2cos (problem 3.3.5)

Average Error: 39.7 → 0.5

Time: 9.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0023808861058280564:\\ \;\;\;\;\left(t_0 - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0027000306911697434:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

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 -0.0023808861058280564)
     (- (- t_0 (* (sin x) (sin eps))) (cos x))
     (if (<= eps 0.0027000306911697434)
       (+
        (*
         (cos x)
         (fma 0.041666666666666664 (pow eps 4.0) (* (* eps eps) -0.5)))
        (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps)))
       (- t_0 (fma (sin eps) (sin x) (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 <= -0.0023808861058280564) {
		tmp = (t_0 - (sin(x) * sin(eps))) - cos(x);
	} else if (eps <= 0.0027000306911697434) {
		tmp = (cos(x) * fma(0.041666666666666664, pow(eps, 4.0), ((eps * eps) * -0.5))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps));
	} else {
		tmp = t_0 - fma(sin(eps), sin(x), 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 <= -0.0023808861058280564)
		tmp = Float64(Float64(t_0 - Float64(sin(x) * sin(eps))) - cos(x));
	elseif (eps <= 0.0027000306911697434)
		tmp = Float64(Float64(cos(x) * fma(0.041666666666666664, (eps ^ 4.0), Float64(Float64(eps * eps) * -0.5))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)));
	else
		tmp = Float64(t_0 - fma(sin(eps), sin(x), 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, -0.0023808861058280564], N[(N[(t$95$0 - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0027000306911697434], N[(N[(N[Cos[x], $MachinePrecision] * N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if eps < -0.00238088610582805641

   1. Initial program 30.6

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

   2. Applied cos-sum_binary64 0.8

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

   if -0.00238088610582805641 < eps < 0.00270003069116974337

   1. Initial program 49.2

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

   2. Taylor expanded in eps around 0 0.1

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

   3. Simplified 0.1

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

   if 0.00270003069116974337 < eps

   1. Initial program 30.6

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

   2. Applied cos-sum_binary64 0.9

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

   3. Applied associate--l-_binary64 0.9

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

   4. Simplified 0.9

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0023808861058280564:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0027000306911697434:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))