2cos (problem 3.3.5)

Average Error: 39.6 → 0.3

Time: 19.7s

Precision: binary64

Cost: 32768

Math

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

↓

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

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (- (* (* (tan (* eps 0.5)) (- (sin eps))) (cos x)) (* (sin eps) (sin x))))

C

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

↓

double code(double x, double eps) {
	return ((tan((eps * 0.5)) * -sin(eps)) * cos(x)) - (sin(eps) * sin(x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = cos((x + eps)) - cos(x)
end function

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((tan((eps * 0.5d0)) * -sin(eps)) * cos(x)) - (sin(eps) * sin(x))
end function

Java

public static double code(double x, double eps) {
	return Math.cos((x + eps)) - Math.cos(x);
}

↓

public static double code(double x, double eps) {
	return ((Math.tan((eps * 0.5)) * -Math.sin(eps)) * Math.cos(x)) - (Math.sin(eps) * Math.sin(x));
}

Python

def code(x, eps):
	return math.cos((x + eps)) - math.cos(x)

↓

def code(x, eps):
	return ((math.tan((eps * 0.5)) * -math.sin(eps)) * math.cos(x)) - (math.sin(eps) * math.sin(x))

Julia

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

↓

function code(x, eps)
	return Float64(Float64(Float64(tan(Float64(eps * 0.5)) * Float64(-sin(eps))) * cos(x)) - Float64(sin(eps) * sin(x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = cos((x + eps)) - cos(x);
end

↓

function tmp = code(x, eps)
	tmp = ((tan((eps * 0.5)) * -sin(eps)) * cos(x)) - (sin(eps) * sin(x));
end

Wolfram

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

↓

code[x_, eps_] := N[(N[(N[(N[Tan[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.6

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

2. Applied egg-rr 25.0

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

3. Simplified 6.3

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

   [Start] 25.0	\[ \cos x \cdot \cos \varepsilon + \left(\sin \varepsilon \cdot \left(-\sin x\right) + \left(-\cos x\right)\right) \]
   +-commutative [=>] 25.0	\[ \cos x \cdot \cos \varepsilon + \color{blue}{\left(\left(-\cos x\right) + \sin \varepsilon \cdot \left(-\sin x\right)\right)} \]
   *-commutative [=>] 25.0	\[ \cos x \cdot \cos \varepsilon + \left(\left(-\cos x\right) + \color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon}\right) \]
   distribute-lft-neg-in [<=] 25.0	\[ \cos x \cdot \cos \varepsilon + \left(\left(-\cos x\right) + \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)}\right) \]
   associate-+r+ [=>] 6.3	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right) + \left(-\sin x \cdot \sin \varepsilon\right)} \]
   +-commutative [<=] 6.3	\[ \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right)} + \left(-\sin x \cdot \sin \varepsilon\right) \]
   +-commutative [=>] 6.3	\[ \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right) + \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right)} \]
   distribute-rgt-neg-in [=>] 6.3	\[ \color{blue}{\sin x \cdot \left(-\sin \varepsilon\right)} + \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) \]
   fma-def [=>] 6.3	\[ \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right)} \]
   +-commutative [=>] 6.3	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos x \cdot \cos \varepsilon + \left(-\cos x\right)}\right) \]
   *-commutative [=>] 6.3	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos \varepsilon \cdot \cos x} + \left(-\cos x\right)\right) \]
   neg-mul-1 [=>] 6.3	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos \varepsilon \cdot \cos x + \color{blue}{-1 \cdot \cos x}\right) \]
   distribute-rgt-out [=>] 6.3	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)}\right) \]

4. Taylor expanded in x around inf 6.3

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

5. Simplified 6.3

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

   [Start] 6.3	\[ -1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos x \cdot \left(\cos \varepsilon - 1\right) \]
   +-commutative [=>] 6.3	\[ \color{blue}{\cos x \cdot \left(\cos \varepsilon - 1\right) + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)} \]
   *-commutative [=>] 6.3	\[ \color{blue}{\left(\cos \varepsilon - 1\right) \cdot \cos x} + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right) \]
   sub-neg [=>] 6.3	\[ \color{blue}{\left(\cos \varepsilon + \left(-1\right)\right)} \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right) \]
   metadata-eval [=>] 6.3	\[ \left(\cos \varepsilon + \color{blue}{-1}\right) \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right) \]
   mul-1-neg [=>] 6.3	\[ \left(\cos \varepsilon + -1\right) \cdot \cos x + \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)} \]
   sub-neg [<=] 6.3	\[ \color{blue}{\left(\cos \varepsilon + -1\right) \cdot \cos x - \sin x \cdot \sin \varepsilon} \]
   *-commutative [=>] 6.3	\[ \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)} - \sin x \cdot \sin \varepsilon \]
   *-commutative [=>] 6.3	\[ \cos x \cdot \left(\cos \varepsilon + -1\right) - \color{blue}{\sin \varepsilon \cdot \sin x} \]

6. Applied egg-rr 0.6

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

7. Simplified 0.3

   \[\leadsto \color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \cdot \cos x} - \sin \varepsilon \cdot \sin x \]
   Proof

   [Start] 0.6	\[ \frac{\cos x \cdot {\sin \varepsilon}^{2}}{-1 - \cos \varepsilon} - \sin \varepsilon \cdot \sin x \]
   sub-neg [=>] 0.6	\[ \frac{\cos x \cdot {\sin \varepsilon}^{2}}{\color{blue}{-1 + \left(-\cos \varepsilon\right)}} - \sin \varepsilon \cdot \sin x \]
   +-commutative [=>] 0.6	\[ \frac{\cos x \cdot {\sin \varepsilon}^{2}}{\color{blue}{\left(-\cos \varepsilon\right) + -1}} - \sin \varepsilon \cdot \sin x \]
   metadata-eval [<=] 0.6	\[ \frac{\cos x \cdot {\sin \varepsilon}^{2}}{\left(-\cos \varepsilon\right) + \color{blue}{\left(-1\right)}} - \sin \varepsilon \cdot \sin x \]
   distribute-neg-in [<=] 0.6	\[ \frac{\cos x \cdot {\sin \varepsilon}^{2}}{\color{blue}{-\left(\cos \varepsilon + 1\right)}} - \sin \varepsilon \cdot \sin x \]
   *-commutative [<=] 0.6	\[ \frac{\color{blue}{{\sin \varepsilon}^{2} \cdot \cos x}}{-\left(\cos \varepsilon + 1\right)} - \sin \varepsilon \cdot \sin x \]
   associate-/l* [=>] 0.6	\[ \color{blue}{\frac{{\sin \varepsilon}^{2}}{\frac{-\left(\cos \varepsilon + 1\right)}{\cos x}}} - \sin \varepsilon \cdot \sin x \]
   associate-/r/ [=>] 0.6	\[ \color{blue}{\frac{{\sin \varepsilon}^{2}}{-\left(\cos \varepsilon + 1\right)} \cdot \cos x} - \sin \varepsilon \cdot \sin x \]
   unpow2 [=>] 0.6	\[ \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{-\left(\cos \varepsilon + 1\right)} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
   neg-mul-1 [=>] 0.6	\[ \frac{\sin \varepsilon \cdot \sin \varepsilon}{\color{blue}{-1 \cdot \left(\cos \varepsilon + 1\right)}} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
   times-frac [=>] 0.7	\[ \color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \frac{\sin \varepsilon}{\cos \varepsilon + 1}\right)} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
   +-commutative [=>] 0.7	\[ \left(\frac{\sin \varepsilon}{-1} \cdot \frac{\sin \varepsilon}{\color{blue}{1 + \cos \varepsilon}}\right) \cdot \cos x - \sin \varepsilon \cdot \sin x \]
   hang-0p-tan [=>] 0.3	\[ \left(\frac{\sin \varepsilon}{-1} \cdot \color{blue}{\tan \left(\frac{\varepsilon}{2}\right)}\right) \cdot \cos x - \sin \varepsilon \cdot \sin x \]

8. Applied egg-rr 0.4

   \[\leadsto \color{blue}{\frac{\tan \left(\varepsilon \cdot 0.5\right)}{\frac{-1}{\sin \varepsilon}}} \cdot \cos x - \sin \varepsilon \cdot \sin x \]

9. Simplified 0.3

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

   [Start] 0.4	\[ \frac{\tan \left(\varepsilon \cdot 0.5\right)}{\frac{-1}{\sin \varepsilon}} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
   associate-/l* [<=] 0.3	\[ \color{blue}{\frac{\tan \left(\varepsilon \cdot 0.5\right) \cdot \sin \varepsilon}{-1}} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
   *-commutative [=>] 0.3	\[ \frac{\color{blue}{\sin \varepsilon \cdot \tan \left(\varepsilon \cdot 0.5\right)}}{-1} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
   associate-/l* [=>] 0.4	\[ \color{blue}{\frac{\sin \varepsilon}{\frac{-1}{\tan \left(\varepsilon \cdot 0.5\right)}}} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
   metadata-eval [<=] 0.4	\[ \frac{\sin \varepsilon}{\frac{\color{blue}{\frac{1}{-1}}}{\tan \left(\varepsilon \cdot 0.5\right)}} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
   associate-/r* [<=] 0.4	\[ \frac{\sin \varepsilon}{\color{blue}{\frac{1}{-1 \cdot \tan \left(\varepsilon \cdot 0.5\right)}}} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
   associate-/r/ [=>] 0.3	\[ \color{blue}{\left(\frac{\sin \varepsilon}{1} \cdot \left(-1 \cdot \tan \left(\varepsilon \cdot 0.5\right)\right)\right)} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
   /-rgt-identity [=>] 0.3	\[ \left(\color{blue}{\sin \varepsilon} \cdot \left(-1 \cdot \tan \left(\varepsilon \cdot 0.5\right)\right)\right) \cdot \cos x - \sin \varepsilon \cdot \sin x \]
   mul-1-neg [=>] 0.3	\[ \left(\sin \varepsilon \cdot \color{blue}{\left(-\tan \left(\varepsilon \cdot 0.5\right)\right)}\right) \cdot \cos x - \sin \varepsilon \cdot \sin x \]

10. Final simplification 0.3

    \[\leadsto \left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(-\sin \varepsilon\right)\right) \cdot \cos x - \sin \varepsilon \cdot \sin x \]

Alternatives

Alternative 1
Error	13.9
Cost	33604

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -0.005:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.008333333333333333 + -0.16666666666666666\right) + \frac{-2}{\varepsilon \cdot \varepsilon}} - t_0\\ \end{array} \]

Alternative 2
Error	14.2
Cost	33220

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -0.005:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x}{-0.16666666666666666 + \frac{-2}{\varepsilon \cdot \varepsilon}} - t_0\\ \end{array} \]

Alternative 3
Error	0.5
Cost	32840

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.027:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.027:\\ \;\;\;\;\frac{\cos x}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.008333333333333333 + -0.16666666666666666\right) + \frac{-2}{\varepsilon \cdot \varepsilon}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + t_0\right)\\ \end{array} \]

Alternative 4
Error	0.5
Cost	32840

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.027:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.027:\\ \;\;\;\;\frac{\cos x}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.008333333333333333 + -0.16666666666666666\right) + \frac{-2}{\varepsilon \cdot \varepsilon}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - t_0\right) - \cos x\\ \end{array} \]

Alternative 5
Error	0.5
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.028:\\ \;\;\;\;\left(t_0 - \cos x\right) - t_1\\ \mathbf{elif}\;\varepsilon \leq 0.027:\\ \;\;\;\;\frac{\cos x}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.008333333333333333 + -0.16666666666666666\right) + \frac{-2}{\varepsilon \cdot \varepsilon}} - t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \end{array} \]

Alternative 6
Error	0.5
Cost	32777

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.027 \lor \neg \left(\varepsilon \leq 0.0255\right):\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.008333333333333333 + -0.16666666666666666\right) + \frac{-2}{\varepsilon \cdot \varepsilon}} - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 7
Error	0.5
Cost	32776

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.027:\\ \;\;\;\;\mathsf{fma}\left(t_0, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0255:\\ \;\;\;\;\frac{\cos x}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.008333333333333333 + -0.16666666666666666\right) + \frac{-2}{\varepsilon \cdot \varepsilon}} - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot t_0\right)\\ \end{array} \]

Alternative 8
Error	0.5
Cost	26441

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.027 \lor \neg \left(\varepsilon \leq 0.0255\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.008333333333333333 + -0.16666666666666666\right) + \frac{-2}{\varepsilon \cdot \varepsilon}} - t_0\\ \end{array} \]

Alternative 9
Error	14.0
Cost	20168

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0102:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0045:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) - t_0\\ \end{array} \]

Alternative 10
Error	14.1
Cost	19912

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0105:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00325:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 11
Error	19.3
Cost	13776

\[\begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_1 := t_0 \cdot \left(\sin x \cdot -2\right)\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-59}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 - \varepsilon \cdot x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	14.2
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0085 \lor \neg \left(\varepsilon \leq 0.0115\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 13
Error	17.6
Cost	13644

\[\begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_1 := t_0 \cdot \left(\sin x \cdot -2\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-112}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \mathbf{elif}\;x \leq 0.00175:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	14.2
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.076 \lor \neg \left(\varepsilon \leq 0.0105\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.5\right) - \sin x\right)\\ \end{array} \]

Alternative 15
Error	20.4
Cost	13448

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;x \leq -5.7 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-89}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-55}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 - \varepsilon \cdot x\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-8}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	20.2
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0066 \lor \neg \left(\varepsilon \leq 0.00325\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 17
Error	20.7
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0066 \lor \neg \left(\varepsilon \leq 0.00325\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 18
Error	30.2
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0066 \lor \neg \left(\varepsilon \leq 0.00016\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 - \varepsilon \cdot x\\ \end{array} \]

Alternative 19
Error	48.8
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{-120} \lor \neg \left(x \leq 3.5 \cdot 10^{-109}\right):\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \]

Alternative 20
Error	47.3
Cost	576

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

Alternative 21
Error	52.9
Cost	256

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

Reproduce

herbie shell --seed 2023045 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))