2cos (problem 3.3.5)

Percentage Accurate: 37.6% → 99.5%

Time: 22.3s

Precision: binary64

Cost: 39360

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 34.5%

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

2. Applied egg-rr 41.9%

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

   [Start] 34.5%	\[ \cos \left(x + \varepsilon\right) - \cos x \]
   diff-cos [=>] 41.9%	\[ \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]
   div-inv [=>] 41.9%	\[ -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
   metadata-eval [=>] 41.9%	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
   div-inv [=>] 41.9%	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]
   +-commutative [=>] 41.9%	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]
   metadata-eval [=>] 41.9%	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

3. Simplified 73.4%

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

   [Start] 41.9%	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   *-commutative [=>] 41.9%	\[ -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   +-commutative [<=] 41.9%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   associate--l+ [=>] 73.2%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   *-commutative [=>] 73.2%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]
   associate-+r+ [=>] 73.4%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)}\right)\right) \]
   +-commutative [=>] 73.4%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x + x\right)\right)}\right)\right) \]

4. Applied egg-rr 99.4%

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

   [Start] 73.4%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right) \]
   distribute-rgt-in [=>] 73.4%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \color{blue}{\left(\varepsilon \cdot 0.5 + \left(x + x\right) \cdot 0.5\right)}\right) \]
   sin-sum [=>] 99.4%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \cos \left(\left(x + x\right) \cdot 0.5\right) + \cos \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(\left(x + x\right) \cdot 0.5\right)\right)}\right) \]

5. Taylor expanded in eps around inf 99.4%

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

6. Simplified 99.4%

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

   [Start] 99.4%	\[ -2 \cdot \left(\left(\cos x \cdot \sin \left(0.5 \cdot \varepsilon\right) + \sin x \cdot \cos \left(0.5 \cdot \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [=>] 99.4%	\[ -2 \cdot \color{blue}{\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \left(\cos x \cdot \sin \left(0.5 \cdot \varepsilon\right) + \sin x \cdot \cos \left(0.5 \cdot \varepsilon\right)\right)\right)} \]
   +-commutative [=>] 99.4%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \color{blue}{\left(\sin x \cdot \cos \left(0.5 \cdot \varepsilon\right) + \cos x \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)}\right) \]
   fma-def [=>] 99.4%	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(\sin x, \cos \left(0.5 \cdot \varepsilon\right), \cos x \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)}\right) \]

7. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	39360

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

Alternative 2
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 3
Accuracy	99.1%
Cost	32841

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015 \lor \neg \left(\varepsilon \leq 0.00018\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \end{array} \]

Alternative 4
Accuracy	99.1%
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00018:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00018:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \end{array} \]

Alternative 5
Accuracy	76.9%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3500 \lor \neg \left(\varepsilon \leq 0.0115\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 6
Accuracy	76.6%
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 7
Accuracy	69.4%
Cost	13513

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -195 \lor \neg \left(x \leq 3 \cdot 10^{-72}\right):\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \]

Alternative 8
Accuracy	69.5%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -195 \lor \neg \left(x \leq 2.8 \cdot 10^{-72}\right):\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\\ \end{array} \]

Alternative 9
Accuracy	66.9%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -190000000000:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 3.4 \cdot 10^{-36}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \end{array} \]

Alternative 10
Accuracy	67.0%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -190000000000 \lor \neg \left(\varepsilon \leq 0.00046\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 11
Accuracy	48.4%
Cost	7120

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -2.2 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 1.32 \cdot 10^{-103}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00014:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Accuracy	66.8%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3500 \lor \neg \left(\varepsilon \leq 0.00065\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 13
Accuracy	24.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-116}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \end{array} \]

Alternative 14
Accuracy	18.5%
Cost	256

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

Alternative 15
Accuracy	13.3%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023277 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))