2cos (problem 3.3.5)

Average Error: 40.1 → 0.6

Time: 25.3s

Precision: binary64

Cost: 32776

Math

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

↓

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \cos x, -\sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;\left(\sin \left(\left(\left(x + \varepsilon\right) + x\right) \cdot 0.5\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot t_0\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (cos eps) -1.0)))
   (if (<= x -2.4e-11)
     (fma t_0 (cos x) (- (* (sin x) (sin eps))))
     (if (<= x 8.5e-17)
       (* (* (sin (* (+ (+ x eps) x) 0.5)) (sin (/ eps 2.0))) -2.0)
       (fma (sin x) (- (sin eps)) (* (cos x) t_0))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = cos(eps) + -1.0;
	double tmp;
	if (x <= -2.4e-11) {
		tmp = fma(t_0, cos(x), -(sin(x) * sin(eps)));
	} else if (x <= 8.5e-17) {
		tmp = (sin((((x + eps) + x) * 0.5)) * sin((eps / 2.0))) * -2.0;
	} else {
		tmp = fma(sin(x), -sin(eps), (cos(x) * 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(cos(eps) + -1.0)
	tmp = 0.0
	if (x <= -2.4e-11)
		tmp = fma(t_0, cos(x), Float64(-Float64(sin(x) * sin(eps))));
	elseif (x <= 8.5e-17)
		tmp = Float64(Float64(sin(Float64(Float64(Float64(x + eps) + x) * 0.5)) * sin(Float64(eps / 2.0))) * -2.0);
	else
		tmp = fma(sin(x), Float64(-sin(eps)), Float64(cos(x) * 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[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -2.4e-11], N[(t$95$0 * N[Cos[x], $MachinePrecision] + (-N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[x, 8.5e-17], N[(N[(N[Sin[N[(N[(N[(x + eps), $MachinePrecision] + x), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4000000000000001e-11

   1. Initial program 59.0

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

   2. Applied egg-rr 0.8

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

   3. Simplified 0.8

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

   4. Applied egg-rr 0.8

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

   if -2.4000000000000001e-11 < x < 8.5e-17

   1. Initial program 19.1

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

   2. Applied egg-rr 0.3

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

   3. Simplified 0.3

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

   if 8.5e-17 < x

   1. Initial program 58.5

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

   2. Applied egg-rr 0.8

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

   3. Simplified 0.8

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

   4. Applied egg-rr 0.8

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

Alternatives

Alternative 1
Error	0.6
Cost	39300

\[\cos x \cdot \begin{array}{l} \mathbf{if}\;-2 \ne 0:\\ \;\;\;\;\frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} - \sin x \cdot \sin \varepsilon \]

Alternative 2
Error	0.6
Cost	32776

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + -1\right)\right)\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\left(\sin \left(\left(\left(x + \varepsilon\right) + x\right) \cdot 0.5\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.6
Cost	26440

\[\begin{array}{l} t_0 := \cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-15}:\\ \;\;\;\;\left(\sin \left(\left(\left(x + \varepsilon\right) + x\right) \cdot 0.5\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	15.3
Cost	13632

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

Alternative 5
Error	21.1
Cost	13448

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \varepsilon\right)}^{2} \cdot -2\\ \mathbf{if}\;\varepsilon \leq -1.5 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-27}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;\varepsilon \ne 0:\\ \;\;\;\;\frac{-\sin x}{\frac{1}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\varepsilon\right) \cdot \sin x\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	20.8
Cost	13256

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -1.15 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;\varepsilon \ne 0:\\ \;\;\;\;\frac{-\sin x}{\frac{1}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\varepsilon\right) \cdot \sin x\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	21.2
Cost	7180

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;\varepsilon \ne 0:\\ \;\;\;\;\frac{-\sin x}{\frac{1}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\varepsilon\right) \cdot \sin x\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	21.2
Cost	6920

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.26 \cdot 10^{-7}:\\ \;\;\;\;\left(-\varepsilon\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	34.6
Cost	6856

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -0.00016:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.00015:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	51.0
Cost	320

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

Reproduce

herbie shell --seed 2023010 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))