2cos (problem 3.3.5)

Average Error: 62.03% → 0.83%

Time: 18.7s

Precision: binary64

Cost: 39044

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-16}:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.6e-15)
   (- (fma (cos x) (cos eps) (- (cos x))) (* (sin eps) (sin x)))
   (if (<= x 1.8e-16)
     (* (sin (/ (+ eps (- x x)) 2.0)) (* -2.0 (sin (/ (+ eps (+ x x)) 2.0))))
     (fma (+ -1.0 (cos eps)) (cos x) (* (sin eps) (- (sin x)))))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if (x <= -3.6e-15) {
		tmp = fma(cos(x), cos(eps), -cos(x)) - (sin(eps) * sin(x));
	} else if (x <= 1.8e-16) {
		tmp = sin(((eps + (x - x)) / 2.0)) * (-2.0 * sin(((eps + (x + x)) / 2.0)));
	} else {
		tmp = fma((-1.0 + cos(eps)), cos(x), (sin(eps) * -sin(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 (x <= -3.6e-15)
		tmp = Float64(fma(cos(x), cos(eps), Float64(-cos(x))) - Float64(sin(eps) * sin(x)));
	elseif (x <= 1.8e-16)
		tmp = Float64(sin(Float64(Float64(eps + Float64(x - x)) / 2.0)) * Float64(-2.0 * sin(Float64(Float64(eps + Float64(x + x)) / 2.0))));
	else
		tmp = fma(Float64(-1.0 + cos(eps)), cos(x), Float64(sin(eps) * Float64(-sin(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[x, -3.6e-15], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + (-N[Cos[x], $MachinePrecision])), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8e-16], N[(N[Sin[N[(N[(eps + N[(x - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * N[Sin[N[(N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.6000000000000001e-15

   1. Initial program 91.82

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

   2. Applied egg-rr 1.13

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

   3. Applied egg-rr 1.13

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

   4. Applied egg-rr 1.1

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

   if -3.6000000000000001e-15 < x < 1.79999999999999991e-16

   1. Initial program 29.68

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

   2. Applied egg-rr 29.9

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x + \varepsilon\right)\right)\right)} - \cos x \]

   3. Applied egg-rr 10.27

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

   4. Simplified 0.41

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

      [Start] 10.27	\[ -2 \cdot \left(\sin \left(\frac{x + \left(\varepsilon - x\right)}{2}\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right) \]
      *-commutative [=>] 10.27	\[ \color{blue}{\left(\sin \left(\frac{x + \left(\varepsilon - x\right)}{2}\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right) \cdot -2} \]
      associate-*l* [=>] 10.26	\[ \color{blue}{\sin \left(\frac{x + \left(\varepsilon - x\right)}{2}\right) \cdot \left(\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot -2\right)} \]
      associate-+r- [=>] 10.26	\[ \sin \left(\frac{\color{blue}{\left(x + \varepsilon\right) - x}}{2}\right) \cdot \left(\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot -2\right) \]
      +-commutative [=>] 10.26	\[ \sin \left(\frac{\color{blue}{\left(\varepsilon + x\right)} - x}{2}\right) \cdot \left(\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot -2\right) \]
      associate--l+ [=>] 0.42	\[ \sin \left(\frac{\color{blue}{\varepsilon + \left(x - x\right)}}{2}\right) \cdot \left(\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot -2\right) \]
      *-commutative [=>] 0.42	\[ \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \color{blue}{\left(-2 \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)} \]
      associate-+r+ [=>] 0.41	\[ \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\color{blue}{\left(x + x\right) + \varepsilon}}{2}\right)\right) \]
      +-commutative [=>] 0.41	\[ \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\color{blue}{\varepsilon + \left(x + x\right)}}{2}\right)\right) \]

   if 1.79999999999999991e-16 < x

   1. Initial program 90.99

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

   2. Applied egg-rr 1.38

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

   3. Applied egg-rr 1.36

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

   4. Taylor expanded in x around inf 1.36

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

   5. Simplified 1.33

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

      [Start] 1.36	\[ -1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos x \cdot \left(\cos \varepsilon - 1\right) \]
      +-commutative [=>] 1.36	\[ \color{blue}{\cos x \cdot \left(\cos \varepsilon - 1\right) + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)} \]
      *-commutative [=>] 1.36	\[ \color{blue}{\left(\cos \varepsilon - 1\right) \cdot \cos x} + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right) \]
      sub-neg [=>] 1.36	\[ \color{blue}{\left(\cos \varepsilon + \left(-1\right)\right)} \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right) \]
      metadata-eval [=>] 1.36	\[ \left(\cos \varepsilon + \color{blue}{-1}\right) \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right) \]
      *-commutative [<=] 1.36	\[ \left(\cos \varepsilon + -1\right) \cdot \cos x + -1 \cdot \color{blue}{\left(\sin \varepsilon \cdot \sin x\right)} \]
      mul-1-neg [=>] 1.36	\[ \left(\cos \varepsilon + -1\right) \cdot \cos x + \color{blue}{\left(-\sin \varepsilon \cdot \sin x\right)} \]
      sub-neg [<=] 1.36	\[ \color{blue}{\left(\cos \varepsilon + -1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x} \]
      fma-neg [=>] 1.33	\[ \color{blue}{\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, -\sin \varepsilon \cdot \sin x\right)} \]
      distribute-rgt-neg-in [=>] 1.33	\[ \mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.83

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-16}:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.02%
Cost	39168

\[\frac{{\sin \varepsilon}^{2} \cdot \cos x}{-1 - \cos \varepsilon} - \sin \varepsilon \cdot \sin x \]

Alternative 2
Error	0.84%
Cost	32776

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{-15}:\\ \;\;\;\;\cos x \cdot t_0 - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-16}:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternative 3
Error	0.86%
Cost	26568

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-16}:\\ \;\;\;\;\cos x \cdot t_0 - t_1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x}{\frac{1}{t_0}} - t_1\\ \end{array} \]

Alternative 4
Error	0.85%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-16} \lor \neg \left(x \leq 2.15 \cdot 10^{-16}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \end{array} \]

Alternative 5
Error	23.63%
Cost	13888

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

Alternative 6
Error	23.16%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0148 \lor \neg \left(\varepsilon \leq 0.028\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 7
Error	23.39%
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00275 \lor \neg \left(\varepsilon \leq 0.0075\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 8
Error	32.43%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 6.2 \cdot 10^{-5}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 9
Error	32.99%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 6 \cdot 10^{-6}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 10
Error	53.07%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015 \lor \neg \left(\varepsilon \leq 1.65 \cdot 10^{-6}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \]

Alternative 11
Error	78.9%
Cost	320

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

Reproduce

herbie shell --seed 2023115 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))