2cos (problem 3.3.5)

Average Error: 39.5 → 0.5

Time: 22.1s

Precision: binary64

Cost: 47048

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{if}\;\varepsilon \leq -0.01587428569448531:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0008569044353900222:\\ \;\;\;\;\left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \left(-0.125 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \left(-0.020833333333333332 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right) + \left(\sin x + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma (sin x) (sin eps) (cos x))))
   (if (<= eps -0.01587428569448531)
     (fma (cos x) (cos eps) (- t_0))
     (if (<= eps 0.0008569044353900222)
       (*
        (*
         (sin (* (+ eps (- x x)) 0.5))
         (+
          (* -0.125 (* (sin x) (pow eps 2.0)))
          (+
           (* -0.020833333333333332 (* (cos x) (pow eps 3.0)))
           (+ (sin x) (* 0.5 (* eps (cos x)))))))
        -2.0)
       (- (* (cos x) (cos eps)) t_0)))))

C

double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}

↓

double code(double x, double eps) {
	double t_0 = fma(sin(x), sin(eps), cos(x));
	double tmp;
	if (eps <= -0.01587428569448531) {
		tmp = fma(cos(x), cos(eps), -t_0);
	} else if (eps <= 0.0008569044353900222) {
		tmp = (sin(((eps + (x - x)) * 0.5)) * ((-0.125 * (sin(x) * pow(eps, 2.0))) + ((-0.020833333333333332 * (cos(x) * pow(eps, 3.0))) + (sin(x) + (0.5 * (eps * cos(x))))))) * -2.0;
	} else {
		tmp = (cos(x) * cos(eps)) - t_0;
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	t_0 = fma(sin(x), sin(eps), cos(x))
	tmp = 0.0
	if (eps <= -0.01587428569448531)
		tmp = fma(cos(x), cos(eps), Float64(-t_0));
	elseif (eps <= 0.0008569044353900222)
		tmp = Float64(Float64(sin(Float64(Float64(eps + Float64(x - x)) * 0.5)) * Float64(Float64(-0.125 * Float64(sin(x) * (eps ^ 2.0))) + Float64(Float64(-0.020833333333333332 * Float64(cos(x) * (eps ^ 3.0))) + Float64(sin(x) + Float64(0.5 * Float64(eps * cos(x))))))) * -2.0);
	else
		tmp = Float64(Float64(cos(x) * cos(eps)) - t_0);
	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] + N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.01587428569448531], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + (-t$95$0)), $MachinePrecision], If[LessEqual[eps, 0.0008569044353900222], N[(N[(N[Sin[N[(N[(eps + N[(x - x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(-0.125 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.020833333333333332 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] + N[(0.5 * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.015874285694485311

   1. Initial program 30.2

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

   2. Applied egg-rr 0.8

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

   if -0.015874285694485311 < eps < 8.56904435390022233e-4

   1. Initial program 49.1

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

   2. Applied egg-rr 0.6

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

   3. Taylor expanded in eps around 0 0.1

      \[\leadsto \left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \color{blue}{\left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + \sin x\right)\right)\right)}\right) \cdot -2 \]

   if 8.56904435390022233e-4 < eps

   1. Initial program 30.4

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

   2. Applied egg-rr 61.2

      \[\leadsto \color{blue}{\sqrt{{\left(\cos \left(x + \varepsilon\right) - \cos x\right)}^{2}}} \]

   3. Applied egg-rr 0.8

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.01587428569448531:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0008569044353900222:\\ \;\;\;\;\left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \left(-0.125 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \left(-0.020833333333333332 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right) + \left(\sin x + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	45316

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{if}\;\varepsilon \leq -0.01587428569448531:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0008569044353900222:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) - \sin x \cdot \mathsf{fma}\left({\varepsilon}^{3}, -0.16666666666666666, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - t_0\\ \end{array} \]

Alternative 2
Error	0.5
Cost	39112

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.01587428569448531:\\ \;\;\;\;\left(t_0 - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0008569044353900222:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) - \sin x \cdot \mathsf{fma}\left({\varepsilon}^{3}, -0.16666666666666666, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \end{array} \]

Alternative 3
Error	0.5
Cost	39112

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.01587428569448531:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0008569044353900222:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) - \sin x \cdot \mathsf{fma}\left({\varepsilon}^{3}, -0.16666666666666666, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \end{array} \]

Alternative 4
Error	0.5
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.01587428569448531:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0008569044353900222:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) - \sin x \cdot \mathsf{fma}\left({\varepsilon}^{3}, -0.16666666666666666, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.5
Cost	32840

\[\begin{array}{l} t_0 := \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.01587428569448531:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0008569044353900222:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) - \sin x \cdot \mathsf{fma}\left({\varepsilon}^{3}, -0.16666666666666666, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	15.4
Cost	13888

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

Alternative 7
Error	15.2
Cost	13768

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0004093905845400458:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{elif}\;\varepsilon \leq 0.0008569044353900222:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 8
Error	15.2
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0004093905845400458:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{elif}\;\varepsilon \leq 0.0008569044353900222:\\ \;\;\;\;\varepsilon \cdot \left(\cos x \cdot \left(\varepsilon \cdot -0.5\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 9
Error	20.8
Cost	13448

\[\begin{array}{l} t_0 := \sin x \cdot \left(-\varepsilon\right)\\ \mathbf{if}\;x \leq -9.663103832920895 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.765655092283564 \cdot 10^{-33}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	21.1
Cost	13256

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.0004093905845400458:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.4478825126045238 \cdot 10^{-27}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	21.6
Cost	6920

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.0004093905845400458:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.4478825126045238 \cdot 10^{-27}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	34.0
Cost	6856

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.0004093905845400458:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0008569044353900222:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	50.7
Cost	320

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

Reproduce

herbie shell --seed 2022308 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))