2cos (problem 3.3.5)

Average Error: 39.5 → 0.5

Time: 19.3s

Precision: binary64

Cost: 58569

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-17} \lor \neg \left(x \leq 3.1 \cdot 10^{-14}\right):\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \mathsf{fma}\left(\cos x, \cos \varepsilon, -\cos x\right)\right) + \left(\cos x - \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.6e-17) (not (<= x 3.1e-14)))
   (+
    (fma (sin x) (- (sin eps)) (fma (cos x) (cos eps) (- (cos x))))
    (- (cos x) (cos x)))
   (* (sin (* 0.5 (+ eps (- x x)))) (* -2.0 (sin (* 0.5 (+ eps (+ x x))))))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.6e-17) || !(x <= 3.1e-14)) {
		tmp = fma(sin(x), -sin(eps), fma(cos(x), cos(eps), -cos(x))) + (cos(x) - cos(x));
	} else {
		tmp = sin((0.5 * (eps + (x - x)))) * (-2.0 * sin((0.5 * (eps + (x + 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 <= -1.6e-17) || !(x <= 3.1e-14))
		tmp = Float64(fma(sin(x), Float64(-sin(eps)), fma(cos(x), cos(eps), Float64(-cos(x)))) + Float64(cos(x) - cos(x)));
	else
		tmp = Float64(sin(Float64(0.5 * Float64(eps + Float64(x - x)))) * Float64(-2.0 * sin(Float64(0.5 * Float64(eps + Float64(x + 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[Or[LessEqual[x, -1.6e-17], N[Not[LessEqual[x, 3.1e-14]], $MachinePrecision]], N[(N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + (-N[Cos[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(0.5 * N[(eps + N[(x - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * N[Sin[N[(0.5 * N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001e-17 or 3.10000000000000004e-14 < x

   1. Initial program 58.6

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

   2. Applied egg-rr 29.8

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

   3. Applied egg-rr 29.8

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

   4. Simplified 0.8

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

      [Start] 29.8	\[ \cos x \cdot \cos \varepsilon + \left(\left(\left(-\cos x\right) + \sin x \cdot \left(-\sin \varepsilon\right)\right) + \mathsf{fma}\left(-\cos x, 1, \cos x\right)\right) \]
      associate-+r+ [=>] 29.8	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(\left(-\cos x\right) + \sin x \cdot \left(-\sin \varepsilon\right)\right)\right) + \mathsf{fma}\left(-\cos x, 1, \cos x\right)} \]
      associate-+r+ [=>] 0.8	\[ \color{blue}{\left(\left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right) + \sin x \cdot \left(-\sin \varepsilon\right)\right)} + \mathsf{fma}\left(-\cos x, 1, \cos x\right) \]
      +-commutative [<=] 0.8	\[ \color{blue}{\left(\sin x \cdot \left(-\sin \varepsilon\right) + \left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right)\right)} + \mathsf{fma}\left(-\cos x, 1, \cos x\right) \]
      +-commutative [<=] 0.8	\[ \left(\sin x \cdot \left(-\sin \varepsilon\right) + \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right)}\right) + \mathsf{fma}\left(-\cos x, 1, \cos x\right) \]
      fma-def [=>] 0.8	\[ \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right)} + \mathsf{fma}\left(-\cos x, 1, \cos x\right) \]
      +-commutative [=>] 0.8	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos x \cdot \cos \varepsilon + \left(-\cos x\right)}\right) + \mathsf{fma}\left(-\cos x, 1, \cos x\right) \]
      fma-def [=>] 0.8	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\cos x\right)}\right) + \mathsf{fma}\left(-\cos x, 1, \cos x\right) \]
      fma-udef [=>] 0.8	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \mathsf{fma}\left(\cos x, \cos \varepsilon, -\cos x\right)\right) + \color{blue}{\left(\left(-\cos x\right) \cdot 1 + \cos x\right)} \]
      *-rgt-identity [=>] 0.8	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \mathsf{fma}\left(\cos x, \cos \varepsilon, -\cos x\right)\right) + \left(\color{blue}{\left(-\cos x\right)} + \cos x\right) \]

   if -1.6000000000000001e-17 < x < 3.10000000000000004e-14

   1. Initial program 19.2

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

   2. Applied egg-rr 19.4

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\cos \left(x + \varepsilon\right) - \cos x\right)}^{3}}} \]

   3. Applied egg-rr 6.7

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

   4. Simplified 0.3

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

      [Start] 6.7	\[ -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
      *-commutative [=>] 6.7	\[ \color{blue}{\left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \cdot -2} \]
      associate-*l* [=>] 6.7	\[ \color{blue}{\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \left(\sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right) \cdot -2\right)} \]
      *-commutative [=>] 6.7	\[ \sin \color{blue}{\left(0.5 \cdot \left(x + \left(\varepsilon - x\right)\right)\right)} \cdot \left(\sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right) \cdot -2\right) \]
      associate-+r- [=>] 6.7	\[ \sin \left(0.5 \cdot \color{blue}{\left(\left(x + \varepsilon\right) - x\right)}\right) \cdot \left(\sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right) \cdot -2\right) \]
      +-commutative [=>] 6.7	\[ \sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \left(\sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right) \cdot -2\right) \]
      associate--l+ [=>] 0.3	\[ \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \left(\sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right) \cdot -2\right) \]
      *-commutative [=>] 0.3	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \color{blue}{\left(-2 \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
      *-commutative [=>] 0.3	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \left(-2 \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]
      associate-+r+ [=>] 0.3	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)}\right)\right) \]
      +-commutative [=>] 0.3	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x + x\right)\right)}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-17} \lor \neg \left(x \leq 3.1 \cdot 10^{-14}\right):\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \mathsf{fma}\left(\cos x, \cos \varepsilon, -\cos x\right)\right) + \left(\cos x - \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	45700

\[\begin{array}{l} t_0 := -\sin \varepsilon\\ t_1 := \cos \varepsilon + -1\\ \mathbf{if}\;x \leq -9 \cdot 10^{-20}:\\ \;\;\;\;\left(\cos x - \cos x\right) + \mathsf{fma}\left(\sin x, t_0, \cos x \cdot t_1\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-14}:\\ \;\;\;\;\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \cos x, \sin x \cdot t_0\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	32776

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-16}:\\ \;\;\;\;\cos x \cdot t_0 - \sin x \cdot \sin \varepsilon\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-14}:\\ \;\;\;\;\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \cos x, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \end{array} \]

Alternative 3
Error	0.6
Cost	32776

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-14}:\\ \;\;\;\;\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	26441

\[\begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-17} \lor \neg \left(x \leq 4.3 \cdot 10^{-14}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	14.7
Cost	13888

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

Alternative 6
Error	14.4
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0017 \lor \neg \left(\varepsilon \leq 0.002\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x \cdot \varepsilon\\ \end{array} \]

Alternative 7
Error	21.6
Cost	13576

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.0045:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -5.2 \cdot 10^{-123}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 7.2 \cdot 10^{-7}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	21.7
Cost	13388

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.00026:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -4.7 \cdot 10^{-123}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 7.5 \cdot 10^{-7}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	22.1
Cost	7052

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.000102:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -5.2 \cdot 10^{-123}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	33.8
Cost	6988

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.000102:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.12 \cdot 10^{-126}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 3.7 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	48.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-125}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 12
Error	52.0
Cost	256

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

Alternative 13
Error	55.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023066 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))