2cos (problem 3.3.5)

Percentage Accurate: 38.2% → 99.3%

Time: 20.6s

Precision: binary64

Cost: 39616

Math

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

↓

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

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

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

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

Derivation

1. Initial program 42.5%

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

2. Applied egg-rr 50.0%

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

   [Start] 42.5	\[ \cos \left(x + \varepsilon\right) - \cos x \]
   diff-cos [=>] 50.0	\[ \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]
   div-inv [=>] 50.0	\[ -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
   metadata-eval [=>] 50.0	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
   div-inv [=>] 50.0	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]
   +-commutative [=>] 50.0	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]
   metadata-eval [=>] 50.0	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

3. Simplified 74.7%

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

   [Start] 50.0	\[ -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   *-commutative [=>] 50.0	\[ -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   +-commutative [<=] 50.0	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   associate--l+ [=>] 74.7	\[ -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   +-inverses [=>] 74.7	\[ -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \color{blue}{0}\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   distribute-lft-in [=>] 74.7	\[ -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \varepsilon + 0.5 \cdot 0\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   metadata-eval [=>] 74.7	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + \color{blue}{0}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
   *-commutative [=>] 74.7	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]
   +-commutative [<=] 74.7	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \color{blue}{\left(\varepsilon + x\right)}\right)\right)\right) \]

4. Applied egg-rr 43.5%

   \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \color{blue}{{\left(\sqrt[3]{x + \left(x + \varepsilon\right)}\right)}^{3}}\right)\right) \]
   Step-by-step derivation

   [Start] 74.7	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right) \]
   add-cube-cbrt [=>] 42.9	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\left(\sqrt[3]{x + \left(\varepsilon + x\right)} \cdot \sqrt[3]{x + \left(\varepsilon + x\right)}\right) \cdot \sqrt[3]{x + \left(\varepsilon + x\right)}\right)}\right)\right) \]
   pow3 [=>] 43.5	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \color{blue}{{\left(\sqrt[3]{x + \left(\varepsilon + x\right)}\right)}^{3}}\right)\right) \]
   +-commutative [=>] 43.5	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot {\left(\sqrt[3]{x + \color{blue}{\left(x + \varepsilon\right)}}\right)}^{3}\right)\right) \]

5. Applied egg-rr 99.4%

   \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \left(x + x\right)\right) \cdot \cos \left(0.5 \cdot \varepsilon\right) + \cos \left(0.5 \cdot \left(x + x\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)}\right) \]
   Step-by-step derivation

   [Start] 43.5	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot {\left(\sqrt[3]{x + \left(x + \varepsilon\right)}\right)}^{3}\right)\right) \]
   unpow3 [=>] 42.9	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\left(\sqrt[3]{x + \left(x + \varepsilon\right)} \cdot \sqrt[3]{x + \left(x + \varepsilon\right)}\right) \cdot \sqrt[3]{x + \left(x + \varepsilon\right)}\right)}\right)\right) \]
   add-cube-cbrt [<=] 74.7	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)}\right)\right) \]
   associate-+r+ [=>] 74.6	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)}\right)\right) \]
   distribute-lft-in [=>] 74.6	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + x\right) + 0.5 \cdot \varepsilon\right)}\right) \]
   sin-sum [=>] 99.4	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \left(x + x\right)\right) \cdot \cos \left(0.5 \cdot \varepsilon\right) + \cos \left(0.5 \cdot \left(x + x\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)}\right) \]

6. Simplified 99.4%

   \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(0.5 \cdot \left(x + x\right)\right), \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x + x\right)\right)\right)}\right) \]
   Step-by-step derivation

   [Start] 99.4	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \left(\sin \left(0.5 \cdot \left(x + x\right)\right) \cdot \cos \left(0.5 \cdot \varepsilon\right) + \cos \left(0.5 \cdot \left(x + x\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right) \]
   fma-def [=>] 99.4	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(0.5 \cdot \left(x + x\right)\right), \cos \left(0.5 \cdot \varepsilon\right), \cos \left(0.5 \cdot \left(x + x\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)}\right) \]
   *-commutative [=>] 99.4	\[ -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \mathsf{fma}\left(\sin \left(0.5 \cdot \left(x + x\right)\right), \cos \left(0.5 \cdot \varepsilon\right), \color{blue}{\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x + x\right)\right)}\right)\right) \]

7. Taylor expanded in eps around inf 99.4%

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

8. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	33088

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

Alternative 2
Accuracy	99.0%
Cost	32776

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ t_1 := \cos x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-48}:\\ \;\;\;\;t_1 - \sin x \cdot \sin \varepsilon\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(t_0 + x \cdot \cos \left(0.5 \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\sin x, \sin \varepsilon, t_1\right)\\ \end{array} \]

Alternative 3
Accuracy	99.0%
Cost	26441

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{-48} \lor \neg \left(x \leq 2.25 \cdot 10^{-9}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(t_0 + x \cdot \cos \left(0.5 \cdot \varepsilon\right)\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	77.6%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.009 \lor \neg \left(\varepsilon \leq 0.0152\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 5
Accuracy	76.9%
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	76.9%
Cost	13632

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

Alternative 7
Accuracy	70.6%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-48} \lor \neg \left(x \leq 8.5 \cdot 10^{-29}\right):\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon \cdot \tan \left(\frac{-\varepsilon}{2}\right)\\ \end{array} \]

Alternative 8
Accuracy	67.3%
Cost	13449

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-72} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-36}\right):\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 9
Accuracy	67.4%
Cost	13449

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2 \cdot 10^{-73} \lor \neg \left(\varepsilon \leq 1.38 \cdot 10^{-36}\right):\\ \;\;\;\;\sin \varepsilon \cdot \tan \left(\frac{-\varepsilon}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 10
Accuracy	67.9%
Cost	13388

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.0078:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -6.5 \cdot 10^{-71}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Accuracy	67.1%
Cost	7052

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.0078:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -3.1 \cdot 10^{-72}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 2.25 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Accuracy	47.2%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0078 \lor \neg \left(\varepsilon \leq 2.4 \cdot 10^{-13}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 13
Accuracy	22.2%
Cost	320

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

Reproduce

herbie shell --seed 2023161 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))