2cos (problem 3.3.5)

Average Error: 38.9 → 0.8

Time: 19.1s

Precision: binary64

Cost: 39176

Math

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

↓

\[\begin{array}{l} t_0 := -\sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0019391995777601862:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, t_0 - \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.002847402336945821:\\ \;\;\;\;\left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, t_0\right) - \cos x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (* (sin x) (sin eps)))))
   (if (<= eps -0.0019391995777601862)
     (fma (cos x) (cos eps) (- t_0 (cos x)))
     (if (<= eps 0.002847402336945821)
       (* (* (sin (* (+ eps (- x x)) 0.5)) (sin (* 0.5 (+ x (+ eps x))))) -2.0)
       (- (fma (cos x) (cos 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(x) * sin(eps));
	double tmp;
	if (eps <= -0.0019391995777601862) {
		tmp = fma(cos(x), cos(eps), (t_0 - cos(x)));
	} else if (eps <= 0.002847402336945821) {
		tmp = (sin(((eps + (x - x)) * 0.5)) * sin((0.5 * (x + (eps + x))))) * -2.0;
	} else {
		tmp = fma(cos(x), cos(eps), t_0) - cos(x);
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	t_0 = Float64(-Float64(sin(x) * sin(eps)))
	tmp = 0.0
	if (eps <= -0.0019391995777601862)
		tmp = fma(cos(x), cos(eps), Float64(t_0 - cos(x)));
	elseif (eps <= 0.002847402336945821)
		tmp = Float64(Float64(sin(Float64(Float64(eps + Float64(x - x)) * 0.5)) * sin(Float64(0.5 * Float64(x + Float64(eps + x))))) * -2.0);
	else
		tmp = Float64(fma(cos(x), cos(eps), t_0) - cos(x));
	end
	return tmp
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[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[eps, -0.0019391995777601862], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(t$95$0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.002847402336945821], N[(N[(N[Sin[N[(N[(eps + N[(x - x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * N[(x + N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + t$95$0), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00193919957776018622

   1. Initial program 30.5

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

   2. Applied egg-rr 0.9

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

   3. Applied egg-rr 0.9

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

   if -0.00193919957776018622 < eps < 0.0028474023369458208

   1. Initial program 48.3

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

   2. Applied egg-rr 0.6

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

   if 0.0028474023369458208 < eps

   1. Initial program 29.3

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

   2. Applied egg-rr 46.6

      \[\leadsto \color{blue}{e^{\log \cos \left(x + \varepsilon\right)}} - \cos x \]

   3. Applied egg-rr 0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0019391995777601862:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.002847402336945821:\\ \;\;\;\;\left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	39176

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)\\ \mathbf{if}\;\varepsilon \leq -0.0019391995777601862:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.002847402336945821:\\ \;\;\;\;\left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.8
Cost	32840

\[\begin{array}{l} t_0 := \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.0019391995777601862:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.002847402336945821:\\ \;\;\;\;\left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	14.9
Cost	13640

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.9303548950018607:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.1268120518731243 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x \cdot \left(\varepsilon \cdot -0.5\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	15.1
Cost	13632

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

Alternative 5
Error	21.6
Cost	13448

\[\begin{array}{l} t_0 := -2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{if}\;\varepsilon \leq -1.1063543140823851 \cdot 10^{-53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.904703325838417 \cdot 10^{-111}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	20.7
Cost	13388

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.001140205663377386:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.904703325838417 \cdot 10^{-111}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.1268120518731243 \cdot 10^{-6}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2} - \varepsilon \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	21.1
Cost	7308

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.001140205663377386:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.904703325838417 \cdot 10^{-111}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.1268120518731243 \cdot 10^{-6}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2} - \varepsilon \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	21.1
Cost	7180

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.001140205663377386:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.904703325838417 \cdot 10^{-111}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.1268120518731243 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(-0.5, \varepsilon, -x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	21.6
Cost	6988

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.001140205663377386:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.904703325838417 \cdot 10^{-111}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.1268120518731243 \cdot 10^{-6}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	33.3
Cost	6856

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.001140205663377386:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.1268120518731243 \cdot 10^{-6}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	50.3
Cost	320

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

Alternative 12
Error	55.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022300 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))