2cos (problem 3.3.5)

Average Error: 39.4 → 0.4

Time: 14.5s

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 39.4

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

2. Applied egg-rr 14.9

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

3. Taylor expanded in eps around inf 14.9

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

4. Applied egg-rr 0.4

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

5. Taylor expanded in eps around inf 0.4

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

6. Simplified 0.4

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

7. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
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 2
Error	0.8
Cost	32840

\[\begin{array}{l} t_0 := \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\varepsilon \leq -8476.33423968978:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.84481990926427 \cdot 10^{-6}:\\ \;\;\;\;-2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.08333333333333333\right)\right) + \varepsilon \cdot \left(\left(\varepsilon \cdot \cos x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.020833333333333332 + 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.8
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -8476.33423968978:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5.84481990926427 \cdot 10^{-6}:\\ \;\;\;\;-2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.08333333333333333\right)\right) + \varepsilon \cdot \left(\left(\varepsilon \cdot \cos x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.020833333333333332 + 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \end{array} \]

Alternative 4
Error	14.1
Cost	26432

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

Alternative 5
Error	14.9
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8476.33423968978:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5.84481990926427 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.5\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \end{array} \]

Alternative 6
Error	14.9
Cost	13504

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

Alternative 7
Error	15.1
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8476.33423968978:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5.84481990926427 \cdot 10^{-6}:\\ \;\;\;\;-2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.08333333333333333\right) + \varepsilon \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \end{array} \]

Alternative 8
Error	15.1
Cost	13124

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8476.33423968978:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5.84481990926427 \cdot 10^{-6}:\\ \;\;\;\;-2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.08333333333333333\right) + \varepsilon \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 9
Error	15.3
Cost	7752

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -8476.33423968978:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.84481990926427 \cdot 10^{-6}:\\ \;\;\;\;-2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.08333333333333333\right) + \varepsilon \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	32.5
Cost	7120

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{if}\;\varepsilon \leq -7.182334796830594 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -4.639383753835457 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 8.095387675205318 \cdot 10^{-113}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 5.84481990926427 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	21.2
Cost	6920

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -5183697436676.89:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.84481990926427 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	48.4
Cost	584

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -5.992340108701455 \cdot 10^{-82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.657910631795708 \cdot 10^{-101}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	52.3
Cost	256

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

Alternative 14
Error	55.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022228 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))