2cos (problem 3.3.5)

Average Error: 39.5 → 0.7

Time: 14.5s

Precision: binary64

Cost: 32776

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos x, \cos \varepsilon + -1, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ t_1 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;\left(x \cdot \cos \left(\varepsilon \cdot 0.5\right) + t_1\right) \cdot \left(t_1 \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma (cos x) (+ (cos eps) -1.0) (* (sin eps) (- (sin x)))))
        (t_1 (sin (* eps 0.5))))
   (if (<= x -1.25e-53)
     t_0
     (if (<= x 1.5e-8) (* (+ (* x (cos (* eps 0.5))) t_1) (* t_1 -2.0)) 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) + -1.0), (sin(eps) * -sin(x)));
	double t_1 = sin((eps * 0.5));
	double tmp;
	if (x <= -1.25e-53) {
		tmp = t_0;
	} else if (x <= 1.5e-8) {
		tmp = ((x * cos((eps * 0.5))) + t_1) * (t_1 * -2.0);
	} 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), Float64(cos(eps) + -1.0), Float64(sin(eps) * Float64(-sin(x))))
	t_1 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if (x <= -1.25e-53)
		tmp = t_0;
	elseif (x <= 1.5e-8)
		tmp = Float64(Float64(Float64(x * cos(Float64(eps * 0.5))) + t_1) * Float64(t_1 * -2.0));
	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[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.25e-53], t$95$0, If[LessEqual[x, 1.5e-8], N[(N[(N[(x * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] * N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e-53 or 1.49999999999999987e-8 < x

   1. Initial program 57.0

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

   2. Applied egg-rr 28.7

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

   3. Taylor expanded in x around inf 28.8

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

   4. Simplified 1.1

      \[\leadsto \color{blue}{\left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin \varepsilon \cdot \sin x} \]
      Proof
      (-.f64 (-.f64 (*.f64 (cos.f64 eps) (cos.f64 x)) (cos.f64 x)) (*.f64 (sin.f64 eps) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 x) (cos.f64 eps))) (cos.f64 x)) (*.f64 (sin.f64 eps) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (*.f64 (cos.f64 x) (cos.f64 eps)) (cos.f64 x)) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 x) (sin.f64 eps)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (-.f64 (*.f64 (cos.f64 x) (cos.f64 eps)) (cos.f64 x)) (neg.f64 (*.f64 (sin.f64 x) (sin.f64 eps))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (*.f64 (cos.f64 x) (cos.f64 eps)) (cos.f64 x)) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (*.f64 (sin.f64 x) (sin.f64 eps))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (*.f64 (cos.f64 x) (cos.f64 eps)) (cos.f64 x)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) -1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) -1) (-.f64 (*.f64 (cos.f64 x) (cos.f64 eps)) (cos.f64 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) -1) (*.f64 (cos.f64 x) (cos.f64 eps))) (cos.f64 x))): 105 points increase in error, 18 points decrease in error
      (-.f64 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 -1 (*.f64 (sin.f64 x) (sin.f64 eps)))) (*.f64 (cos.f64 x) (cos.f64 eps))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 -1 (*.f64 (sin.f64 x) (sin.f64 eps))) (Rewrite=> *-commutative_binary64 (*.f64 (cos.f64 eps) (cos.f64 x)))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 1.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon - 1, \sin x \cdot \left(-\sin \varepsilon\right)\right)} \]

   if -1.25e-53 < x < 1.49999999999999987e-8

   1. Initial program 18.4

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

   2. Applied egg-rr 0.4

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

   3. Simplified 0.4

      \[\leadsto \color{blue}{\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)} \]
      Proof
      (*.f64 (sin.f64 (*.f64 1/2 (+.f64 eps (+.f64 x x)))) (*.f64 -2 (sin.f64 (*.f64 eps 1/2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 1/2 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 x x) eps)))) (*.f64 -2 (sin.f64 (*.f64 eps 1/2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 1/2 (Rewrite<= associate-+r+_binary64 (+.f64 x (+.f64 x eps))))) (*.f64 -2 (sin.f64 (*.f64 eps 1/2)))): 2 points increase in error, 1 points decrease in error
      (*.f64 (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 x (+.f64 x eps)) 1/2))) (*.f64 -2 (sin.f64 (*.f64 eps 1/2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (+.f64 x (+.f64 x eps)) 1/2)) (*.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -2) (sqrt.f64 -2))) (sin.f64 (*.f64 eps 1/2)))): 256 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (+.f64 x (+.f64 x eps)) 1/2)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -2) 2)) (sin.f64 (*.f64 eps 1/2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (+.f64 x (+.f64 x eps)) 1/2)) (*.f64 (pow.f64 (sqrt.f64 -2) 2) (sin.f64 (*.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 eps 0)) 1/2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (+.f64 x (+.f64 x eps)) 1/2)) (*.f64 (pow.f64 (sqrt.f64 -2) 2) (sin.f64 (*.f64 (+.f64 eps (Rewrite<= +-inverses_binary64 (-.f64 x x))) 1/2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (pow.f64 (sqrt.f64 -2) 2) (sin.f64 (*.f64 (+.f64 eps (-.f64 x x)) 1/2))) (sin.f64 (*.f64 (+.f64 x (+.f64 x eps)) 1/2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 (sqrt.f64 -2) 2) (*.f64 (sin.f64 (*.f64 (+.f64 eps (-.f64 x x)) 1/2)) (sin.f64 (*.f64 (+.f64 x (+.f64 x eps)) 1/2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> unpow2_binary64 (*.f64 (sqrt.f64 -2) (sqrt.f64 -2))) (*.f64 (sin.f64 (*.f64 (+.f64 eps (-.f64 x x)) 1/2)) (sin.f64 (*.f64 (+.f64 x (+.f64 x eps)) 1/2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> rem-square-sqrt_binary64 -2) (*.f64 (sin.f64 (*.f64 (+.f64 eps (-.f64 x x)) 1/2)) (sin.f64 (*.f64 (+.f64 x (+.f64 x eps)) 1/2)))): 0 points increase in error, 256 points decrease in error

   4. Taylor expanded in x around 0 0.3

      \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot x + \sin \left(0.5 \cdot \varepsilon\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon + -1, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;\left(x \cdot \cos \left(\varepsilon \cdot 0.5\right) + \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon + -1, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	26440

\[\begin{array}{l} t_0 := \cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ t_1 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\left(x \cdot \cos \left(\varepsilon \cdot 0.5\right) + t_1\right) \cdot \left(t_1 \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	15.6
Cost	13768

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0076:\\ \;\;\;\;\cos \left(x + \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.5\right)\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 3
Error	15.6
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.017:\\ \;\;\;\;\cos \left(x + \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(-0.5 \cdot \left(\cos x \cdot \varepsilon\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 4
Error	15.4
Cost	13632

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

Alternative 5
Error	15.8
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.004:\\ \;\;\;\;\cos \left(x + \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 6
Error	15.8
Cost	13384

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0023:\\ \;\;\;\;\cos \left(x + \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0024:\\ \;\;\;\;\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon + -1\\ \end{array} \]

Alternative 7
Error	15.8
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082:\\ \;\;\;\;\cos \left(x + \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0048:\\ \;\;\;\;\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 8
Error	15.4
Cost	13124

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.002:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0033:\\ \;\;\;\;\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 9
Error	15.6
Cost	7304

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.009:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0018:\\ \;\;\;\;\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	21.4
Cost	6920

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	30.3
Cost	6856

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -9.6 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.00019:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	48.5
Cost	584

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -1.22 \cdot 10^{-166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.12 \cdot 10^{-156}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	46.7
Cost	448

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

Alternative 14
Error	52.3
Cost	256

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

Alternative 15
Error	55.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022328 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))