2cos (problem 3.3.5)

Average Accuracy: 38.4% → 99.2%

Time: 21.4s

Precision: binary64

Cost: 40008

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.003:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0032:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.003)
   (fma (cos x) (cos eps) (- (- (cos x)) (* (sin x) (sin eps))))
   (if (<= eps 0.0032)
     (fma
      0.041666666666666664
      (* (cos x) (pow eps 4.0))
      (+
       (* (cos x) (* -0.5 (* eps eps)))
       (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps))))
     (- (fma (cos x) (cos eps) (* (sin eps) (- (sin x)))) (cos x)))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.003) {
		tmp = fma(cos(x), cos(eps), (-cos(x) - (sin(x) * sin(eps))));
	} else if (eps <= 0.0032) {
		tmp = fma(0.041666666666666664, (cos(x) * pow(eps, 4.0)), ((cos(x) * (-0.5 * (eps * eps))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps))));
	} else {
		tmp = fma(cos(x), cos(eps), (sin(eps) * -sin(x))) - cos(x);
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.003)
		tmp = fma(cos(x), cos(eps), Float64(Float64(-cos(x)) - Float64(sin(x) * sin(eps))));
	elseif (eps <= 0.0032)
		tmp = fma(0.041666666666666664, Float64(cos(x) * (eps ^ 4.0)), Float64(Float64(cos(x) * Float64(-0.5 * Float64(eps * eps))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps))));
	else
		tmp = Float64(fma(cos(x), cos(eps), Float64(sin(eps) * Float64(-sin(x)))) - 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_] := If[LessEqual[eps, -0.003], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[((-N[Cos[x], $MachinePrecision]) - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0032], N[(0.041666666666666664 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0030000000000000001

   1. Initial program 54.0%

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

   2. Applied egg-rr 98.7%

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

      [Start] 54.0	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      sub-neg [=>] 54.0	\[ \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]
      cos-sum [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} + \left(-\cos x\right) \]
      associate-+l- [=>] 98.7	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)} \]
      fma-neg [=>] 98.7	\[ \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)\right)} \]

   if -0.0030000000000000001 < eps < 0.00320000000000000015

   1. Initial program 22.5%

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

   2. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(\left(-\varepsilon\right) + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right)} \]
      Proof

      [Start] 99.8	\[ 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right) \]
      fma-def [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4} \cdot \cos x, 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)} \]
      *-commutative [=>] 99.8	\[ \mathsf{fma}\left(0.041666666666666664, \color{blue}{\cos x \cdot {\varepsilon}^{4}}, 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right) \]
      +-commutative [=>] 99.8	\[ \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)}\right) \]
      associate-+l+ [=>] 99.8	\[ \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)}\right) \]
      associate-*r* [=>] 99.8	\[ \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \color{blue}{\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x} + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right) \]
      *-commutative [=>] 99.8	\[ \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)} + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right) \]
      unpow2 [=>] 99.8	\[ \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \cos x \cdot \left(-0.5 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right) \]
      associate-*r* [=>] 99.8	\[ \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \left(\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right) \]
      associate-*r* [=>] 99.8	\[ \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \left(\left(-1 \cdot \varepsilon\right) \cdot \sin x + \color{blue}{\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x}\right)\right) \]
      distribute-rgt-out [=>] 99.8	\[ \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \color{blue}{\sin x \cdot \left(-1 \cdot \varepsilon + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)}\right) \]
      mul-1-neg [=>] 99.8	\[ \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(\color{blue}{\left(-\varepsilon\right)} + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right) \]

   if 0.00320000000000000015 < eps

   1. Initial program 52.9%

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

   2. Applied egg-rr 98.7%

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

      [Start] 52.9	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      cos-sum [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]
      cancel-sign-sub-inv [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]
      fma-def [=>] 98.7	\[ \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.003:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0032:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	39176

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000145:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000175:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	39176

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000145:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000175:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \]

Alternative 3
Accuracy	99.1%
Cost	32841

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000145 \lor \neg \left(\varepsilon \leq 0.000175\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right)\\ \end{array} \]

Alternative 4
Accuracy	99.1%
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.000145:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000175:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \end{array} \]

Alternative 5
Accuracy	77.0%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0068 \lor \neg \left(\varepsilon \leq 0.0062\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 6
Accuracy	76.5%
Cost	13632

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

Alternative 7
Accuracy	76.5%
Cost	13632

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

Alternative 8
Accuracy	76.7%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0028 \lor \neg \left(\varepsilon \leq 0.0033\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right) \cdot \left(\varepsilon \cdot 0.5\right)\\ \end{array} \]

Alternative 9
Accuracy	76.1%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0027 \lor \neg \left(\varepsilon \leq 0.0043\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right) \cdot \left(\varepsilon \cdot 0.5\right)\\ \end{array} \]

Alternative 10
Accuracy	49.6%
Cost	7120

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.000145:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -4.3 \cdot 10^{-157}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-153}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000175:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Accuracy	67.3%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 1.35 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 12
Accuracy	23.7%
Cost	717

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-76} \lor \neg \left(x \leq 5 \cdot 10^{-81}\right) \land x \leq 95000:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 13
Accuracy	23.7%
Cost	716

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-81}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 23500:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 14
Accuracy	17.0%
Cost	256

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

Reproduce

herbie shell --seed 2023151 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))