2cos (problem 3.3.5)

Percentage Accurate: 38.0% → 99.4%

Time: 23.7s

Precision: binary64

Cost: 39360

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• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

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• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

function code(x, eps)
	t_0 = sin(Float64(0.5 * eps))
	return Float64(-2.0 * Float64(t_0 * fma(sin(x), cos(Float64(0.5 * eps)), Float64(t_0 * cos(x)))))
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[(0.5 * eps), $MachinePrecision]], $MachinePrecision]}, N[(-2.0 * N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] * N[Cos[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 38.1%

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

2. Applied egg-rr 45.3%

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

   [Start] 38.1	\[ \cos \left(x + \varepsilon\right) - \cos x \]
   diff-cos [=>] 45.3	\[ \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]
   div-inv [=>] 45.3	\[ -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
   metadata-eval [=>] 45.3	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
   div-inv [=>] 45.3	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]
   +-commutative [=>] 45.3	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]
   metadata-eval [=>] 45.3	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

3. Simplified 76.5%

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

   [Start] 45.3	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   *-commutative [=>] 45.3	\[ -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   +-commutative [<=] 45.3	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   associate--l+ [=>] 76.4	\[ -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   *-commutative [=>] 76.4	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]
   associate-+r+ [=>] 76.5	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)}\right)\right) \]
   +-commutative [=>] 76.5	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x + x\right)\right)}\right)\right) \]

4. Taylor expanded in x around -inf 76.5%

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

5. Applied egg-rr 99.5%

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

   [Start] 76.5	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   sub-neg [=>] 76.5	\[ -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(--2 \cdot x\right)\right)}\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   distribute-rgt-in [=>] 76.5	\[ -2 \cdot \left(\sin \color{blue}{\left(\varepsilon \cdot 0.5 + \left(--2 \cdot x\right) \cdot 0.5\right)} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [<=] 76.5	\[ -2 \cdot \left(\sin \left(\color{blue}{0.5 \cdot \varepsilon} + \left(--2 \cdot x\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   sin-sum [=>] 99.5	\[ -2 \cdot \left(\color{blue}{\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(--2 \cdot x\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(--2 \cdot x\right) \cdot 0.5\right)\right)} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [=>] 99.5	\[ -2 \cdot \left(\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(-\color{blue}{x \cdot -2}\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(--2 \cdot x\right) \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   distribute-rgt-neg-in [=>] 99.5	\[ -2 \cdot \left(\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\color{blue}{\left(x \cdot \left(--2\right)\right)} \cdot 0.5\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(--2 \cdot x\right) \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   metadata-eval [=>] 99.5	\[ -2 \cdot \left(\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x \cdot \color{blue}{2}\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(--2 \cdot x\right) \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [=>] 99.5	\[ -2 \cdot \left(\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x \cdot 2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(-\color{blue}{x \cdot -2}\right) \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   distribute-rgt-neg-in [=>] 99.5	\[ -2 \cdot \left(\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x \cdot 2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\color{blue}{\left(x \cdot \left(--2\right)\right)} \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   metadata-eval [=>] 99.5	\[ -2 \cdot \left(\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x \cdot 2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(x \cdot \color{blue}{2}\right) \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]

6. Applied egg-rr 99.5%

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

   [Start] 99.5	\[ -2 \cdot \left(\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x \cdot 2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(x \cdot 2\right) \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   +-commutative [=>] 99.5	\[ -2 \cdot \left(\color{blue}{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(x \cdot 2\right) \cdot 0.5\right) + \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x \cdot 2\right) \cdot 0.5\right)\right)} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [=>] 99.5	\[ -2 \cdot \left(\left(\color{blue}{\sin \left(\left(x \cdot 2\right) \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \varepsilon\right)} + \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x \cdot 2\right) \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   fma-def [=>] 99.5	\[ -2 \cdot \left(\color{blue}{\mathsf{fma}\left(\sin \left(\left(x \cdot 2\right) \cdot 0.5\right), \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x \cdot 2\right) \cdot 0.5\right)\right)} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   associate-*l* [=>] 99.5	\[ -2 \cdot \left(\mathsf{fma}\left(\sin \color{blue}{\left(x \cdot \left(2 \cdot 0.5\right)\right)}, \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x \cdot 2\right) \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   metadata-eval [=>] 99.5	\[ -2 \cdot \left(\mathsf{fma}\left(\sin \left(x \cdot \color{blue}{1}\right), \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x \cdot 2\right) \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-rgt-identity [=>] 99.5	\[ -2 \cdot \left(\mathsf{fma}\left(\sin \color{blue}{x}, \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(x \cdot 2\right) \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   associate-*l* [=>] 99.5	\[ -2 \cdot \left(\mathsf{fma}\left(\sin x, \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \color{blue}{\left(x \cdot \left(2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   metadata-eval [=>] 99.5	\[ -2 \cdot \left(\mathsf{fma}\left(\sin x, \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(x \cdot \color{blue}{1}\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-rgt-identity [=>] 99.5	\[ -2 \cdot \left(\mathsf{fma}\left(\sin x, \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \color{blue}{x}\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]

7. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	33088

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

Alternative 2
Accuracy	99.1%
Cost	32841

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.09 \lor \neg \left(\varepsilon \leq 0.00013\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(\sin x + t_0 \cdot \cos \left(0.5 \cdot \left(x \cdot 2\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	77.6%
Cost	26688

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

Alternative 4
Accuracy	76.9%
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0031 \lor \neg \left(\varepsilon \leq 8.5\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.5\right) - \sin x\right)\\ \end{array} \]

Alternative 5
Accuracy	76.4%
Cost	13632

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

Alternative 6
Accuracy	76.7%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00255 \lor \neg \left(\varepsilon \leq 8.5\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	76.1%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00285 \lor \neg \left(\varepsilon \leq 8.5\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	49.3%
Cost	7120

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.000135:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -3.7 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{-160}:\\ \;\;\;\;-x \cdot \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00013:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	67.5%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.46 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 8.5\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 10
Accuracy	29.0%
Cost	969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-157} \lor \neg \left(\varepsilon \leq 4.6 \cdot 10^{-160}\right):\\ \;\;\;\;-2 \cdot \frac{1}{0.3333333333333333 + \frac{\frac{4}{\varepsilon}}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \varepsilon\\ \end{array} \]

Alternative 11
Accuracy	28.1%
Cost	848

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -30:\\ \;\;\;\;-0.001953125\\ \mathbf{elif}\;\varepsilon \leq -3.7 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-160}:\\ \;\;\;\;-x \cdot \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.65 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.001953125\\ \end{array} \]

Alternative 12
Accuracy	28.5%
Cost	848

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -11:\\ \;\;\;\;-0.015625\\ \mathbf{elif}\;\varepsilon \leq -3.7 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-160}:\\ \;\;\;\;-x \cdot \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.65 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.015625\\ \end{array} \]

Alternative 13
Accuracy	28.9%
Cost	848

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -4.6:\\ \;\;\;\;-0.0625\\ \mathbf{elif}\;\varepsilon \leq -3.7 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{-160}:\\ \;\;\;\;-x \cdot \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.65 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.0625\\ \end{array} \]

Alternative 14
Accuracy	29.2%
Cost	848

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -4.5:\\ \;\;\;\;-0.125\\ \mathbf{elif}\;\varepsilon \leq -3.7 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.1 \cdot 10^{-160}:\\ \;\;\;\;-x \cdot \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4.7 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.125\\ \end{array} \]

Alternative 15
Accuracy	29.4%
Cost	848

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -3.8:\\ \;\;\;\;-0.25\\ \mathbf{elif}\;\varepsilon \leq -3.7 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-160}:\\ \;\;\;\;-x \cdot \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4.7 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.25\\ \end{array} \]

Alternative 16
Accuracy	29.7%
Cost	848

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -3.3:\\ \;\;\;\;-0.5\\ \mathbf{elif}\;\varepsilon \leq -3.7 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-160}:\\ \;\;\;\;-x \cdot \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4.7 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5\\ \end{array} \]

Alternative 17
Accuracy	30.1%
Cost	848

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -1.45:\\ \;\;\;\;-1\\ \mathbf{elif}\;\varepsilon \leq -3.7 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-161}:\\ \;\;\;\;-x \cdot \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.4:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 18
Accuracy	30.2%
Cost	848

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -1.75:\\ \;\;\;\;-1.5\\ \mathbf{elif}\;\varepsilon \leq -3.7 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-160}:\\ \;\;\;\;-x \cdot \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.4:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 19
Accuracy	30.1%
Cost	848

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -2:\\ \;\;\;\;-2\\ \mathbf{elif}\;\varepsilon \leq -3.7 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{-160}:\\ \;\;\;\;-x \cdot \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.4:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 20
Accuracy	24.5%
Cost	585

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-157} \lor \neg \left(\varepsilon \leq 2.65 \cdot 10^{-160}\right):\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \varepsilon\\ \end{array} \]

Alternative 21
Accuracy	18.8%
Cost	256

\[-x \cdot \varepsilon \]

Reproduce

herbie shell --seed 2023160 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))