2cos (problem 3.3.5)

Average Error: 39.3 → 0.5

Time: 9.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin x \cdot \left(-\sin \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.00016:\\ \;\;\;\;\mathsf{fma}\left(1, t_0, t_1\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.000165:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \cos x\right) \cdot -0.5, \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t_0, t_1 - \cos x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (cos eps))) (t_1 (* (sin x) (- (sin eps)))))
   (if (<= eps -0.00016)
     (- (fma 1.0 t_0 t_1) (cos x))
     (if (<= eps 0.000165)
       (fma
        eps
        (* (* eps (cos x)) -0.5)
        (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps)))
       (fma 1.0 t_0 (- t_1 (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 t_1 = sin(x) * -sin(eps);
	double tmp;
	if (eps <= -0.00016) {
		tmp = fma(1.0, t_0, t_1) - cos(x);
	} else if (eps <= 0.000165) {
		tmp = fma(eps, ((eps * cos(x)) * -0.5), (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps)));
	} else {
		tmp = fma(1.0, t_0, (t_1 - 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))
	t_1 = Float64(sin(x) * Float64(-sin(eps)))
	tmp = 0.0
	if (eps <= -0.00016)
		tmp = Float64(fma(1.0, t_0, t_1) - cos(x));
	elseif (eps <= 0.000165)
		tmp = fma(eps, Float64(Float64(eps * cos(x)) * -0.5), Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)));
	else
		tmp = fma(1.0, t_0, Float64(t_1 - 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]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[eps, -0.00016], N[(N[(1.0 * t$95$0 + t$95$1), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.000165], N[(eps * N[(N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * t$95$0 + N[(t$95$1 - 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 < -1.60000000000000013e-4

   1. Initial program 28.8

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

   2. Applied egg-rr 0.9

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

   3. Applied egg-rr 1.0

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

   4. Applied egg-rr 0.9

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

   if -1.60000000000000013e-4 < eps < 1.65e-4

   1. Initial program 49.6

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

   2. Taylor expanded in eps around 0 0.2

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

   3. Simplified 0.2

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

   if 1.65e-4 < eps

   1. Initial program 29.8

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

   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 0.9

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

   4. Applied egg-rr 0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \cos x \cdot \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00016:\\ \;\;\;\;\mathsf{fma}\left(1, \cos x \cdot \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.000165:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \cos x\right) \cdot -0.5, \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \cos x \cdot \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right) - \cos x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022165 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))