2cos (problem 3.3.5)

Average Error: 39.7 → 0.5

Time: 20.4s

Precision: binary64

Cost: 39432

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0052:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0045:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \frac{\sin \varepsilon \cdot {\sin x}^{2}}{\sin x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0052)
   (- (fma (cos x) (cos eps) (* (sin eps) (- (sin x)))) (cos x))
   (if (<= eps 0.0045)
     (-
      (*
       (cos x)
       (+ (* -0.5 (* eps eps)) (* 0.041666666666666664 (pow eps 4.0))))
      (* (sin eps) (sin x)))
     (-
      (* (cos x) (+ (cos eps) -1.0))
      (/ (* (sin eps) (pow (sin x) 2.0)) (sin x))))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0052) {
		tmp = fma(cos(x), cos(eps), (sin(eps) * -sin(x))) - cos(x);
	} else if (eps <= 0.0045) {
		tmp = (cos(x) * ((-0.5 * (eps * eps)) + (0.041666666666666664 * pow(eps, 4.0)))) - (sin(eps) * sin(x));
	} else {
		tmp = (cos(x) * (cos(eps) + -1.0)) - ((sin(eps) * pow(sin(x), 2.0)) / sin(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 <= -0.0052)
		tmp = Float64(fma(cos(x), cos(eps), Float64(sin(eps) * Float64(-sin(x)))) - cos(x));
	elseif (eps <= 0.0045)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.5 * Float64(eps * eps)) + Float64(0.041666666666666664 * (eps ^ 4.0)))) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - Float64(Float64(sin(eps) * (sin(x) ^ 2.0)) / sin(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, -0.0052], 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], If[LessEqual[eps, 0.0045], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[eps], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0051999999999999998

   1. Initial program 30.4

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

   2. Applied egg-rr 0.8

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

   if -0.0051999999999999998 < eps < 0.00449999999999999966

   1. Initial program 48.9

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

   2. Applied egg-rr 11.8

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

   3. Taylor expanded in x around inf 48.3

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x} \]

   4. Simplified 11.8

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

      [Start] 48.3	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x \]
      +-commutative [=>] 48.3	\[ \color{blue}{\left(\cos \varepsilon \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
      *-commutative [=>] 48.3	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right) - \cos x \]
      *-commutative [<=] 48.3	\[ \left(\cos x \cdot \cos \varepsilon + -1 \cdot \color{blue}{\left(\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      mul-1-neg [=>] 48.3	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(-\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      sub0-neg [<=] 48.3	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(0 - \sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      associate-+r- [=>] 48.3	\[ \color{blue}{\left(\left(\cos x \cdot \cos \varepsilon + 0\right) - \sin \varepsilon \cdot \sin x\right)} - \cos x \]
      +-rgt-identity [=>] 48.3	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin \varepsilon \cdot \sin x\right) - \cos x \]
      associate--r+ [<=] 48.3	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin \varepsilon \cdot \sin x + \cos x\right)} \]
      +-commutative [<=] 48.3	\[ \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin \varepsilon \cdot \sin x\right)} \]
      associate--r+ [=>] 11.8	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x} \]

   5. Taylor expanded in eps around 0 0.1

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

   6. Simplified 0.1

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

      [Start] 0.1	\[ \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) - \sin x \cdot \sin \varepsilon \]
      +-commutative [=>] 0.1	\[ \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right)} - \sin x \cdot \sin \varepsilon \]
      associate-*r* [=>] 0.1	\[ \left(\color{blue}{\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) - \sin x \cdot \sin \varepsilon \]
      associate-*r* [=>] 0.1	\[ \left(\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x + \color{blue}{\left(0.041666666666666664 \cdot {\varepsilon}^{4}\right) \cdot \cos x}\right) - \sin x \cdot \sin \varepsilon \]
      distribute-rgt-out [=>] 0.1	\[ \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)} - \sin x \cdot \sin \varepsilon \]
      unpow2 [=>] 0.1	\[ \cos x \cdot \left(-0.5 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin x \cdot \sin \varepsilon \]

   if 0.00449999999999999966 < eps

   1. Initial program 30.3

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

   2. Applied egg-rr 0.8

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

   3. Applied egg-rr 0.8

      \[\leadsto \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]

   4. Applied egg-rr 0.8

      \[\leadsto \cos x \cdot \left(\cos \varepsilon + -1\right) + \color{blue}{\frac{\left(0 - {\sin x}^{2}\right) \cdot \sin \varepsilon}{\sin x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0052:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0045:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \frac{\sin \varepsilon \cdot {\sin x}^{2}}{\sin x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	39296

\[\cos x \cdot \left({\sin \varepsilon}^{2} \cdot \frac{-1}{1 + \cos \varepsilon}\right) - \sin \varepsilon \cdot \sin x \]

Alternative 2
Error	0.5
Cost	39044

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0052:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.005:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - t_0\\ \end{array} \]

Alternative 3
Error	0.5
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0052:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0055:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 - \cos x\right) - t_1\\ \end{array} \]

Alternative 4
Error	0.5
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0052:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0045:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 - \cos x\right) - t_1\\ \end{array} \]

Alternative 5
Error	0.5
Cost	32708

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0052:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0045:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \end{array} \]

Alternative 6
Error	0.5
Cost	26889

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0052 \lor \neg \left(\varepsilon \leq 0.0045\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \end{array} \]

Alternative 7
Error	0.7
Cost	26441

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -9 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 0.00031\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right) \cdot -2\\ \end{array} \]

Alternative 8
Error	14.6
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.031 \lor \neg \left(\varepsilon \leq 0.0003\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 9
Error	15.0
Cost	13632

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

Alternative 10
Error	14.8
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.016 \lor \neg \left(\varepsilon \leq 0.0003\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	15.2
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.031 \lor \neg \left(\varepsilon \leq 0.0003\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	20.9
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0022 \lor \neg \left(\varepsilon \leq 1.16 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 13
Error	33.9
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0052 \lor \neg \left(\varepsilon \leq 0.0045\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-0.16666666666666666 + \frac{-2}{\varepsilon \cdot \varepsilon}}\\ \end{array} \]

Alternative 14
Error	46.9
Cost	576

\[\frac{1}{-0.16666666666666666 + \frac{-2}{\varepsilon \cdot \varepsilon}} \]

Alternative 15
Error	50.2
Cost	320

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

Alternative 16
Error	55.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))