2cos (problem 3.3.5)

Average Error: 39.4 → 0.5

Time: 19.8s

Precision: binary64

Cost: 45380

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0052:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \mathsf{fma}\left(\sin x, -\sin \varepsilon, -\cos x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0053:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0052)
   (fma (cos x) (cos eps) (fma (sin x) (- (sin eps)) (- (cos x))))
   (if (<= eps 0.0053)
     (-
      (*
       (cos x)
       (+ (* -0.5 (* eps eps)) (* 0.041666666666666664 (pow eps 4.0))))
      (* (sin x) (sin eps)))
     (- (* (cos x) (cos eps)) (fma (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 <= -0.0052) {
		tmp = fma(cos(x), cos(eps), fma(sin(x), -sin(eps), -cos(x)));
	} else if (eps <= 0.0053) {
		tmp = (cos(x) * ((-0.5 * (eps * eps)) + (0.041666666666666664 * pow(eps, 4.0)))) - (sin(x) * sin(eps));
	} else {
		tmp = (cos(x) * cos(eps)) - fma(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 <= -0.0052)
		tmp = fma(cos(x), cos(eps), fma(sin(x), Float64(-sin(eps)), Float64(-cos(x))));
	elseif (eps <= 0.0053)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.5 * Float64(eps * eps)) + Float64(0.041666666666666664 * (eps ^ 4.0)))) - Float64(sin(x) * sin(eps)));
	else
		tmp = Float64(Float64(cos(x) * cos(eps)) - fma(sin(eps), 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, -0.0052], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision]) + (-N[Cos[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0053], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0051999999999999998

   1. Initial program 30.2

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

   2. Applied egg-rr 42.3

      \[\leadsto \color{blue}{\sqrt{{\cos \left(x + \varepsilon\right)}^{2}}} - \cos x \]

   3. Simplified 42.3

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

      [Start] 42.3	\[ \sqrt{{\cos \left(x + \varepsilon\right)}^{2}} - \cos x \]
      unpow2 [=>] 42.3	\[ \sqrt{\color{blue}{\cos \left(x + \varepsilon\right) \cdot \cos \left(x + \varepsilon\right)}} - \cos x \]
      rem-sqrt-square [=>] 42.3	\[ \color{blue}{\left|\cos \left(x + \varepsilon\right)\right|} - \cos x \]
      +-commutative [<=] 42.3	\[ \left|\cos \color{blue}{\left(\varepsilon + x\right)}\right| - \cos x \]

   4. Applied egg-rr 0.8

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

   5. Simplified 0.8

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

      [Start] 0.8	\[ \cos x \cdot \cos \varepsilon + \left(\sin x \cdot \left(-\sin \varepsilon\right) + \left(-\cos x\right)\right) \]
      sub-neg [<=] 0.8	\[ \cos x \cdot \cos \varepsilon + \color{blue}{\left(\sin x \cdot \left(-\sin \varepsilon\right) - \cos x\right)} \]
      fma-def [=>] 0.8	\[ \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right) - \cos x\right)} \]
      fma-neg [=>] 0.8	\[ \mathsf{fma}\left(\cos x, \cos \varepsilon, \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, -\cos x\right)}\right) \]

   if -0.0051999999999999998 < eps < 0.00530000000000000002

   1. Initial program 49.2

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

   2. Applied egg-rr 11.2

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

   3. Taylor expanded in x around inf 48.4

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

   4. Simplified 11.2

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

      [Start] 48.4	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x \]
      +-commutative [=>] 48.4	\[ \color{blue}{\left(\cos \varepsilon \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
      *-commutative [=>] 48.4	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right) - \cos x \]
      *-commutative [<=] 48.4	\[ \left(\cos x \cdot \cos \varepsilon + -1 \cdot \color{blue}{\left(\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      mul-1-neg [=>] 48.4	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(-\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      sub0-neg [<=] 48.4	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(0 - \sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      associate-+r- [=>] 48.4	\[ \color{blue}{\left(\left(\cos x \cdot \cos \varepsilon + 0\right) - \sin \varepsilon \cdot \sin x\right)} - \cos x \]
      +-rgt-identity [=>] 48.4	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin \varepsilon \cdot \sin x\right) - \cos x \]
      associate--r+ [<=] 48.4	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin \varepsilon \cdot \sin x + \cos x\right)} \]
      +-commutative [<=] 48.4	\[ \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin \varepsilon \cdot \sin x\right)} \]
      associate--r+ [=>] 11.2	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x} \]

   5. Taylor expanded in eps around 0 0.2

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

   6. Simplified 0.2

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

      [Start] 0.2	\[ \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) - \sin x \cdot \sin \varepsilon \]
      +-commutative [=>] 0.2	\[ \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right)} - \sin x \cdot \sin \varepsilon \]
      associate-*r* [=>] 0.2	\[ \left(\color{blue}{\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) - \sin x \cdot \sin \varepsilon \]
      associate-*r* [=>] 0.2	\[ \left(\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x + \color{blue}{\left(0.041666666666666664 \cdot {\varepsilon}^{4}\right) \cdot \cos x}\right) - \sin x \cdot \sin \varepsilon \]
      distribute-rgt-out [=>] 0.2	\[ \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)} - \sin x \cdot \sin \varepsilon \]
      unpow2 [=>] 0.2	\[ \cos x \cdot \left(-0.5 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin x \cdot \sin \varepsilon \]

   if 0.00530000000000000002 < eps

   1. Initial program 30.0

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

   2. Applied egg-rr 0.8

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

   3. Simplified 0.8

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

      [Start] 0.8	\[ \cos x \cdot \cos \varepsilon + \left(\sin \varepsilon \cdot \left(-\sin x\right) + \left(-\cos x\right)\right) \]
      *-commutative [=>] 0.8	\[ \cos x \cdot \cos \varepsilon + \left(\color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon} + \left(-\cos x\right)\right) \]
      distribute-lft-neg-in [<=] 0.8	\[ \cos x \cdot \cos \varepsilon + \left(\color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)} + \left(-\cos x\right)\right) \]
      distribute-neg-out [=>] 0.8	\[ \cos x \cdot \cos \varepsilon + \color{blue}{\left(-\left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \]
      unsub-neg [=>] 0.8	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \]
      *-commutative [=>] 0.8	\[ \color{blue}{\cos \varepsilon \cdot \cos x} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \]
      *-commutative [=>] 0.8	\[ \cos \varepsilon \cdot \cos x - \left(\color{blue}{\sin \varepsilon \cdot \sin x} + \cos x\right) \]
      fma-def [=>] 0.8	\[ \cos \varepsilon \cdot \cos x - \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0052:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \mathsf{fma}\left(\sin x, -\sin \varepsilon, -\cos x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0053:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	39112

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0052:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0053:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	39112

\[\begin{array}{l} t_0 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0052:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0053:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

Alternative 3
Error	0.5
Cost	32840

\[\begin{array}{l} t_0 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0052:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0053:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + t_0\right)\\ \end{array} \]

Alternative 4
Error	0.5
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0052:\\ \;\;\;\;\left(t_0 - \cos x\right) - t_1\\ \mathbf{elif}\;\varepsilon \leq 0.0052:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \end{array} \]

Alternative 5
Error	0.5
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0052:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0053:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \end{array} \]

Alternative 6
Error	0.5
Cost	26889

\[\begin{array}{l} t_0 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0052 \lor \neg \left(\varepsilon \leq 0.0052\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \end{array} \]

Alternative 7
Error	0.6
Cost	26441

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-25} \lor \neg \left(x \leq 2.65 \cdot 10^{-20}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \end{array} \]

Alternative 8
Error	15.4
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 9
Error	15.0
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0004 \lor \neg \left(\varepsilon \leq 0.012\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.5\right) - \sin x\right)\\ \end{array} \]

Alternative 10
Error	15.1
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0004 \lor \neg \left(\varepsilon \leq 0.0027\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 11
Error	15.6
Cost	7241

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0004 \lor \neg \left(\varepsilon \leq 0.00165\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 12
Error	20.9
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.8 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 0.00126\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 13
Error	30.9
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 8.8 \cdot 10^{-10}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 - x\right)\\ \end{array} \]

Alternative 14
Error	49.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-162} \lor \neg \left(\varepsilon \leq 2.3 \cdot 10^{-126}\right):\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \end{array} \]

Alternative 15
Error	47.8
Cost	448

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

Alternative 16
Error	52.8
Cost	256

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

Reproduce

herbie shell --seed 2023018 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))