2cos (problem 3.3.5)

Average Error: 61.74% → 0.75%

Time: 15.9s

Precision: binary64

Cost: 33161

Math

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

↓

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.005 \lor \neg \left(\varepsilon \leq 0.0047\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))))
   (if (or (<= eps -0.005) (not (<= eps 0.0047)))
     (- (* (cos x) (+ (cos eps) -1.0)) t_0)
     (-
      (* (cos x) (fma 0.041666666666666664 (pow eps 4.0) (* -0.5 (* eps eps))))
      t_0))))

C

double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}

↓

double code(double x, double eps) {
	double t_0 = sin(eps) * sin(x);
	double tmp;
	if ((eps <= -0.005) || !(eps <= 0.0047)) {
		tmp = (cos(x) * (cos(eps) + -1.0)) - t_0;
	} else {
		tmp = (cos(x) * fma(0.041666666666666664, pow(eps, 4.0), (-0.5 * (eps * eps)))) - t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end

↓

function code(x, eps)
	t_0 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if ((eps <= -0.005) || !(eps <= 0.0047))
		tmp = Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - t_0);
	else
		tmp = Float64(Float64(cos(x) * fma(0.041666666666666664, (eps ^ 4.0), Float64(-0.5 * Float64(eps * eps)))) - 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[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -0.005], N[Not[LessEqual[eps, 0.0047]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision] + N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0050000000000000001 or 0.00470000000000000018 < eps

   1. Initial program 47.82

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

   2. Applied egg-rr 1.28

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

   3. Applied egg-rr 1.26

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

   4. Applied egg-rr 1.26

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

   if -0.0050000000000000001 < eps < 0.00470000000000000018

   1. Initial program 75.95

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

   2. Applied egg-rr 17.83

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

   3. Applied egg-rr 17.82

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

   4. Taylor expanded in eps around 0 0.23

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

   5. Simplified 0.23

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

      [Start] 0.23	\[ \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      fma-def [=>] 0.23	\[ \cos x \cdot \color{blue}{\mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot {\varepsilon}^{2}\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      unpow2 [=>] 0.23	\[ \cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.75

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.005 \lor \neg \left(\varepsilon \leq 0.0047\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternatives

Alternative 1
Error	1.24%
Cost	58688

\[\begin{array}{l} t_0 := \sqrt{\cos \varepsilon + 1}\\ \frac{\frac{-{\sin \varepsilon}^{2} \cdot \cos x}{t_0}}{t_0} - \sin \varepsilon \cdot \sin x \end{array} \]

Alternative 2
Error	0.85%
Cost	26441

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00015 \lor \neg \left(\varepsilon \leq 0.00018\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - t_0\\ \end{array} \]

Alternative 3
Error	23.98%
Cost	13888

\[-2 \cdot \left(\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right) \]

Alternative 4
Error	23.45%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -830 \lor \neg \left(\varepsilon \leq 0.0132\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 5
Error	32.12%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 2.4 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 6
Error	32.8%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 6.5 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 7
Error	53.42%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015 \lor \neg \left(\varepsilon \leq 2.5 \cdot 10^{-12}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 8
Error	78.94%
Cost	320

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

Reproduce

herbie shell --seed 2023090 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))