2cos (problem 3.3.5)

Average Error: 61.67% → 0.79%

Time: 17.4s

Precision: binary64

Cost: 39176

Math

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

↓

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00013:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \cos x\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 0.000165:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))))
   (if (<= eps -0.00013)
     (- (- (* (cos eps) (cos x)) (cos x)) t_0)
     (if (<= eps 0.000165)
       (- (* (* eps eps) (* (cos x) -0.5)) t_0)
       (- (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 t_0 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.00013) {
		tmp = ((cos(eps) * cos(x)) - cos(x)) - t_0;
	} else if (eps <= 0.000165) {
		tmp = ((eps * eps) * (cos(x) * -0.5)) - t_0;
	} 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)
	t_0 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.00013)
		tmp = Float64(Float64(Float64(cos(eps) * cos(x)) - cos(x)) - t_0);
	elseif (eps <= 0.000165)
		tmp = Float64(Float64(Float64(eps * eps) * Float64(cos(x) * -0.5)) - t_0);
	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_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.00013], N[(N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[eps, 0.000165], N[(N[(N[(eps * eps), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - t$95$0), $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 < -1.29999999999999989e-4

   1. Initial program 46.92

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

   2. Applied egg-rr 1.27

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

   3. Taylor expanded in x around inf 1.29

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

   4. Simplified 1.27

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

      [Start] 1.29	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x \]
      +-commutative [=>] 1.29	\[ \color{blue}{\left(\cos \varepsilon \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
      *-commutative [=>] 1.29	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right) - \cos x \]
      *-commutative [<=] 1.29	\[ \left(\cos x \cdot \cos \varepsilon + -1 \cdot \color{blue}{\left(\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      mul-1-neg [=>] 1.29	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(-\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      sub0-neg [<=] 1.29	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(0 - \sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      associate-+r- [=>] 1.29	\[ \color{blue}{\left(\left(\cos x \cdot \cos \varepsilon + 0\right) - \sin \varepsilon \cdot \sin x\right)} - \cos x \]
      +-rgt-identity [=>] 1.29	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin \varepsilon \cdot \sin x\right) - \cos x \]
      associate--r+ [<=] 1.35	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin \varepsilon \cdot \sin x + \cos x\right)} \]
      +-commutative [<=] 1.35	\[ \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin \varepsilon \cdot \sin x\right)} \]
      associate--r+ [=>] 1.27	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x} \]

   if -1.29999999999999989e-4 < eps < 1.65e-4

   1. Initial program 76.32

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

   2. Applied egg-rr 18.63

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

   3. Taylor expanded in eps around 0 0.29

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

   4. Simplified 0.29

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

      [Start] 0.29	\[ -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      *-commutative [=>] 0.29	\[ \color{blue}{\left({\varepsilon}^{2} \cdot \cos x\right) \cdot -0.5} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      associate-*l* [=>] 0.29	\[ \color{blue}{{\varepsilon}^{2} \cdot \left(\cos x \cdot -0.5\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      unpow2 [=>] 0.29	\[ \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\cos x \cdot -0.5\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]

   if 1.65e-4 < eps

   1. Initial program 48.3

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

   2. Applied egg-rr 1.27

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

4. Final simplification 0.79

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00013:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.000165:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \]

Alternatives

Alternative 1
Error	1.01%
Cost	39168

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

Alternative 2
Error	0.79%
Cost	32708

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00014:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \cos x\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 0.000165:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \end{array} \]

Alternative 3
Error	0.79%
Cost	26441

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00013 \lor \neg \left(\varepsilon \leq 0.000165\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - t_0\\ \end{array} \]

Alternative 4
Error	0.82%
Cost	26440

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00013:\\ \;\;\;\;\frac{\cos x}{\frac{1}{t_0}} - t_1\\ \mathbf{elif}\;\varepsilon \leq 0.000165:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot t_0 - t_1\\ \end{array} \]

Alternative 5
Error	22.1%
Cost	26313

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0075 \lor \neg \left(\varepsilon \leq 0.0225\right):\\ \;\;\;\;\left(\cos \varepsilon - \cos x\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - t_0\\ \end{array} \]

Alternative 6
Error	24.06%
Cost	13888

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

Alternative 7
Error	23.4%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.023 \lor \neg \left(\varepsilon \leq 0.011\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 8
Error	23.69%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0037 \lor \neg \left(\varepsilon \leq 0.00225\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 9
Error	24.4%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0033 \lor \neg \left(\varepsilon \leq 0.009\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 10
Error	34.18%
Cost	6988

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -7.8 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{-78}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000165:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	52.92%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00013 \lor \neg \left(\varepsilon \leq 0.000165\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \]

Alternative 12
Error	78.77%
Cost	320

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

Alternative 13
Error	86.94%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023125 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))