2cos (problem 3.3.5)

Average Accuracy: 38.7% → 99.4%

Time: 17.8s

Precision: binary64

Cost: 39360

Math

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

↓

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

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 (cos x) t_0 (* (sin x) (cos (* 0.5 eps))))))))

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(cos(x), t_0, (sin(x) * cos((0.5 * eps)))));
}

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(cos(x), t_0, Float64(sin(x) * cos(Float64(0.5 * eps))))))
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[Cos[x], $MachinePrecision] * t$95$0 + N[(N[Sin[x], $MachinePrecision] * N[Cos[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 38.7%

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

2. Applied egg-rr 47.4%

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

   [Start] 38.7	\[ \cos \left(x + \varepsilon\right) - \cos x \]
   diff-cos [=>] 47.4	\[ \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]
   div-inv [=>] 47.4	\[ -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
   metadata-eval [=>] 47.4	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
   div-inv [=>] 47.4	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]
   +-commutative [=>] 47.4	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]
   metadata-eval [=>] 47.4	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

3. Simplified 76.2%

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

   [Start] 47.4	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   associate-*r* [=>] 47.4	\[ \color{blue}{\left(-2 \cdot \sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)} \]
   *-commutative [=>] 47.4	\[ \color{blue}{\left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot -2\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right) \]
   associate-*l* [=>] 47.4	\[ \color{blue}{\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \left(-2 \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
   *-commutative [=>] 47.4	\[ \sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)} \cdot \left(-2 \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   +-commutative [<=] 47.4	\[ \sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \left(-2 \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   associate--l+ [=>] 76.2	\[ \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \left(-2 \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   +-inverses [=>] 76.2	\[ \sin \left(0.5 \cdot \left(\varepsilon + \color{blue}{0}\right)\right) \cdot \left(-2 \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   distribute-lft-in [=>] 76.2	\[ \sin \color{blue}{\left(0.5 \cdot \varepsilon + 0.5 \cdot 0\right)} \cdot \left(-2 \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   metadata-eval [=>] 76.2	\[ \sin \left(0.5 \cdot \varepsilon + \color{blue}{0}\right) \cdot \left(-2 \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   *-commutative [=>] 76.2	\[ \sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \left(-2 \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]
   +-commutative [<=] 76.2	\[ \sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(x + \color{blue}{\left(\varepsilon + x\right)}\right)\right)\right) \]

4. Taylor expanded in x around -inf 76.2%

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

5. Applied egg-rr 99.3%

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

   [Start] 76.2	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   sub-neg [=>] 76.2	\[ -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(--2 \cdot x\right)\right)}\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   distribute-lft-in [=>] 76.2	\[ -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \varepsilon + 0.5 \cdot \left(--2 \cdot x\right)\right)} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   sin-sum [=>] 99.3	\[ -2 \cdot \left(\color{blue}{\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(--2 \cdot x\right)\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(--2 \cdot x\right)\right)\right)} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [=>] 99.3	\[ -2 \cdot \left(\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(-\color{blue}{x \cdot -2}\right)\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(--2 \cdot x\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   distribute-rgt-neg-in [=>] 99.3	\[ -2 \cdot \left(\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \color{blue}{\left(x \cdot \left(--2\right)\right)}\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(--2 \cdot x\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   metadata-eval [=>] 99.3	\[ -2 \cdot \left(\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x \cdot \color{blue}{2}\right)\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(--2 \cdot x\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [=>] 99.3	\[ -2 \cdot \left(\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x \cdot 2\right)\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(-\color{blue}{x \cdot -2}\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   distribute-rgt-neg-in [=>] 99.3	\[ -2 \cdot \left(\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x \cdot 2\right)\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(x \cdot \left(--2\right)\right)}\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   metadata-eval [=>] 99.3	\[ -2 \cdot \left(\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x \cdot 2\right)\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot \color{blue}{2}\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]

6. Simplified 99.4%

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

   [Start] 99.3	\[ -2 \cdot \left(\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x \cdot 2\right)\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot 2\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [=>] 99.3	\[ -2 \cdot \left(\left(\color{blue}{\cos \left(0.5 \cdot \left(x \cdot 2\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)} + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot 2\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [<=] 99.3	\[ -2 \cdot \left(\left(\cos \color{blue}{\left(\left(x \cdot 2\right) \cdot 0.5\right)} \cdot \sin \left(0.5 \cdot \varepsilon\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot 2\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [<=] 99.3	\[ -2 \cdot \left(\left(\cos \left(\left(x \cdot 2\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \varepsilon\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \color{blue}{\left(\left(x \cdot 2\right) \cdot 0.5\right)}\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   fma-def [=>] 99.3	\[ -2 \cdot \left(\color{blue}{\mathsf{fma}\left(\cos \left(\left(x \cdot 2\right) \cdot 0.5\right), \sin \left(0.5 \cdot \varepsilon\right), \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(x \cdot 2\right) \cdot 0.5\right)\right)} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   associate-*l* [=>] 99.3	\[ -2 \cdot \left(\mathsf{fma}\left(\cos \color{blue}{\left(x \cdot \left(2 \cdot 0.5\right)\right)}, \sin \left(0.5 \cdot \varepsilon\right), \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(x \cdot 2\right) \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   metadata-eval [=>] 99.3	\[ -2 \cdot \left(\mathsf{fma}\left(\cos \left(x \cdot \color{blue}{1}\right), \sin \left(0.5 \cdot \varepsilon\right), \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(x \cdot 2\right) \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-rgt-identity [=>] 99.3	\[ -2 \cdot \left(\mathsf{fma}\left(\cos \color{blue}{x}, \sin \left(0.5 \cdot \varepsilon\right), \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(x \cdot 2\right) \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-commutative [=>] 99.3	\[ -2 \cdot \left(\mathsf{fma}\left(\cos x, \sin \left(0.5 \cdot \varepsilon\right), \color{blue}{\sin \left(\left(x \cdot 2\right) \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \varepsilon\right)}\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   associate-*l* [=>] 99.4	\[ -2 \cdot \left(\mathsf{fma}\left(\cos x, \sin \left(0.5 \cdot \varepsilon\right), \sin \color{blue}{\left(x \cdot \left(2 \cdot 0.5\right)\right)} \cdot \cos \left(0.5 \cdot \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   metadata-eval [=>] 99.4	\[ -2 \cdot \left(\mathsf{fma}\left(\cos x, \sin \left(0.5 \cdot \varepsilon\right), \sin \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(0.5 \cdot \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
   *-rgt-identity [=>] 99.4	\[ -2 \cdot \left(\mathsf{fma}\left(\cos x, \sin \left(0.5 \cdot \varepsilon\right), \sin \color{blue}{x} \cdot \cos \left(0.5 \cdot \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]

7. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	33088

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

Alternative 2
Accuracy	99.2%
Cost	32841

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0027 \lor \neg \left(\varepsilon \leq 0.005\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \left(\sin x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.125\right) + \cos x \cdot \left(\varepsilon \cdot \left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.020833333333333332\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	99.2%
Cost	32841

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0027 \lor \neg \left(\varepsilon \leq 0.0029\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \left(\sin x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.125\right) + \cos x \cdot \left(\varepsilon \cdot \left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.020833333333333332\right)\right)\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	76.4%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.24:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0021:\\ \;\;\;\;\varepsilon \cdot \left(-0.5 \cdot \left(\cos x \cdot \varepsilon\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \end{array} \]

Alternative 5
Accuracy	76.2%
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	70.3%
Cost	13513

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-7} \lor \neg \left(x \leq 2.2 \cdot 10^{-16}\right):\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \]

Alternative 7
Accuracy	66.9%
Cost	13449

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-69} \lor \neg \left(\varepsilon \leq 8.5 \cdot 10^{-24}\right):\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 8
Accuracy	67.2%
Cost	13124

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.16:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 9
Accuracy	66.9%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.16 \lor \neg \left(\varepsilon \leq 1.9 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 10
Accuracy	47.4%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.16 \lor \neg \left(\varepsilon \leq 0.000145\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \]

Alternative 11
Accuracy	21.4%
Cost	320

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

Alternative 12
Accuracy	13.3%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023150 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))