2cos (problem 3.3.5)

Average Error: 39.8 → 0.5

Time: 17.5s

Precision: binary64

Cost: 39176

Math

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

↓

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0053:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0054:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\cos x\right) - t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))))
   (if (<= eps -0.0053)
     (- (* (cos x) (+ -1.0 (cos eps))) t_0)
     (if (<= eps 0.0054)
       (-
        (*
         (cos x)
         (+ (* -0.5 (* eps eps)) (* 0.041666666666666664 (pow eps 4.0))))
        t_0)
       (- (fma (cos x) (cos eps) (- (cos x))) t_0)))))

C

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

↓

double code(double x, double eps) {
	double t_0 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.0053) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - t_0;
	} else if (eps <= 0.0054) {
		tmp = (cos(x) * ((-0.5 * (eps * eps)) + (0.041666666666666664 * pow(eps, 4.0)))) - t_0;
	} else {
		tmp = fma(cos(x), cos(eps), -cos(x)) - t_0;
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	t_0 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.0053)
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - t_0);
	elseif (eps <= 0.0054)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.5 * Float64(eps * eps)) + Float64(0.041666666666666664 * (eps ^ 4.0)))) - t_0);
	else
		tmp = Float64(fma(cos(x), cos(eps), Float64(-cos(x))) - 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[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0053], N[(N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[eps, 0.0054], 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] - t$95$0), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + (-N[Cos[x], $MachinePrecision])), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00530000000000000002

   1. Initial program 30.4

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

   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

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

   if -0.00530000000000000002 < eps < 0.0054000000000000003

   1. Initial program 49.6

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

   2. Applied egg-rr 12.3

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

   3. Taylor expanded in x around inf 48.8

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

   4. Simplified 12.3

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

      [Start] 48.8	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x \]
      +-commutative [=>] 48.8	\[ \color{blue}{\left(\cos \varepsilon \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
      *-commutative [=>] 48.8	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right) - \cos x \]
      *-commutative [<=] 48.8	\[ \left(\cos x \cdot \cos \varepsilon + -1 \cdot \color{blue}{\left(\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      mul-1-neg [=>] 48.8	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(-\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      sub0-neg [<=] 48.8	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(0 - \sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      associate-+r- [=>] 48.8	\[ \color{blue}{\left(\left(\cos x \cdot \cos \varepsilon + 0\right) - \sin \varepsilon \cdot \sin x\right)} - \cos x \]
      +-rgt-identity [=>] 48.8	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin \varepsilon \cdot \sin x\right) - \cos x \]
      associate--r+ [<=] 48.8	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin \varepsilon \cdot \sin x + \cos x\right)} \]
      +-commutative [<=] 48.8	\[ \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin \varepsilon \cdot \sin x\right)} \]
      associate--r+ [=>] 12.3	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x} \]

   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 0.0054000000000000003 < eps

   1. Initial program 29.6

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

   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. Taylor expanded in x around inf 0.8

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

   4. Simplified 0.8

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

      [Start] 0.8	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x \]
      +-commutative [=>] 0.8	\[ \color{blue}{\left(\cos \varepsilon \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
      *-commutative [=>] 0.8	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right) - \cos x \]
      *-commutative [<=] 0.8	\[ \left(\cos x \cdot \cos \varepsilon + -1 \cdot \color{blue}{\left(\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      mul-1-neg [=>] 0.8	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(-\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      sub0-neg [<=] 0.8	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(0 - \sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      associate-+r- [=>] 0.8	\[ \color{blue}{\left(\left(\cos x \cdot \cos \varepsilon + 0\right) - \sin \varepsilon \cdot \sin x\right)} - \cos x \]
      +-rgt-identity [=>] 0.8	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin \varepsilon \cdot \sin x\right) - \cos x \]
      associate--r+ [<=] 0.8	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin \varepsilon \cdot \sin x + \cos x\right)} \]
      +-commutative [<=] 0.8	\[ \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin \varepsilon \cdot \sin x\right)} \]
      associate--r+ [=>] 0.8	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x} \]

   5. Taylor expanded in eps around inf 0.8

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

   6. Simplified 0.8

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

      [Start] 0.8	\[ \left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin x \cdot \sin \varepsilon \]
      fma-neg [=>] 0.8	\[ \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\cos x\right)} - \sin x \cdot \sin \varepsilon \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0053:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.0054:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\cos x\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	39168

\[\frac{{\sin \varepsilon}^{2}}{\frac{-1 - \cos \varepsilon}{\cos x}} - \sin \varepsilon \cdot \sin x \]

Alternative 2
Error	0.5
Cost	26889

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0053 \lor \neg \left(\varepsilon \leq 0.0054\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \end{array} \]

Alternative 3
Error	0.5
Cost	26441

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-8} \lor \neg \left(x \leq 8.6 \cdot 10^{-8}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(x \cdot \cos \left(\varepsilon \cdot 0.5\right) + \sin \left(\varepsilon \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	15.0
Cost	13888

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

Alternative 5
Error	14.6
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0062 \lor \neg \left(\varepsilon \leq 0.00042\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 6
Error	14.8
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0058 \lor \neg \left(\varepsilon \leq 0.00042\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 7
Error	21.5
Cost	7252

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -0.0058:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 8 \cdot 10^{-145}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 2.15 \cdot 10^{-48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 0.00019:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	15.3
Cost	7241

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0058 \lor \neg \left(\varepsilon \leq 0.00042\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 9
Error	48.9
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -12500000000 \lor \neg \left(\varepsilon \leq 370000000\right):\\ \;\;\;\;1 - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 10
Error	34.0
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0058 \lor \neg \left(\varepsilon \leq 0.00019\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 11
Error	50.5
Cost	320

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

Alternative 12
Error	50.5
Cost	320

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

Reproduce

herbie shell --seed 2023039 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))