2cos (problem 3.3.5)

Average Error: 39.3 → 1.3

Time: 28.0s

Precision: binary64

Cost: 46600

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -66854077635.15686:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos x \cdot {\varepsilon}^{4}\right) + \left(0.16666666666666666 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right) + \left(-0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \frac{-1}{\frac{1}{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -66854077635.15686)
   (- (fma (sin x) (- (sin eps)) (* (cos x) (cos eps))) (cos x))
   (if (<= eps 5.573415221579436e-13)
     (+
      (* 0.041666666666666664 (* (cos x) (pow eps 4.0)))
      (+
       (* 0.16666666666666666 (* (sin x) (pow eps 3.0)))
       (- (* -0.5 (* (cos x) (pow eps 2.0))) (* eps (sin x)))))
     (fma
      (cos x)
      (cos eps)
      (/ -1.0 (/ 1.0 (fma (sin x) (sin eps) (cos x))))))))

C

double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}

↓

double code(double x, double eps) {
	double tmp;
	if (eps <= -66854077635.15686) {
		tmp = fma(sin(x), -sin(eps), (cos(x) * cos(eps))) - cos(x);
	} else if (eps <= 5.573415221579436e-13) {
		tmp = (0.041666666666666664 * (cos(x) * pow(eps, 4.0))) + ((0.16666666666666666 * (sin(x) * pow(eps, 3.0))) + ((-0.5 * (cos(x) * pow(eps, 2.0))) - (eps * sin(x))));
	} else {
		tmp = fma(cos(x), cos(eps), (-1.0 / (1.0 / fma(sin(x), sin(eps), cos(x)))));
	}
	return tmp;
}

Julia

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end

↓

function code(x, eps)
	tmp = 0.0
	if (eps <= -66854077635.15686)
		tmp = Float64(fma(sin(x), Float64(-sin(eps)), Float64(cos(x) * cos(eps))) - cos(x));
	elseif (eps <= 5.573415221579436e-13)
		tmp = Float64(Float64(0.041666666666666664 * Float64(cos(x) * (eps ^ 4.0))) + Float64(Float64(0.16666666666666666 * Float64(sin(x) * (eps ^ 3.0))) + Float64(Float64(-0.5 * Float64(cos(x) * (eps ^ 2.0))) - Float64(eps * sin(x)))));
	else
		tmp = fma(cos(x), cos(eps), Float64(-1.0 / Float64(1.0 / fma(sin(x), sin(eps), 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_] := If[LessEqual[eps, -66854077635.15686], N[(N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.573415221579436e-13], N[(N[(0.041666666666666664 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.16666666666666666 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(-1.0 / N[(1.0 / N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -66854077635.15686

   1. Initial program 29.2

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

   2. Applied egg-rr 0.8

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

   3. Applied egg-rr 0.8

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

   if -66854077635.15686 < eps < 5.57341522157943627e-13

   1. Initial program 48.4

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

   2. Taylor expanded in eps around 0 1.3

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)} \]

   if 5.57341522157943627e-13 < eps

   1. Initial program 31.4

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

   2. Applied egg-rr 1.7

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

   3. Applied egg-rr 1.7

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -66854077635.15686:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos x \cdot {\varepsilon}^{4}\right) + \left(0.16666666666666666 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right) + \left(-0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \frac{-1}{\frac{1}{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	45640

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -66854077635.15686:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \frac{-1}{\frac{1}{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)}}\right)\\ \end{array} \]

Alternative 2
Error	1.3
Cost	45448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -66854077635.15686:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array} \]

Alternative 3
Error	1.3
Cost	39176

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -66854077635.15686:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right) - \cos x\right)\\ \end{array} \]

Alternative 4
Error	1.3
Cost	39176

\[\begin{array}{l} t_0 := -\sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -66854077635.15686:\\ \;\;\;\;\mathsf{fma}\left(\sin x, t_0, \cos x \cdot \cos \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot t_0 - \cos x\right)\\ \end{array} \]

Alternative 5
Error	1.3
Cost	39112

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -66854077635.15686:\\ \;\;\;\;\left(t_0 - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \end{array} \]

Alternative 6
Error	1.3
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -66854077635.15686:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.3
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -66854077635.15686:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \end{array} \]

Alternative 8
Error	21.0
Cost	13648

\[\begin{array}{l} t_0 := \cos x \cdot \left(\cos \varepsilon + -1\right)\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -0.0052071745123544975:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -5.367921965559038 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq -3.5205277515653727 \cdot 10^{-143}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	14.7
Cost	13640

\[\begin{array}{l} t_0 := \cos x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{if}\;\varepsilon \leq -0.0052071745123544975:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x \cdot \left(\varepsilon \cdot -0.5\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	15.2
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 11
Error	21.4
Cost	13520

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -0.0052071745123544975:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -5.367921965559038 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq -3.5205277515653727 \cdot 10^{-143}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	21.9
Cost	7184

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -0.0052071745123544975:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -5.367921965559038 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq -3.5205277515653727 \cdot 10^{-143}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	33.9
Cost	6856

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -9.454167810978805 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	50.4
Cost	320

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

Alternative 15
Error	55.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022317 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))