2cos (problem 3.3.5)

Average Error: 39.6 → 0.6

Time: 17.8s

Precision: binary64

Cost: 39176

Math

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

↓

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.196625606945672:\\ \;\;\;\;\left(t_0 - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00024298140312862157:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, t_0\right) - \cos x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (cos eps))))
   (if (<= eps -1.196625606945672)
     (- (- t_0 (* (sin x) (sin eps))) (cos x))
     (if (<= eps 0.00024298140312862157)
       (+
        (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps))
        (*
         (cos x)
         (+ (* 0.041666666666666664 (pow eps 4.0)) (* -0.5 (* eps eps)))))
       (- (fma (sin x) (- (sin eps)) t_0) (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 tmp;
	if (eps <= -1.196625606945672) {
		tmp = (t_0 - (sin(x) * sin(eps))) - cos(x);
	} else if (eps <= 0.00024298140312862157) {
		tmp = (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps)) + (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) + (-0.5 * (eps * eps))));
	} else {
		tmp = fma(sin(x), -sin(eps), t_0) - 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))
	tmp = 0.0
	if (eps <= -1.196625606945672)
		tmp = Float64(Float64(t_0 - Float64(sin(x) * sin(eps))) - cos(x));
	elseif (eps <= 0.00024298140312862157)
		tmp = Float64(Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)) + Float64(cos(x) * Float64(Float64(0.041666666666666664 * (eps ^ 4.0)) + Float64(-0.5 * Float64(eps * eps)))));
	else
		tmp = Float64(fma(sin(x), Float64(-sin(eps)), t_0) - 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]}, If[LessEqual[eps, -1.196625606945672], N[(N[(t$95$0 - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00024298140312862157], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision]) + t$95$0), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.196625606945672

   1. Initial program 30.0

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

   2. Applied egg-rr 0.8

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

   if -1.196625606945672 < eps < 2.42981403128621573e-4

   1. Initial program 49.0

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

   2. Taylor expanded in eps around 0 0.3

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

   3. Simplified 0.3

      \[\leadsto \color{blue}{\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
      Proof
      (+.f64 (*.f64 (sin.f64 x) (-.f64 (*.f64 1/6 (pow.f64 eps 3)) eps)) (*.f64 (cos.f64 x) (+.f64 (*.f64 1/24 (pow.f64 eps 4)) (*.f64 -1/2 (*.f64 eps eps))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 (*.f64 1/6 (pow.f64 eps 3)) (sin.f64 x)) (*.f64 eps (sin.f64 x)))) (*.f64 (cos.f64 x) (+.f64 (*.f64 1/24 (pow.f64 eps 4)) (*.f64 -1/2 (*.f64 eps eps))))): 1 points increase in error, 1 points decrease in error
      (+.f64 (-.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x)))) (*.f64 eps (sin.f64 x))) (*.f64 (cos.f64 x) (+.f64 (*.f64 1/24 (pow.f64 eps 4)) (*.f64 -1/2 (*.f64 eps eps))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x))) (neg.f64 (*.f64 eps (sin.f64 x))))) (*.f64 (cos.f64 x) (+.f64 (*.f64 1/24 (pow.f64 eps 4)) (*.f64 -1/2 (*.f64 eps eps))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 eps (sin.f64 x))))) (*.f64 (cos.f64 x) (+.f64 (*.f64 1/24 (pow.f64 eps 4)) (*.f64 -1/2 (*.f64 eps eps))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1 (*.f64 eps (sin.f64 x))) (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x))))) (*.f64 (cos.f64 x) (+.f64 (*.f64 1/24 (pow.f64 eps 4)) (*.f64 -1/2 (*.f64 eps eps))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (*.f64 eps (sin.f64 x))) (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x)))) (*.f64 (cos.f64 x) (+.f64 (*.f64 1/24 (pow.f64 eps 4)) (*.f64 -1/2 (Rewrite<= unpow2_binary64 (pow.f64 eps 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (*.f64 eps (sin.f64 x))) (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x)))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 1/24 (pow.f64 eps 4)) (cos.f64 x)) (*.f64 (*.f64 -1/2 (pow.f64 eps 2)) (cos.f64 x))))): 0 points increase in error, 1 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (*.f64 eps (sin.f64 x))) (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x)))) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/24 (*.f64 (pow.f64 eps 4) (cos.f64 x)))) (*.f64 (*.f64 -1/2 (pow.f64 eps 2)) (cos.f64 x)))): 5 points increase in error, 4 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (*.f64 eps (sin.f64 x))) (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x)))) (+.f64 (*.f64 1/24 (*.f64 (pow.f64 eps 4) (cos.f64 x))) (Rewrite<= associate-*r*_binary64 (*.f64 -1/2 (*.f64 (pow.f64 eps 2) (cos.f64 x)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 1/24 (*.f64 (pow.f64 eps 4) (cos.f64 x))) (*.f64 -1/2 (*.f64 (pow.f64 eps 2) (cos.f64 x)))) (+.f64 (*.f64 -1 (*.f64 eps (sin.f64 x))) (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 1/24 (*.f64 (pow.f64 eps 4) (cos.f64 x))) (+.f64 (*.f64 -1/2 (*.f64 (pow.f64 eps 2) (cos.f64 x))) (+.f64 (*.f64 -1 (*.f64 eps (sin.f64 x))) (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x))))))): 1 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/24 (*.f64 (pow.f64 eps 4) (cos.f64 x))) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 -1/2 (*.f64 (pow.f64 eps 2) (cos.f64 x))) (*.f64 -1 (*.f64 eps (sin.f64 x)))) (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/24 (*.f64 (pow.f64 eps 4) (cos.f64 x))) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x))) (+.f64 (*.f64 -1/2 (*.f64 (pow.f64 eps 2) (cos.f64 x))) (*.f64 -1 (*.f64 eps (sin.f64 x))))))): 0 points increase in error, 0 points decrease in error

   if 2.42981403128621573e-4 < eps

   1. Initial program 30.2

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

   2. Applied egg-rr 0.9

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

   3. Applied egg-rr 0.9

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.196625606945672:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00024298140312862157:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon\right) - \cos x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	39112

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.196625606945672:\\ \;\;\;\;\left(t_0 - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00024298140312862157:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \end{array} \]

Alternative 2
Error	0.6
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -1.196625606945672:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.00024298140312862157:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.6
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.196625606945672:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00024298140312862157:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \end{array} \]

Alternative 4
Error	15.0
Cost	13888

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

Alternative 5
Error	14.9
Cost	13640

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -11055402534.511545:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.102764668462208 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x \cdot \left(\varepsilon \cdot -0.5\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	20.7
Cost	13448

\[\begin{array}{l} t_0 := -2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{if}\;\varepsilon \leq -9.537407858767459 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.2791814135071852 \cdot 10^{-24}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	20.3
Cost	13256

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -9.537407858767459 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.102764668462208 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	20.7
Cost	6920

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -9.537407858767459 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.102764668462208 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	34.5
Cost	6856

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -11055402534.511545:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.8343464697285863 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	58.2
Cost	320

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

Alternative 11
Error	50.7
Cost	320

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

Reproduce

herbie shell --seed 2022295 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))