2cos (problem 3.3.5)

Percentage Accurate: 38.0% → 99.5%

Time: 17.7s

Precision: binary64

Cost: 39360

Math

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

↓

\[\begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathsf{fma}\left(\sin x, \cos \left(\varepsilon \cdot 0.5\right), \cos x \cdot t_0\right) \cdot \left(-2 \cdot t_0\right) \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (* (fma (sin x) (cos (* eps 0.5)) (* (cos x) t_0)) (* -2.0 t_0))))

C

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

↓

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	return fma(sin(x), cos((eps * 0.5)), (cos(x) * t_0)) * (-2.0 * t_0);
}

Julia

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

↓

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	return Float64(fma(sin(x), cos(Float64(eps * 0.5)), Float64(cos(x) * t_0)) * Float64(-2.0 * t_0))
end

Wolfram

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

↓

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Sin[x], $MachinePrecision] * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 40.2%

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

2. Applied egg-rr 44.3%

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

   [Start] 40.2%	\[ \cos \left(x + \varepsilon\right) - \cos x \]
   diff-cos [=>] 44.3%	\[ \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]
   div-inv [=>] 44.3%	\[ -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
   metadata-eval [=>] 44.3%	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
   div-inv [=>] 44.3%	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]
   +-commutative [=>] 44.3%	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]
   metadata-eval [=>] 44.3%	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

3. Simplified 72.7%

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

   [Start] 44.3%	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   associate-*r* [=>] 44.3%	\[ \color{blue}{\left(-2 \cdot \sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)} \]
   *-commutative [=>] 44.3%	\[ \color{blue}{\sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right) \cdot \left(-2 \cdot \sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right)\right)} \]
   *-commutative [=>] 44.3%	\[ \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right)\right) \]
   associate-+r+ [=>] 44.3%	\[ \sin \left(0.5 \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)}\right) \cdot \left(-2 \cdot \sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right)\right) \]
   +-commutative [=>] 44.3%	\[ \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x + x\right)\right)}\right) \cdot \left(-2 \cdot \sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right)\right) \]
   *-commutative [=>] 44.3%	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)}\right) \]
   +-commutative [=>] 44.3%	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right)\right) \]
   associate--l+ [=>] 72.7%	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right)\right) \]
   +-inverses [=>] 72.7%	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + \color{blue}{0}\right)\right)\right) \]

4. Applied egg-rr 99.3%

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

   [Start] 72.7%	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   distribute-lft-in [=>] 72.7%	\[ \sin \color{blue}{\left(0.5 \cdot \varepsilon + 0.5 \cdot \left(x + x\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   +-rgt-identity [<=] 72.7%	\[ \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + 0\right)} + 0.5 \cdot \left(x + x\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   sin-sum [=>] 99.3%	\[ \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right) \cdot \cos \left(0.5 \cdot \left(x + x\right)\right) + \cos \left(0.5 \cdot \left(\varepsilon + 0\right)\right) \cdot \sin \left(0.5 \cdot \left(x + x\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   +-rgt-identity [=>] 99.3%	\[ \left(\sin \left(0.5 \cdot \color{blue}{\varepsilon}\right) \cdot \cos \left(0.5 \cdot \left(x + x\right)\right) + \cos \left(0.5 \cdot \left(\varepsilon + 0\right)\right) \cdot \sin \left(0.5 \cdot \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   +-rgt-identity [=>] 99.3%	\[ \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x + x\right)\right) + \cos \left(0.5 \cdot \color{blue}{\varepsilon}\right) \cdot \sin \left(0.5 \cdot \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]

5. Simplified 99.3%

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

   [Start] 99.3%	\[ \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x + x\right)\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   +-commutative [=>] 99.3%	\[ \color{blue}{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x + x\right)\right) + \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x + x\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   *-commutative [=>] 99.3%	\[ \left(\color{blue}{\sin \left(0.5 \cdot \left(x + x\right)\right) \cdot \cos \left(0.5 \cdot \varepsilon\right)} + \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   count-2 [=>] 99.3%	\[ \left(\sin \left(0.5 \cdot \color{blue}{\left(2 \cdot x\right)}\right) \cdot \cos \left(0.5 \cdot \varepsilon\right) + \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   *-commutative [<=] 99.3%	\[ \left(\sin \color{blue}{\left(\left(2 \cdot x\right) \cdot 0.5\right)} \cdot \cos \left(0.5 \cdot \varepsilon\right) + \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   count-2 [<=] 99.3%	\[ \left(\sin \left(\color{blue}{\left(x + x\right)} \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \varepsilon\right) + \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   count-2 [=>] 99.3%	\[ \left(\sin \left(\left(x + x\right) \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \varepsilon\right) + \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \color{blue}{\left(2 \cdot x\right)}\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   *-commutative [<=] 99.3%	\[ \left(\sin \left(\left(x + x\right) \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \varepsilon\right) + \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \color{blue}{\left(\left(2 \cdot x\right) \cdot 0.5\right)}\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   count-2 [<=] 99.3%	\[ \left(\sin \left(\left(x + x\right) \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \varepsilon\right) + \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\color{blue}{\left(x + x\right)} \cdot 0.5\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   fma-def [=>] 99.3%	\[ \color{blue}{\mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot 0.5\right), \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x + x\right) \cdot 0.5\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]

6. Taylor expanded in x around inf 99.3%

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

7. Simplified 99.3%

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

   [Start] 99.3%	\[ -2 \cdot \left(\left(\cos x \cdot \sin \left(0.5 \cdot \varepsilon\right) + \sin x \cdot \cos \left(0.5 \cdot \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [=>] 99.3%	\[ \color{blue}{\left(\left(\cos x \cdot \sin \left(0.5 \cdot \varepsilon\right) + \sin x \cdot \cos \left(0.5 \cdot \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot -2} \]
   +-commutative [<=] 99.3%	\[ \left(\color{blue}{\left(\sin x \cdot \cos \left(0.5 \cdot \varepsilon\right) + \cos x \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot -2 \]
   fma-def [=>] 99.3%	\[ \left(\color{blue}{\mathsf{fma}\left(\sin x, \cos \left(0.5 \cdot \varepsilon\right), \cos x \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot -2 \]
   associate-*r* [<=] 99.3%	\[ \color{blue}{\mathsf{fma}\left(\sin x, \cos \left(0.5 \cdot \varepsilon\right), \cos x \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot -2\right)} \]
   *-commutative [<=] 99.3%	\[ \mathsf{fma}\left(\sin x, \cos \left(0.5 \cdot \varepsilon\right), \cos x \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot \color{blue}{\left(-2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \]
   fma-def [<=] 99.3%	\[ \color{blue}{\left(\sin x \cdot \cos \left(0.5 \cdot \varepsilon\right) + \cos x \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   +-commutative [=>] 99.3%	\[ \color{blue}{\left(\cos x \cdot \sin \left(0.5 \cdot \varepsilon\right) + \sin x \cdot \cos \left(0.5 \cdot \varepsilon\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [=>] 99.3%	\[ \left(\color{blue}{\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos x} + \sin x \cdot \cos \left(0.5 \cdot \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   fma-def [=>] 99.3%	\[ \color{blue}{\mathsf{fma}\left(\sin \left(0.5 \cdot \varepsilon\right), \cos x, \sin x \cdot \cos \left(0.5 \cdot \varepsilon\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [=>] 99.3%	\[ \mathsf{fma}\left(\sin \left(0.5 \cdot \varepsilon\right), \cos x, \color{blue}{\cos \left(0.5 \cdot \varepsilon\right) \cdot \sin x}\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]

8. Taylor expanded in eps around inf 99.3%

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

9. Simplified 99.3%

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

   [Start] 99.3%	\[ \left(\cos x \cdot \sin \left(0.5 \cdot \varepsilon\right) + \sin x \cdot \cos \left(0.5 \cdot \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   +-commutative [=>] 99.3%	\[ \color{blue}{\left(\sin x \cdot \cos \left(0.5 \cdot \varepsilon\right) + \cos x \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   fma-def [=>] 99.3%	\[ \color{blue}{\mathsf{fma}\left(\sin x, \cos \left(0.5 \cdot \varepsilon\right), \cos x \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [=>] 99.3%	\[ \mathsf{fma}\left(\sin x, \cos \color{blue}{\left(\varepsilon \cdot 0.5\right)}, \cos x \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [=>] 99.3%	\[ \mathsf{fma}\left(\sin x, \cos \left(\varepsilon \cdot 0.5\right), \cos x \cdot \sin \color{blue}{\left(\varepsilon \cdot 0.5\right)}\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]

10. Final simplification 99.3%

    \[\leadsto \mathsf{fma}\left(\sin x, \cos \left(\varepsilon \cdot 0.5\right), \cos x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	39360

\[\begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathsf{fma}\left(\sin x, \cos \left(\varepsilon \cdot 0.5\right), \cos x \cdot t_0\right) \cdot \left(-2 \cdot t_0\right) \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	33088

\[\begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ -2 \cdot \left(t_0 \cdot \left(\cos x \cdot t_0 + \sin x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right)\right) \end{array} \]

Alternative 3
Accuracy	99.1%
Cost	32968

\[\begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.56:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_1\\ \mathbf{elif}\;\varepsilon \leq 0.00015:\\ \;\;\;\;\left(-2 \cdot t_0\right) \cdot \mathsf{fma}\left(t_0, \cos x, \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - t_1\right) - \cos x\\ \end{array} \]

Alternative 4
Accuracy	99.1%
Cost	32840

\[\begin{array}{l} t_0 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.8 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 4.1 \cdot 10^{-5}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin x + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - t_0\right) - \cos x\\ \end{array} \]

Alternative 5
Accuracy	99.1%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.2 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin x + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	67.6%
Cost	26436

\[\begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;\cos \left(x + \varepsilon\right) - \cos x \leq -2 \cdot 10^{-15}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-2 \cdot t_0\right)\\ \end{array} \]

Alternative 7
Accuracy	76.4%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00043 \lor \neg \left(\varepsilon \leq 0.0073\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x \cdot \varepsilon\\ \end{array} \]

Alternative 8
Accuracy	75.8%
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 9
Accuracy	66.8%
Cost	13449

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.2 \cdot 10^{-64} \lor \neg \left(\varepsilon \leq 1.35 \cdot 10^{-25}\right):\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 10
Accuracy	67.4%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 8.8 \cdot 10^{-8}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 11
Accuracy	46.3%
Cost	6988

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.45 \cdot 10^{-120}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Accuracy	66.8%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 6.5 \cdot 10^{-8}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 13
Accuracy	23.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.55 \cdot 10^{-121} \lor \neg \left(\varepsilon \leq 5.2 \cdot 10^{-75}\right):\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 14
Accuracy	18.3%
Cost	256

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

Alternative 15
Accuracy	13.0%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023263 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))