2cos (problem 3.3.5)

Average Accuracy: 38.4% → 99.1%

Time: 21.3s

Precision: binary64

Cost: 39176

Math

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

↓

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -0.00014:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000155:\\ \;\;\;\;-0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, t_0 - \cos x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (- (sin x)))))
   (if (<= eps -0.00014)
     (fma (+ (cos eps) -1.0) (cos x) t_0)
     (if (<= eps 0.000155)
       (- (* -0.5 (* (cos x) (* eps eps))) (* (sin x) (sin eps)))
       (fma (cos x) (cos 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(eps) * -sin(x);
	double tmp;
	if (eps <= -0.00014) {
		tmp = fma((cos(eps) + -1.0), cos(x), t_0);
	} else if (eps <= 0.000155) {
		tmp = (-0.5 * (cos(x) * (eps * eps))) - (sin(x) * sin(eps));
	} else {
		tmp = fma(cos(x), cos(eps), (t_0 - cos(x)));
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	t_0 = Float64(sin(eps) * Float64(-sin(x)))
	tmp = 0.0
	if (eps <= -0.00014)
		tmp = fma(Float64(cos(eps) + -1.0), cos(x), t_0);
	elseif (eps <= 0.000155)
		tmp = Float64(Float64(-0.5 * Float64(cos(x) * Float64(eps * eps))) - Float64(sin(x) * sin(eps)));
	else
		tmp = fma(cos(x), cos(eps), Float64(t_0 - cos(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_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[eps, -0.00014], N[(N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[eps, 0.000155], N[(N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(t$95$0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.3999999999999999e-4

   1. Initial program 53.2%

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

   2. Applied egg-rr 98.7%

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

      [Start] 53.2	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      sub-neg [=>] 53.2	\[ \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]
      +-commutative [=>] 53.2	\[ \color{blue}{\left(-\cos x\right) + \cos \left(x + \varepsilon\right)} \]
      cos-sum [=>] 98.6	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} \]
      cancel-sign-sub-inv [=>] 98.6	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} \]
      associate-+r+ [=>] 98.7	\[ \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \left(-\sin x\right) \cdot \sin \varepsilon} \]
      *-commutative [=>] 98.7	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} \]

   3. Taylor expanded in x around inf 98.6%

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

   4. Simplified 98.7%

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

      [Start] 98.6	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x \]
      +-commutative [=>] 98.6	\[ \color{blue}{\left(\cos \varepsilon \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
      *-commutative [=>] 98.6	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right) - \cos x \]
      *-commutative [<=] 98.6	\[ \left(\cos x \cdot \cos \varepsilon + -1 \cdot \color{blue}{\left(\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      mul-1-neg [=>] 98.6	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(-\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      sub0-neg [<=] 98.6	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(0 - \sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      associate-+r- [=>] 98.6	\[ \color{blue}{\left(\left(\cos x \cdot \cos \varepsilon + 0\right) - \sin \varepsilon \cdot \sin x\right)} - \cos x \]
      +-rgt-identity [=>] 98.6	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin \varepsilon \cdot \sin x\right) - \cos x \]
      associate--r+ [<=] 98.6	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin \varepsilon \cdot \sin x + \cos x\right)} \]
      +-commutative [<=] 98.6	\[ \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin \varepsilon \cdot \sin x\right)} \]
      associate--r+ [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x} \]

   5. Applied egg-rr 98.7%

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

      [Start] 98.7	\[ \left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin x \cdot \sin \varepsilon \]
      *-un-lft-identity [=>] 98.7	\[ \left(\cos \varepsilon \cdot \cos x - \color{blue}{1 \cdot \cos x}\right) - \sin x \cdot \sin \varepsilon \]
      distribute-rgt-out-- [=>] 98.7	\[ \color{blue}{\cos x \cdot \left(\cos \varepsilon - 1\right)} - \sin x \cdot \sin \varepsilon \]

   6. Applied egg-rr 98.7%

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

      [Start] 98.7	\[ \cos x \cdot \left(\cos \varepsilon - 1\right) - \sin x \cdot \sin \varepsilon \]
      *-commutative [=>] 98.7	\[ \color{blue}{\left(\cos \varepsilon - 1\right) \cdot \cos x} - \sin x \cdot \sin \varepsilon \]
      fma-neg [=>] 98.7	\[ \color{blue}{\mathsf{fma}\left(\cos \varepsilon - 1, \cos x, -\sin x \cdot \sin \varepsilon\right)} \]
      sub-neg [=>] 98.7	\[ \mathsf{fma}\left(\color{blue}{\cos \varepsilon + \left(-1\right)}, \cos x, -\sin x \cdot \sin \varepsilon\right) \]
      metadata-eval [=>] 98.7	\[ \mathsf{fma}\left(\cos \varepsilon + \color{blue}{-1}, \cos x, -\sin x \cdot \sin \varepsilon\right) \]
      distribute-rgt-neg-in [=>] 98.7	\[ \mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \color{blue}{\sin x \cdot \left(-\sin \varepsilon\right)}\right) \]

   if -1.3999999999999999e-4 < eps < 1.55e-4

   1. Initial program 23.7%

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

   2. Applied egg-rr 81.4%

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

      [Start] 23.7	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      sub-neg [=>] 23.7	\[ \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]
      +-commutative [=>] 23.7	\[ \color{blue}{\left(-\cos x\right) + \cos \left(x + \varepsilon\right)} \]
      cos-sum [=>] 24.8	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} \]
      cancel-sign-sub-inv [=>] 24.8	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} \]
      associate-+r+ [=>] 81.4	\[ \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \left(-\sin x\right) \cdot \sin \varepsilon} \]
      *-commutative [=>] 81.4	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} \]

   3. Taylor expanded in x around inf 24.8%

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

   4. Simplified 81.4%

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

      [Start] 24.8	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x \]
      +-commutative [=>] 24.8	\[ \color{blue}{\left(\cos \varepsilon \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
      *-commutative [=>] 24.8	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right) - \cos x \]
      *-commutative [<=] 24.8	\[ \left(\cos x \cdot \cos \varepsilon + -1 \cdot \color{blue}{\left(\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      mul-1-neg [=>] 24.8	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(-\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      sub0-neg [<=] 24.8	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(0 - \sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      associate-+r- [=>] 24.8	\[ \color{blue}{\left(\left(\cos x \cdot \cos \varepsilon + 0\right) - \sin \varepsilon \cdot \sin x\right)} - \cos x \]
      +-rgt-identity [=>] 24.8	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin \varepsilon \cdot \sin x\right) - \cos x \]
      associate--r+ [<=] 24.8	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin \varepsilon \cdot \sin x + \cos x\right)} \]
      +-commutative [<=] 24.8	\[ \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin \varepsilon \cdot \sin x\right)} \]
      associate--r+ [=>] 81.4	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x} \]

   5. Taylor expanded in eps around 0 99.7%

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

   6. Simplified 99.7%

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

      [Start] 99.7	\[ -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) - \sin x \cdot \sin \varepsilon \]
      unpow2 [=>] 99.7	\[ -0.5 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \cos x\right) - \sin x \cdot \sin \varepsilon \]

   if 1.55e-4 < eps

   1. Initial program 52.2%

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

   2. Applied egg-rr 98.5%

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

      [Start] 52.2	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      sub-neg [=>] 52.2	\[ \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]
      +-commutative [=>] 52.2	\[ \color{blue}{\left(-\cos x\right) + \cos \left(x + \varepsilon\right)} \]
      cos-sum [=>] 98.5	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} \]
      cancel-sign-sub-inv [=>] 98.5	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} \]
      associate-+r+ [=>] 98.5	\[ \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \left(-\sin x\right) \cdot \sin \varepsilon} \]
      *-commutative [=>] 98.5	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} \]

   3. Taylor expanded in x around inf 98.5%

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

   4. Simplified 98.5%

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

      [Start] 98.5	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x \]
      +-commutative [=>] 98.5	\[ \color{blue}{\left(\cos \varepsilon \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
      *-commutative [=>] 98.5	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right) - \cos x \]
      *-commutative [<=] 98.5	\[ \left(\cos x \cdot \cos \varepsilon + -1 \cdot \color{blue}{\left(\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      mul-1-neg [=>] 98.5	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(-\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      sub0-neg [<=] 98.5	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(0 - \sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      associate-+r- [=>] 98.5	\[ \color{blue}{\left(\left(\cos x \cdot \cos \varepsilon + 0\right) - \sin \varepsilon \cdot \sin x\right)} - \cos x \]
      +-rgt-identity [=>] 98.5	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin \varepsilon \cdot \sin x\right) - \cos x \]
      associate--r+ [<=] 98.5	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin \varepsilon \cdot \sin x + \cos x\right)} \]
      +-commutative [<=] 98.5	\[ \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin \varepsilon \cdot \sin x\right)} \]
      associate--r+ [=>] 98.5	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x} \]

   5. Applied egg-rr 98.6%

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

      [Start] 98.5	\[ \left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin x \cdot \sin \varepsilon \]
      associate--l- [=>] 98.5	\[ \color{blue}{\cos \varepsilon \cdot \cos x - \left(\cos x + \sin x \cdot \sin \varepsilon\right)} \]
      *-commutative [=>] 98.5	\[ \color{blue}{\cos x \cdot \cos \varepsilon} - \left(\cos x + \sin x \cdot \sin \varepsilon\right) \]
      fma-neg [=>] 98.6	\[ \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\cos x + \sin x \cdot \sin \varepsilon\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00014:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000155:\\ \;\;\;\;-0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right) - \cos x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	32840

\[\begin{array}{l} t_0 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00014:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;-0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \cos x\right) - t_0\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	32777

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00014 \lor \neg \left(\varepsilon \leq 0.00016\right):\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\\ \end{array} \]

Alternative 3
Accuracy	99.1%
Cost	26441

\[\begin{array}{l} t_0 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00014 \lor \neg \left(\varepsilon \leq 0.00016\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_0\\ \end{array} \]

Alternative 4
Accuracy	99.1%
Cost	26440

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00014:\\ \;\;\;\;\frac{\cos x}{\frac{1}{t_0}} - t_1\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;-0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot t_0 - t_1\\ \end{array} \]

Alternative 5
Accuracy	77.0%
Cost	26312

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq 170:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot -2\right)\\ \mathbf{elif}\;\varepsilon \leq 1.3 \cdot 10^{+291}:\\ \;\;\;\;\left(\cos \varepsilon - \cos x\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 6
Accuracy	76.9%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00105 \lor \neg \left(\varepsilon \leq 0.0295\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 7
Accuracy	76.3%
Cost	13632

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

Alternative 8
Accuracy	70.1%
Cost	13580

\[\begin{array}{l} t_0 := \sin x \cdot \left(-\varepsilon\right)\\ \mathbf{if}\;x \leq -0.00165:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-66}:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) - x \cdot \sin \varepsilon\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-10}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	68.2%
Cost	13449

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-49} \lor \neg \left(x \leq 1.5 \cdot 10^{-9}\right):\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 10
Accuracy	67.1%
Cost	13124

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.6 \cdot 10^{-7}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 52:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 11
Accuracy	66.7%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000205 \lor \neg \left(\varepsilon \leq 52\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 12
Accuracy	47.0%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00014 \lor \neg \left(\varepsilon \leq 0.00016\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 13
Accuracy	21.4%
Cost	320

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

Alternative 14
Accuracy	13.1%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))