2cos (problem 3.3.5)

Average Error: 62.01% → 0.74%

Time: 17.7s

Precision: binary64

Cost: 39112

Math

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

↓

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0058:\\ \;\;\;\;\left(t_0 - \cos x\right) - t_1\\ \mathbf{elif}\;\varepsilon \leq 0.0048:\\ \;\;\;\;\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (cos eps))) (t_1 (* (sin eps) (sin x))))
   (if (<= eps -0.0058)
     (- (- t_0 (cos x)) t_1)
     (if (<= eps 0.0048)
       (-
        (*
         (cos x)
         (+ (* eps (* eps -0.5)) (* 0.041666666666666664 (pow eps 4.0))))
        t_1)
       (- t_0 (fma (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 = cos(x) * cos(eps);
	double t_1 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.0058) {
		tmp = (t_0 - cos(x)) - t_1;
	} else if (eps <= 0.0048) {
		tmp = (cos(x) * ((eps * (eps * -0.5)) + (0.041666666666666664 * pow(eps, 4.0)))) - t_1;
	} else {
		tmp = t_0 - fma(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(cos(x) * cos(eps))
	t_1 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.0058)
		tmp = Float64(Float64(t_0 - cos(x)) - t_1);
	elseif (eps <= 0.0048)
		tmp = Float64(Float64(cos(x) * Float64(Float64(eps * Float64(eps * -0.5)) + Float64(0.041666666666666664 * (eps ^ 4.0)))) - t_1);
	else
		tmp = Float64(t_0 - fma(sin(eps), 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[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0058], N[(N[(t$95$0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[eps, 0.0048], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0058

   1. Initial program 47.28

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

   2. Applied egg-rr 1.23

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

   3. Taylor expanded in x around inf 1.22

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

   4. Simplified 1.23

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

      [Start] 1.22	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x \]
      +-commutative [=>] 1.22	\[ \color{blue}{\left(\cos \varepsilon \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
      *-commutative [=>] 1.22	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right) - \cos x \]
      *-commutative [<=] 1.22	\[ \left(\cos x \cdot \cos \varepsilon + -1 \cdot \color{blue}{\left(\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      mul-1-neg [=>] 1.22	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(-\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      sub0-neg [<=] 1.22	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(0 - \sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      associate-+r- [=>] 1.22	\[ \color{blue}{\left(\left(\cos x \cdot \cos \varepsilon + 0\right) - \sin \varepsilon \cdot \sin x\right)} - \cos x \]
      +-rgt-identity [=>] 1.22	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin \varepsilon \cdot \sin x\right) - \cos x \]
      associate--r+ [<=] 1.25	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin \varepsilon \cdot \sin x + \cos x\right)} \]
      +-commutative [<=] 1.25	\[ \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin \varepsilon \cdot \sin x\right)} \]
      associate--r+ [=>] 1.23	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x} \]

   if -0.0058 < eps < 0.00479999999999999958

   1. Initial program 76.91

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

   2. Applied egg-rr 18.53

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

   3. Taylor expanded in x around inf 75.75

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

   4. Simplified 18.53

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

      [Start] 75.75	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x \]
      +-commutative [=>] 75.75	\[ \color{blue}{\left(\cos \varepsilon \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
      *-commutative [=>] 75.75	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right) - \cos x \]
      *-commutative [<=] 75.75	\[ \left(\cos x \cdot \cos \varepsilon + -1 \cdot \color{blue}{\left(\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      mul-1-neg [=>] 75.75	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(-\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      sub0-neg [<=] 75.75	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(0 - \sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      associate-+r- [=>] 75.75	\[ \color{blue}{\left(\left(\cos x \cdot \cos \varepsilon + 0\right) - \sin \varepsilon \cdot \sin x\right)} - \cos x \]
      +-rgt-identity [=>] 75.75	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin \varepsilon \cdot \sin x\right) - \cos x \]
      associate--r+ [<=] 75.75	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin \varepsilon \cdot \sin x + \cos x\right)} \]
      +-commutative [<=] 75.75	\[ \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin \varepsilon \cdot \sin x\right)} \]
      associate--r+ [=>] 18.53	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x} \]

   5. Taylor expanded in eps around 0 0.23

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

   6. Simplified 0.23

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

      [Start] 0.23	\[ \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) - \sin x \cdot \sin \varepsilon \]
      +-commutative [=>] 0.23	\[ \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right)} - \sin x \cdot \sin \varepsilon \]
      associate-*r* [=>] 0.23	\[ \left(\color{blue}{\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) - \sin x \cdot \sin \varepsilon \]
      associate-*r* [=>] 0.23	\[ \left(\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x + \color{blue}{\left(0.041666666666666664 \cdot {\varepsilon}^{4}\right) \cdot \cos x}\right) - \sin x \cdot \sin \varepsilon \]
      distribute-rgt-out [=>] 0.23	\[ \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)} - \sin x \cdot \sin \varepsilon \]
      unpow2 [=>] 0.23	\[ \cos x \cdot \left(-0.5 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin x \cdot \sin \varepsilon \]
      associate-*r* [=>] 0.23	\[ \cos x \cdot \left(\color{blue}{\left(-0.5 \cdot \varepsilon\right) \cdot \varepsilon} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin x \cdot \sin \varepsilon \]

   if 0.00479999999999999958 < eps

   1. Initial program 46.55

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

   2. Applied egg-rr 1.29

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

   3. Simplified 1.27

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

      [Start] 1.29	\[ \cos x \cdot \cos \varepsilon + \left(\sin \varepsilon \cdot \left(-\sin x\right) + \left(-\cos x\right)\right) \]
      *-commutative [=>] 1.29	\[ \cos x \cdot \cos \varepsilon + \left(\color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon} + \left(-\cos x\right)\right) \]
      distribute-lft-neg-in [<=] 1.29	\[ \cos x \cdot \cos \varepsilon + \left(\color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)} + \left(-\cos x\right)\right) \]
      distribute-neg-out [=>] 1.29	\[ \cos x \cdot \cos \varepsilon + \color{blue}{\left(-\left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \]
      unsub-neg [=>] 1.29	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \]
      *-commutative [=>] 1.29	\[ \color{blue}{\cos \varepsilon \cdot \cos x} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \]
      *-commutative [=>] 1.29	\[ \cos \varepsilon \cdot \cos x - \left(\color{blue}{\sin \varepsilon \cdot \sin x} + \cos x\right) \]
      fma-def [=>] 1.27	\[ \cos \varepsilon \cdot \cos x - \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.74

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0058:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.0048:\\ \;\;\;\;\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.15%
Cost	39168

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

Alternative 2
Error	0.7%
Cost	32777

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0058 \lor \neg \left(\varepsilon \leq 0.0045\right):\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, -t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \end{array} \]

Alternative 3
Error	0.71%
Cost	32776

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0058:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0045:\\ \;\;\;\;\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, -t_0\right)\\ \end{array} \]

Alternative 4
Error	0.72%
Cost	26889

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0058 \lor \neg \left(\varepsilon \leq 0.0045\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \end{array} \]

Alternative 5
Error	1.03%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-57} \lor \neg \left(x \leq 8.3 \cdot 10^{-16}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot -2\right)\\ \end{array} \]

Alternative 6
Error	21.51%
Cost	26313

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00175 \lor \neg \left(\varepsilon \leq 0.14\right):\\ \;\;\;\;\cos \varepsilon - \left(\cos x + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right) - t_0\\ \end{array} \]

Alternative 7
Error	21.51%
Cost	26313

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0013 \lor \neg \left(\varepsilon \leq 0.14\right):\\ \;\;\;\;\left(\cos \varepsilon - \cos x\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right) - t_0\\ \end{array} \]

Alternative 8
Error	23.15%
Cost	13888

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

Alternative 9
Error	22.64%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00175 \lor \neg \left(\varepsilon \leq 0.14\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 10
Error	22.88%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.001 \lor \neg \left(\varepsilon \leq 0.14\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 11
Error	23.53%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00165 \lor \neg \left(\varepsilon \leq 0.14\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 12
Error	33.2%
Cost	7184

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := \sin x \cdot \left(-\varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.000115:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -5.5 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq -2.25 \cdot 10^{-136}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	53.27%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.1 \cdot 10^{-14} \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 14
Error	77.49%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq 13.5:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \cos x\\ \end{array} \]

Alternative 15
Error	78.7%
Cost	320

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

Alternative 16
Error	78.7%
Cost	320

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

Alternative 17
Error	86.99%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023102 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))