2cos (problem 3.3.5)

Average Accuracy: 39.1% → 99.2%

Time: 23.3s

Precision: binary64

Cost: 39177

Math

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

↓

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0045 \lor \neg \left(\varepsilon \leq 0.005\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\cos x\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))))
   (if (or (<= eps -0.0045) (not (<= eps 0.005)))
     (- (fma (cos x) (cos eps) (- (cos x))) t_0)
     (-
      (* (cos x) (fma 0.041666666666666664 (pow eps 4.0) (* -0.5 (* eps 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 = sin(eps) * sin(x);
	double tmp;
	if ((eps <= -0.0045) || !(eps <= 0.005)) {
		tmp = fma(cos(x), cos(eps), -cos(x)) - t_0;
	} else {
		tmp = (cos(x) * fma(0.041666666666666664, pow(eps, 4.0), (-0.5 * (eps * 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 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if ((eps <= -0.0045) || !(eps <= 0.005))
		tmp = Float64(fma(cos(x), cos(eps), Float64(-cos(x))) - t_0);
	else
		tmp = Float64(Float64(cos(x) * fma(0.041666666666666664, (eps ^ 4.0), Float64(-0.5 * Float64(eps * 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[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -0.0045], N[Not[LessEqual[eps, 0.005]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + (-N[Cos[x], $MachinePrecision])), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision] + N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00449999999999999966 or 0.0050000000000000001 < eps

   1. Initial program 54.2%

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

   2. Applied egg-rr 98.7%

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

      [Start] 54.2	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      sub-neg [=>] 54.2	\[ \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]
      +-commutative [=>] 54.2	\[ \color{blue}{\left(-\cos x\right) + \cos \left(x + \varepsilon\right)} \]
      cos-sum [=>] 98.7	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} \]
      cancel-sign-sub-inv [=>] 98.7	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} \]
      associate-+r+ [=>] 98.7	\[ \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \left(-\sin x\right) \cdot \sin \varepsilon} \]
      *-commutative [=>] 98.7	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} \]

   3. Applied egg-rr 98.7%

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

      [Start] 98.7	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      +-commutative [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      *-commutative [=>] 98.7	\[ \left(\color{blue}{\cos \varepsilon \cdot \cos x} + \left(-\cos x\right)\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      neg-mul-1 [=>] 98.7	\[ \left(\cos \varepsilon \cdot \cos x + \color{blue}{-1 \cdot \cos x}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      distribute-rgt-out [=>] 98.7	\[ \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]

   4. Applied egg-rr 98.7%

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

      [Start] 98.7	\[ \cos x \cdot \left(\cos \varepsilon + -1\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      distribute-rgt-in [=>] 98.7	\[ \color{blue}{\left(\cos \varepsilon \cdot \cos x + -1 \cdot \cos x\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      *-commutative [=>] 98.7	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} + -1 \cdot \cos x\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      fma-def [=>] 98.7	\[ \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -1 \cdot \cos x\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      mul-1-neg [=>] 98.7	\[ \mathsf{fma}\left(\cos x, \cos \varepsilon, \color{blue}{-\cos x}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]

   if -0.00449999999999999966 < eps < 0.0050000000000000001

   1. Initial program 23.3%

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

   2. Applied egg-rr 81.1%

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

      [Start] 23.3	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      sub-neg [=>] 23.3	\[ \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]
      +-commutative [=>] 23.3	\[ \color{blue}{\left(-\cos x\right) + \cos \left(x + \varepsilon\right)} \]
      cos-sum [=>] 24.3	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} \]
      cancel-sign-sub-inv [=>] 24.3	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} \]
      associate-+r+ [=>] 81.1	\[ \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \left(-\sin x\right) \cdot \sin \varepsilon} \]
      *-commutative [=>] 81.1	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} \]

   3. Applied egg-rr 81.1%

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

      [Start] 81.1	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      +-commutative [=>] 81.1	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      *-commutative [=>] 81.1	\[ \left(\color{blue}{\cos \varepsilon \cdot \cos x} + \left(-\cos x\right)\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      neg-mul-1 [=>] 81.1	\[ \left(\cos \varepsilon \cdot \cos x + \color{blue}{-1 \cdot \cos x}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      distribute-rgt-out [=>] 81.1	\[ \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]

   4. Taylor expanded in eps around 0 99.7%

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

   5. Simplified 99.7%

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

      [Start] 99.7	\[ \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      fma-def [=>] 99.7	\[ \cos x \cdot \color{blue}{\mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot {\varepsilon}^{2}\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      unpow2 [=>] 99.7	\[ \cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0045 \lor \neg \left(\varepsilon \leq 0.005\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.9%
Cost	39296

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

Alternative 2
Accuracy	99.0%
Cost	39168

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

Alternative 3
Accuracy	99.2%
Cost	33160

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0045:\\ \;\;\;\;\frac{\cos x}{\frac{1}{\cos \varepsilon + -1}} - t_0\\ \mathbf{elif}\;\varepsilon \leq 0.005:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - t_0\\ \end{array} \]

Alternative 4
Accuracy	99.0%
Cost	32840

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\cos x}{\frac{1}{\cos \varepsilon + -1}} - t_0\\ \mathbf{elif}\;\varepsilon \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - t_0\\ \end{array} \]

Alternative 5
Accuracy	98.9%
Cost	26696

\[\begin{array}{l} t_0 := \frac{1}{\cos \varepsilon + -1}\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\cos x}{t_0} - t_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-54}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \frac{1}{t_0} - t_1\\ \end{array} \]

Alternative 6
Accuracy	98.9%
Cost	26569

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-31} \lor \neg \left(x \leq 8.5 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\cos x}{\frac{1}{\cos \varepsilon + -1}} - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	98.9%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;x \leq -1.52 \cdot 10^{-33} \lor \neg \left(x \leq 8.5 \cdot 10^{-52}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	72.6%
Cost	13900

\[\begin{array}{l} t_0 := \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(\sin x \cdot -2\right)\\ \mathbf{if}\;x \leq -6 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-124}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{elif}\;x \leq 0.0006:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) - x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	76.9%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0029:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 10
Accuracy	70.8%
Cost	13644

\[\begin{array}{l} t_0 := \sin x \cdot \left(-\varepsilon\right)\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-125}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{elif}\;x \leq 0.0026:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) - x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Accuracy	76.7%
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 12
Accuracy	66.6%
Cost	13516

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-14}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq -6.8 \cdot 10^{-87}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0036:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos x \cdot \cos \varepsilon\\ \end{array} \]

Alternative 13
Accuracy	69.0%
Cost	13449

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-27} \lor \neg \left(x \leq 3.7 \cdot 10^{-15}\right):\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 14
Accuracy	67.0%
Cost	13388

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -2.2 \cdot 10^{-84}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0029:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Accuracy	66.3%
Cost	7052

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.65 \cdot 10^{-84}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0029:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Accuracy	47.4%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-14} \lor \neg \left(\varepsilon \leq 0.0029\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 17
Accuracy	21.0%
Cost	320

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

Reproduce

herbie shell --seed 2023137 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))