2cos (problem 3.3.5)

Average Error: 40.0 → 0.5

Time: 25.1s

Precision: binary64

Cost: 45448

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \mathbf{if}\;\varepsilon \leq -0.01967658489477877:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.430927827935982 \cdot 10^{-5}:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma (cos x) (cos eps) (- (fma (sin x) (sin eps) (cos x))))))
   (if (<= eps -0.01967658489477877)
     t_0
     (if (<= eps 2.430927827935982e-5)
       (+
        (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps))
        (*
         (cos x)
         (+ (* 0.041666666666666664 (pow eps 4.0)) (* -0.5 (* eps 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 = fma(cos(x), cos(eps), -fma(sin(x), sin(eps), cos(x)));
	double tmp;
	if (eps <= -0.01967658489477877) {
		tmp = t_0;
	} else if (eps <= 2.430927827935982e-5) {
		tmp = (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps)) + (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) + (-0.5 * (eps * eps))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	t_0 = fma(cos(x), cos(eps), Float64(-fma(sin(x), sin(eps), cos(x))))
	tmp = 0.0
	if (eps <= -0.01967658489477877)
		tmp = t_0;
	elseif (eps <= 2.430927827935982e-5)
		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 = 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[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + (-N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[eps, -0.01967658489477877], t$95$0, If[LessEqual[eps, 2.430927827935982e-5], 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.019676584894778769 or 2.43092782793598188e-5 < eps

   1. Initial program 30.4

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

   2. Applied egg-rr 0.8

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

   if -0.019676584894778769 < eps < 2.43092782793598188e-5

   1. Initial program 49.5

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

   2. Taylor expanded in eps around 0 0.2

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

   3. Simplified 0.2

      \[\leadsto \color{blue}{\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
      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, 0 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, 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 (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)))): 4 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, 1 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
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.01967658489477877:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 2.430927827935982 \cdot 10^{-5}:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	39176

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.01967658489477877:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.430927827935982 \cdot 10^{-5}:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	39112

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{if}\;\varepsilon \leq -0.01967658489477877:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.430927827935982 \cdot 10^{-5}:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.5
Cost	32840

\[\begin{array}{l} t_0 := \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.01967658489477877:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.430927827935982 \cdot 10^{-5}:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.5
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.01967658489477877:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 2.430927827935982 \cdot 10^{-5}:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \end{array} \]

Alternative 5
Error	15.1
Cost	13888

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

Alternative 6
Error	15.2
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.825915668698037 \cdot 10^{-7}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{elif}\;\varepsilon \leq 2924667556.7783012:\\ \;\;\;\;\varepsilon \cdot \left(\cos x \cdot \left(\varepsilon \cdot -0.5\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 7
Error	20.1
Cost	13448

\[\begin{array}{l} t_0 := \mathsf{expm1}\left(\sin x \cdot \left(-\varepsilon\right)\right)\\ \mathbf{if}\;x \leq -4.365679655594944 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.7657628582656835 \cdot 10^{-67}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	21.3
Cost	13256

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.1706050148096177 \cdot 10^{-11}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq 2924667556.7783012:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 9
Error	32.8
Cost	7120

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{if}\;\varepsilon \leq -0.01967658489477877:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -2.684791900755567 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 8.600544163350966 \cdot 10^{-158}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 9.489344468365535 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	21.5
Cost	6920

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -2.1706050148096177 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2924667556.7783012:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	49.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 6.04785922399166 \cdot 10^{-143}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 12
Error	52.7
Cost	256

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

Reproduce

herbie shell --seed 2022313 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))