x\right)} \]
      *-commutative [<=] 98.9%	\[ \left(-\sin x \cdot \sin \varepsilon\right) + \left(\color{blue}{\cos \varepsilon \cdot \cos x} - \cos x\right) \]
      distribute-rgt-neg-in [=>] 98.9%	\[ \color{blue}{\sin x \cdot \left(-\sin \varepsilon\right)} + \left(\cos \varepsilon \cdot \cos x - \cos x\right) \]
      fma-def [=>] 99.1%	\[ \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos \varepsilon \cdot \cos x - \cos x\right)} \]
      *-commutative [=>] 99.1%	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos x \cdot \cos \varepsilon} - \cos x\right) \]
      *-rgt-identity [<=] 99.1%	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon - \color{blue}{\cos x \cdot 1}\right) \]
      distribute-lft-out-- [=>] 99.1%	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos x \cdot \left(\cos \varepsilon - 1\right)}\right) \]
      sub-neg [=>] 99.1%	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\left(\cos \varepsilon + \left(-1\right)\right)}\right) \]
      metadata-eval [=>] 99.1%	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + \color{blue}{-1}\right)\right) \]
      +-commutative [=>] 99.1%	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\left(-1 + \cos \varepsilon\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00325:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0014:\\ \;\;\;\;\left(-2 \cdot \left(\left(\sin x \cdot {\varepsilon}^{3}\right) \cdot -0.08333333333333333\right) - \varepsilon \cdot \sin x\right) + \cos x \cdot \mathsf{fma}\left({\varepsilon}^{4}, 0.041666666666666664, \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + -1\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	40136

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00325:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0014:\\ \;\;\;\;\left(-2 \cdot \left(\left(\sin x \cdot {\varepsilon}^{3}\right) \cdot -0.08333333333333333\right) - \varepsilon \cdot \sin x\right) + \cos x \cdot \mathsf{fma}\left({\varepsilon}^{4}, 0.041666666666666664, \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + -1\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	40008

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0025:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0036:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + -1\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	99.1%
Cost	39044

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00014:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00014:\\ \;\;\;\;\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right) + \sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + -1\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	99.1%
Cost	32777

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000108 \lor \neg \left(\varepsilon \leq 0.00014\right):\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right) + \sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right)\\ \end{array} \]

Alternative 5
Accuracy	75.6%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-15}:\\ \;\;\;\;\left(-\sin \varepsilon\right) \cdot \tan \left(\frac{\varepsilon}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 6
Accuracy	66.3%
Cost	19844

\[\begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-15}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 7
Accuracy	76.5%
Cost	13632

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

Alternative 8
Accuracy	76.5%
Cost	13632

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

Alternative 9
Accuracy	69.2%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-15} \lor \neg \left(x \leq 2.9 \cdot 10^{-85}\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin \varepsilon\right) \cdot \tan \left(\frac{\varepsilon}{2}\right)\\ \end{array} \]

Alternative 10
Accuracy	67.3%
Cost	13449

\[\begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-13} \lor \neg \left(x \leq 3.5 \cdot 10^{-85}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 11
Accuracy	67.4%
Cost	13449

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-15} \lor \neg \left(x \leq 3.5 \cdot 10^{-85}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin \varepsilon\right) \cdot \tan \left(\frac{\varepsilon}{2}\right)\\ \end{array} \]

Alternative 12
Accuracy	47.3%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.085 \lor \neg \left(\varepsilon \leq 0.00014\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 13
Accuracy	22.0%
Cost	320

\[\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) \]

Alternative 14
Accuracy	13.0%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023165 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))