2cos (problem 3.3.5)

Average Error: 39.0 → 1.3

Time: 14.2s

Precision: binary64

Cost: 45448

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -12297687.609885052:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 8.43316946514483 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\cos x \cdot -0.5\right), \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -12297687.609885052)
   (- (fma (cos x) (cos eps) (* (sin x) (- (sin eps)))) (cos x))
   (if (<= eps 8.43316946514483e-18)
     (fma
      eps
      (* eps (* (cos x) -0.5))
      (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps)))
     (fma (cos x) (cos eps) (- (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 <= -12297687.609885052) {
		tmp = fma(cos(x), cos(eps), (sin(x) * -sin(eps))) - cos(x);
	} else if (eps <= 8.43316946514483e-18) {
		tmp = fma(eps, (eps * (cos(x) * -0.5)), (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps)));
	} else {
		tmp = fma(cos(x), cos(eps), -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 <= -12297687.609885052)
		tmp = Float64(fma(cos(x), cos(eps), Float64(sin(x) * Float64(-sin(eps)))) - cos(x));
	elseif (eps <= 8.43316946514483e-18)
		tmp = fma(eps, Float64(eps * Float64(cos(x) * -0.5)), Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)));
	else
		tmp = fma(cos(x), cos(eps), Float64(-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, -12297687.609885052], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 8.43316946514483e-18], N[(eps * N[(eps * N[(N[Cos[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + (-N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -12297687.609885052

   1. Initial program 29.4

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

   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 -12297687.609885052 < eps < 8.43316946514483051e-18

   1. Initial program 48.6

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

   2. Taylor expanded in eps around 0 1.1

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

   3. Simplified 1.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(-0.5 \cdot \cos x\right), \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)} \]
      Proof
      (fma.f64 eps (*.f64 eps (*.f64 -1/2 (cos.f64 x))) (*.f64 (sin.f64 x) (-.f64 (*.f64 1/6 (pow.f64 eps 3)) eps))): 0 points increase in error, 0 points decrease in error
      (fma.f64 eps (*.f64 eps (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 x) -1/2))) (*.f64 (sin.f64 x) (-.f64 (*.f64 1/6 (pow.f64 eps 3)) eps))): 0 points increase in error, 0 points decrease in error
      (fma.f64 eps (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 (cos.f64 x) -1/2) eps)) (*.f64 (sin.f64 x) (-.f64 (*.f64 1/6 (pow.f64 eps 3)) eps))): 0 points increase in error, 0 points decrease in error
      (fma.f64 eps (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 -1/2 (cos.f64 x))) eps) (*.f64 (sin.f64 x) (-.f64 (*.f64 1/6 (pow.f64 eps 3)) eps))): 0 points increase in error, 0 points decrease in error
      (fma.f64 eps (*.f64 (*.f64 -1/2 (cos.f64 x)) eps) (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 (*.f64 1/6 (pow.f64 eps 3)) (sin.f64 x)) (*.f64 eps (sin.f64 x))))): 1 points increase in error, 0 points decrease in error
      (fma.f64 eps (*.f64 (*.f64 -1/2 (cos.f64 x)) eps) (-.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x)))) (*.f64 eps (sin.f64 x)))): 3 points increase in error, 3 points decrease in error
      (fma.f64 eps (*.f64 (*.f64 -1/2 (cos.f64 x)) eps) (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x))) (neg.f64 (*.f64 eps (sin.f64 x)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 eps (*.f64 (*.f64 -1/2 (cos.f64 x)) eps) (+.f64 (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 eps (sin.f64 x)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 eps (*.f64 (*.f64 -1/2 (cos.f64 x)) eps) (Rewrite=> +-commutative_binary64 (+.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=> fma-udef_binary64 (+.f64 (*.f64 eps (*.f64 (*.f64 -1/2 (cos.f64 x)) eps)) (+.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 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 (*.f64 -1/2 (cos.f64 x)) eps) eps)) (+.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
      (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 -1/2 (cos.f64 x)) (*.f64 eps eps))) (+.f64 (*.f64 -1 (*.f64 eps (sin.f64 x))) (*.f64 1/6 (*.f64 (pow.f64 eps 3) (sin.f64 x))))): 2 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (*.f64 -1/2 (cos.f64 x)) (Rewrite<= unpow2_binary64 (pow.f64 eps 2))) (+.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
      (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1/2 (*.f64 (cos.f64 x) (pow.f64 eps 2)))) (+.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
      (+.f64 (*.f64 -1/2 (Rewrite<= *-commutative_binary64 (*.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-+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
      (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 8.43316946514483051e-18 < eps

   1. Initial program 31.0

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

   2. Applied egg-rr 2.2

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -12297687.609885052:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 8.43316946514483 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\cos x \cdot -0.5\right), \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	15.3
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 2
Error	15.2
Cost	13768

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -12297687.609885052:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0796388076311669:\\ \;\;\;\;\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	21.2
Cost	6920

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -12297687.609885052:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0796388076311669:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	50.9
Cost	320

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

Alternative 5
Error	55.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022289 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))