2cos (problem 3.3.5)

Average Error: 39.7 → 0.5

Time: 18.1s

Precision: binary64

Cost: 39044

Math

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

↓

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -0.0054:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, t_0\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0052:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, t_0\right)\\ \end{array} \]

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 -0.0054)
     (- (fma (cos x) (cos eps) t_0) (cos x))
     (if (<= eps 0.0052)
       (-
        (*
         (cos x)
         (+ (* -0.5 (* eps eps)) (* 0.041666666666666664 (pow eps 4.0))))
        (* (sin eps) (sin x)))
       (fma (+ -1.0 (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 <= -0.0054) {
		tmp = fma(cos(x), cos(eps), t_0) - cos(x);
	} else if (eps <= 0.0052) {
		tmp = (cos(x) * ((-0.5 * (eps * eps)) + (0.041666666666666664 * pow(eps, 4.0)))) - (sin(eps) * sin(x));
	} else {
		tmp = fma((-1.0 + 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) * Float64(-sin(x)))
	tmp = 0.0
	if (eps <= -0.0054)
		tmp = Float64(fma(cos(x), cos(eps), t_0) - cos(x));
	elseif (eps <= 0.0052)
		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 = fma(Float64(-1.0 + cos(eps)), 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, -0.0054], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + t$95$0), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0052], 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[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0054000000000000003

   1. Initial program 30.0

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

   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 -0.0054000000000000003 < eps < 0.0051999999999999998

   1. Initial program 49.0

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

   2. Applied egg-rr 11.5

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

   3. Taylor expanded in x around inf 48.4

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

   4. Simplified 11.5

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

      [Start] 48.4	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x \]
      +-commutative [=>] 48.4	\[ \color{blue}{\left(\cos \varepsilon \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
      *-commutative [=>] 48.4	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right) - \cos x \]
      *-commutative [<=] 48.4	\[ \left(\cos x \cdot \cos \varepsilon + -1 \cdot \color{blue}{\left(\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      mul-1-neg [=>] 48.4	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(-\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      sub0-neg [<=] 48.4	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(0 - \sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      associate-+r- [=>] 48.4	\[ \color{blue}{\left(\left(\cos x \cdot \cos \varepsilon + 0\right) - \sin \varepsilon \cdot \sin x\right)} - \cos x \]
      +-rgt-identity [=>] 48.4	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin \varepsilon \cdot \sin x\right) - \cos x \]
      associate--r+ [<=] 48.4	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin \varepsilon \cdot \sin x + \cos x\right)} \]
      +-commutative [<=] 48.4	\[ \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin \varepsilon \cdot \sin x\right)} \]
      associate--r+ [=>] 11.5	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x} \]

   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 0.0051999999999999998 < eps

   1. Initial program 30.8

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

   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. Applied egg-rr 0.8

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

   6. Applied egg-rr 0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin x \cdot \left(-\sin \varepsilon\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0054:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0052:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	39232

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

Alternative 2
Error	0.8
Cost	39168

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

Alternative 3
Error	0.5
Cost	32777

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0054 \lor \neg \left(\varepsilon \leq 0.0052\right):\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 4
Error	0.5
Cost	32776

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0054:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0052:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternative 5
Error	0.5
Cost	32776

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0054:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - t_0\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0052:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternative 6
Error	0.5
Cost	26889

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0054 \lor \neg \left(\varepsilon \leq 0.0052\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \end{array} \]

Alternative 7
Error	0.5
Cost	26441

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000166 \lor \neg \left(\varepsilon \leq 0.000175\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5 + \varepsilon \cdot \left(\sin x \cdot 0.16666666666666666\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 8
Error	15.1
Cost	13888

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

Alternative 9
Error	14.9
Cost	13768

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.28:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0034:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]

Alternative 10
Error	15.1
Cost	13124

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.28:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0031:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]

Alternative 11
Error	15.3
Cost	7241

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.28 \lor \neg \left(\varepsilon \leq 0.0042\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 12
Error	20.9
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.28 \lor \neg \left(\varepsilon \leq 0.0031\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 13
Error	34.2
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000166 \lor \neg \left(\varepsilon \leq 1.8 \cdot 10^{-8}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 14
Error	50.6
Cost	320

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

Alternative 15
Error	55.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023017 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))