2cos (problem 3.3.5)

Average Error: 39.0 → 0.5

Time: 17.5s

Precision: binary64

Cost: 39044

Math

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

↓

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0054:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0056:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \end{array} \]

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 -0.0054)
     (- (fma (cos x) (cos eps) (* (sin eps) (- (sin x)))) (cos x))
     (if (<= eps 0.0056)
       (-
        (*
         (cos x)
         (fma 0.041666666666666664 (pow eps 4.0) (* -0.5 (* eps eps))))
        t_0)
       (- (* (cos x) (+ -1.0 (cos 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 <= -0.0054) {
		tmp = fma(cos(x), cos(eps), (sin(eps) * -sin(x))) - cos(x);
	} else if (eps <= 0.0056) {
		tmp = (cos(x) * fma(0.041666666666666664, pow(eps, 4.0), (-0.5 * (eps * eps)))) - t_0;
	} else {
		tmp = (cos(x) * (-1.0 + cos(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 <= -0.0054)
		tmp = Float64(fma(cos(x), cos(eps), Float64(sin(eps) * Float64(-sin(x)))) - cos(x));
	elseif (eps <= 0.0056)
		tmp = Float64(Float64(cos(x) * fma(0.041666666666666664, (eps ^ 4.0), Float64(-0.5 * Float64(eps * eps)))) - t_0);
	else
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(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[LessEqual[eps, -0.0054], 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], If[LessEqual[eps, 0.0056], 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], N[(N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0054000000000000003

   1. Initial program 29.3

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

   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 < 0.00559999999999999994

   1. Initial program 48.4

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

   2. Applied egg-rr 10.7

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

   3. Applied egg-rr 10.7

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

   4. Taylor expanded in eps around 0 0.2

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

   5. Simplified 0.2

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

      [Start] 0.2	\[ \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      fma-def [=>] 0.2	\[ \cos x \cdot \color{blue}{\mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot {\varepsilon}^{2}\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      unpow2 [=>] 0.2	\[ \cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]

   if 0.00559999999999999994 < eps

   1. Initial program 30.2

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

   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

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0054:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0056:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	39232

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

Alternative 2
Error	0.7
Cost	39168

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

Alternative 3
Error	0.5
Cost	33160

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0058:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0056:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \end{array} \]

Alternative 4
Error	0.5
Cost	26441

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00018 \lor \neg \left(\varepsilon \leq 0.000185\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - t_0\\ \end{array} \]

Alternative 5
Error	13.9
Cost	26313

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0115 \lor \neg \left(\varepsilon \leq 0.035\right):\\ \;\;\;\;\left(\cos \varepsilon - \cos x\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - t_0\\ \end{array} \]

Alternative 6
Error	14.9
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 7
Error	14.9
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 8
Error	14.6
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.036 \lor \neg \left(\varepsilon \leq 0.014\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 9
Error	20.1
Cost	13388

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.000155:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.7 \cdot 10^{-53}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	32.4
Cost	7120

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.00018:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -2.5 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 9.6 \cdot 10^{-122}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000185:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	20.6
Cost	7052

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00018:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.7 \cdot 10^{-53}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	48.4
Cost	6988

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -2.5 \cdot 10^{-167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.2 \cdot 10^{-122}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 - \cos x\\ \end{array} \]

Alternative 13
Error	49.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.5 \cdot 10^{-167} \lor \neg \left(\varepsilon \leq 5.5 \cdot 10^{-122}\right):\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 14
Error	52.1
Cost	256

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

Reproduce

herbie shell --seed 2023057 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))