2cos (problem 3.3.5)

Average Error: 39.1 → 0.5

Time: 16.7s

Precision: binary64

Cost: 40008

Math

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

↓

\[\begin{array}{l} t_0 := \sin x \cdot \left(-\sin \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.0033:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, t_0\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.003:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, t_0 - \cos x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin x) (- (sin eps)))))
   (if (<= eps -0.0033)
     (- (fma (cos x) (cos eps) t_0) (cos x))
     (if (<= eps 0.003)
       (fma
        0.041666666666666664
        (* (cos x) (pow eps 4.0))
        (+
         (* (cos x) (* -0.5 (* eps eps)))
         (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps))))
       (fma (cos x) (cos 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 = sin(x) * -sin(eps);
	double tmp;
	if (eps <= -0.0033) {
		tmp = fma(cos(x), cos(eps), t_0) - cos(x);
	} else if (eps <= 0.003) {
		tmp = fma(0.041666666666666664, (cos(x) * pow(eps, 4.0)), ((cos(x) * (-0.5 * (eps * eps))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps))));
	} else {
		tmp = fma(cos(x), cos(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(sin(x) * Float64(-sin(eps)))
	tmp = 0.0
	if (eps <= -0.0033)
		tmp = Float64(fma(cos(x), cos(eps), t_0) - cos(x));
	elseif (eps <= 0.003)
		tmp = fma(0.041666666666666664, Float64(cos(x) * (eps ^ 4.0)), Float64(Float64(cos(x) * Float64(-0.5 * Float64(eps * eps))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps))));
	else
		tmp = fma(cos(x), cos(eps), Float64(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[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[eps, -0.0033], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + t$95$0), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.003], N[(0.041666666666666664 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(t$95$0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0033

   1. Initial program 30.1

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

   2. Applied egg-rr 0.8

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

   if -0.0033 < eps < 0.0030000000000000001

   1. Initial program 48.8

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

   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

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(\left(-\varepsilon\right) + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right)} \]
      Proof
      (fma.f64 1/24 (*.f64 (cos.f64 x) (pow.f64 eps 4)) (+.f64 (*.f64 (cos.f64 x) (*.f64 -1/2 (*.f64 eps eps))) (*.f64 (sin.f64 x) (+.f64 (neg.f64 eps) (*.f64 1/6 (pow.f64 eps 3)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/24 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 eps 4) (cos.f64 x))) (+.f64 (*.f64 (cos.f64 x) (*.f64 -1/2 (*.f64 eps eps))) (*.f64 (sin.f64 x) (+.f64 (neg.f64 eps) (*.f64 1/6 (pow.f64 eps 3)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/24 (*.f64 (pow.f64 eps 4) (cos.f64 x)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 -1/2 (*.f64 eps eps)) (cos.f64 x))) (*.f64 (sin.f64 x) (+.f64 (neg.f64 eps) (*.f64 1/6 (pow.f64 eps 3)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/24 (*.f64 (pow.f64 eps 4) (cos.f64 x)) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1/2 (*.f64 (*.f64 eps eps) (cos.f64 x)))) (*.f64 (sin.f64 x) (+.f64 (neg.f64 eps) (*.f64 1/6 (pow.f64 eps 3)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/24 (*.f64 (pow.f64 eps 4) (cos.f64 x)) (+.f64 (*.f64 -1/2 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 eps 2)) (cos.f64 x))) (*.f64 (sin.f64 x) (+.f64 (neg.f64 eps) (*.f64 1/6 (pow.f64 eps 3)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/24 (*.f64 (pow.f64 eps 4) (cos.f64 x)) (+.f64 (*.f64 -1/2 (*.f64 (pow.f64 eps 2) (cos.f64 x))) (*.f64 (sin.f64 x) (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 eps)) (*.f64 1/6 (pow.f64 eps 3)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/24 (*.f64 (pow.f64 eps 4) (cos.f64 x)) (+.f64 (*.f64 -1/2 (*.f64 (pow.f64 eps 2) (cos.f64 x))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 -1 eps) (sin.f64 x)) (*.f64 (*.f64 1/6 (pow.f64 eps 3)) (sin.f64 x)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/24 (*.f64 (pow.f64 eps 4) (cos.f64 x)) (+.f64 (*.f64 -1/2 (*.f64 (pow.f64 eps 2) (cos.f64 x))) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 eps (sin.f64 x)))) (*.f64 (*.f64 1/6 (pow.f64 eps 3)) (sin.f64 x))))): 0 points increase in error, 0 points decrease in error
      (fma.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))) (Rewrite<= associate-*r*_binary64 (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x))))))): 0 points increase in error, 0 points decrease in error
      (fma.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, 1 points decrease in error
      (fma.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
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/24 (*.f64 (pow.f64 eps 4) (cos.f64 x))) (+.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))))))): 4 points increase in error, 1 points decrease in error

   if 0.0030000000000000001 < eps

   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

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0033:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.003:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right) - \cos x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	39176

\[\begin{array}{l} t_0 := \sin x \cdot \left(-\sin \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -2.45 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, t_0\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, t_0 - \cos x\right)\\ \end{array} \]

Alternative 2
Error	0.6
Cost	39044

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]

Alternative 3
Error	0.6
Cost	38980

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-5}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -3.9 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	14.2
Cost	13768

\[\begin{array}{l} t_0 := \cos x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{if}\;\varepsilon \leq -0.0026:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.074:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	14.9
Cost	13632

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

Alternative 7
Error	14.4
Cost	13384

\[\begin{array}{l} t_0 := \cos x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{if}\;\varepsilon \leq -0.0092:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0105:\\ \;\;\;\;\left(-\varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	14.9
Cost	13124

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0116:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0065:\\ \;\;\;\;\left(-\varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 9
Error	15.2
Cost	7304

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.0144:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0045:\\ \;\;\;\;\left(-\varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	21.2
Cost	7184

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.1 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq -1.85 \cdot 10^{-89}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 4.3 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	32.1
Cost	7120

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.000152:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.36 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-137}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000136:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	48.3
Cost	584

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -2.55 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-126}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	48.3
Cost	584

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-126}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	52.4
Cost	256

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

Alternative 15
Error	55.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))