2cos (problem 3.3.5)

Average Error: 39.9 → 0.5

Time: 8.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000185:\\ \;\;\;\;\left(\mathsf{fma}\left(\cos x, \cos \varepsilon, 0\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.000155:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) - \sin x \cdot \mathsf{fma}\left({\varepsilon}^{3}, -0.16666666666666666, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \log \left(1 + \mathsf{expm1}\left(\sin \varepsilon\right)\right) \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.000185)
   (- (- (fma (cos x) (cos eps) 0.0) (* (sin x) (sin eps))) (cos x))
   (if (<= eps 0.000155)
     (-
      (* (cos x) (* (* eps eps) -0.5))
      (* (sin x) (fma (pow eps 3.0) -0.16666666666666666 eps)))
     (-
      (fma (cos x) (cos eps) (* (log (+ 1.0 (expm1 (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 tmp;
	if (eps <= -0.000185) {
		tmp = (fma(cos(x), cos(eps), 0.0) - (sin(x) * sin(eps))) - cos(x);
	} else if (eps <= 0.000155) {
		tmp = (cos(x) * ((eps * eps) * -0.5)) - (sin(x) * fma(pow(eps, 3.0), -0.16666666666666666, eps));
	} else {
		tmp = fma(cos(x), cos(eps), (log((1.0 + expm1(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)
	tmp = 0.0
	if (eps <= -0.000185)
		tmp = Float64(Float64(fma(cos(x), cos(eps), 0.0) - Float64(sin(x) * sin(eps))) - cos(x));
	elseif (eps <= 0.000155)
		tmp = Float64(Float64(cos(x) * Float64(Float64(eps * eps) * -0.5)) - Float64(sin(x) * fma((eps ^ 3.0), -0.16666666666666666, eps)));
	else
		tmp = Float64(fma(cos(x), cos(eps), Float64(log(Float64(1.0 + expm1(sin(eps)))) * Float64(-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_] := If[LessEqual[eps, -0.000185], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + 0.0), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.000155], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[(N[Power[eps, 3.0], $MachinePrecision] * -0.16666666666666666 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Log[N[(1.0 + N[(Exp[N[Sin[eps], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if eps < -1.85e-4

   1. Initial program 31.1

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

   2. Applied egg-rr 0.8

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

   3. Applied egg-rr 0.9

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\cos x, \cos \varepsilon, 0\right) - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

   if -1.85e-4 < eps < 1.55e-4

   1. Initial program 48.9

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

   2. Applied egg-rr 48.3

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

   3. Taylor expanded in eps around 0 0.2

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

   4. Simplified 0.2

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

   if 1.55e-4 < eps

   1. Initial program 29.7

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

   2. Applied egg-rr 0.8

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

   3. Applied egg-rr 0.9

      \[\leadsto \mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin x \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\sin \varepsilon\right)\right)}\right) - \cos x \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000185:\\ \;\;\;\;\left(\mathsf{fma}\left(\cos x, \cos \varepsilon, 0\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.000155:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) - \sin x \cdot \mathsf{fma}\left({\varepsilon}^{3}, -0.16666666666666666, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \log \left(1 + \mathsf{expm1}\left(\sin \varepsilon\right)\right) \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))