2cos (problem 3.3.5)

Percentage Accurate: 38.6% → 99.1%

Time: 17.4s

Precision: binary64

Cost: 51976

Math

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

↓

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.000155:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \log \left(e^{\cos \varepsilon}\right), t_0\right)\\ \end{array} \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (- (cos x)) (* (sin x) (sin eps)))))
   (if (<= eps -0.000155)
     (fma (cos x) (cos eps) t_0)
     (if (<= eps 0.00016)
       (+
        (* -0.5 (* eps (* eps (cos x))))
        (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps)))
       (fma (cos x) (log (exp (cos 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 = -cos(x) - (sin(x) * sin(eps));
	double tmp;
	if (eps <= -0.000155) {
		tmp = fma(cos(x), cos(eps), t_0);
	} else if (eps <= 0.00016) {
		tmp = (-0.5 * (eps * (eps * cos(x)))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps));
	} else {
		tmp = fma(cos(x), log(exp(cos(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(Float64(-cos(x)) - Float64(sin(x) * sin(eps)))
	tmp = 0.0
	if (eps <= -0.000155)
		tmp = fma(cos(x), cos(eps), t_0);
	elseif (eps <= 0.00016)
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)));
	else
		tmp = fma(cos(x), log(exp(cos(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[Cos[x], $MachinePrecision]) - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.000155], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[eps, 0.00016], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[Log[N[Exp[N[Cos[eps], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.55e-4

   1. Initial program 46.4%

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

   2. Applied egg-rr 98.6%

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

      [Start] 46.4%	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      sub-neg [=>] 46.4%	\[ \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]
      cos-sum [=>] 98.5%	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} + \left(-\cos x\right) \]
      associate-+l- [=>] 98.5%	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)} \]
      fma-neg [=>] 98.6%	\[ \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)\right)} \]

   3. Taylor expanded in x around inf 98.6%

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

   if -1.55e-4 < eps < 1.60000000000000013e-4

   1. Initial program 23.2%

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

   2. Taylor expanded in eps around 0 99.7%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(\left(-\varepsilon\right) + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)} \]
      Step-by-step derivation

      [Start] 99.7%	\[ 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]
      +-commutative [=>] 99.7%	\[ \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)} \]
      associate-+l+ [=>] 99.7%	\[ \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)} \]
      unpow2 [=>] 99.7%	\[ -0.5 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]
      associate-*l* [=>] 99.7%	\[ -0.5 \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]
      associate-*r* [=>] 99.7%	\[ -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]
      associate-*r* [=>] 99.7%	\[ -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\left(-1 \cdot \varepsilon\right) \cdot \sin x + \color{blue}{\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x}\right) \]
      distribute-rgt-out [=>] 99.7%	\[ -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \color{blue}{\sin x \cdot \left(-1 \cdot \varepsilon + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)} \]
      mul-1-neg [=>] 99.7%	\[ -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(\color{blue}{\left(-\varepsilon\right)} + 0.16666666666666666 \cdot {\varepsilon}^{3}\right) \]

   if 1.60000000000000013e-4 < eps

   1. Initial program 39.9%

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

   2. Applied egg-rr 99.1%

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

      [Start] 39.9%	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      sub-neg [=>] 39.9%	\[ \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]
      cos-sum [=>] 99.1%	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} + \left(-\cos x\right) \]
      associate-+l- [=>] 99.1%	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)} \]
      fma-neg [=>] 99.1%	\[ \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)\right)} \]

   3. Applied egg-rr 99.2%

      \[\leadsto \mathsf{fma}\left(\cos x, \color{blue}{\log \left(e^{\cos \varepsilon}\right)}, -\left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)\right) \]
      Step-by-step derivation

      [Start] 99.1%	\[ \mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)\right) \]
      add-log-exp [=>] 99.2%	\[ \mathsf{fma}\left(\cos x, \color{blue}{\log \left(e^{\cos \varepsilon}\right)}, -\left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000155:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \log \left(e^{\cos \varepsilon}\right), \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	51976

\[\begin{array}{l} t_0 := \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.000155:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \log \left(e^{\cos \varepsilon}\right), t_0\right)\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	39044

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000165:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.000135:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]

Alternative 3
Accuracy	99.1%
Cost	39044

\[\begin{array}{l} t_0 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.000155:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + t_0\right)\\ \end{array} \]

Alternative 4
Accuracy	99.1%
Cost	38980

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.000175:\\ \;\;\;\;t_0 - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00018:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]

Alternative 5
Accuracy	99.1%
Cost	32841

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000175 \lor \neg \left(\varepsilon \leq 0.000155\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \end{array} \]

Alternative 6
Accuracy	99.1%
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.000155:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \end{array} \]

Alternative 7
Accuracy	75.9%
Cost	26432

\[-2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\varepsilon \cdot 0.5\right)\right)\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right) \]

Alternative 8
Accuracy	75.4%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.9 \cdot 10^{+20}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{elif}\;\varepsilon \leq 0.025:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 9
Accuracy	76.0%
Cost	13632

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

Alternative 10
Accuracy	76.0%
Cost	13632

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

Alternative 11
Accuracy	75.0%
Cost	13316

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.9 \cdot 10^{+20}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{elif}\;\varepsilon \leq 0.0125:\\ \;\;\;\;-2 \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 12
Accuracy	76.0%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1700 \lor \neg \left(\varepsilon \leq 0.0065\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

Alternative 13
Accuracy	74.7%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.9 \cdot 10^{+20} \lor \neg \left(\varepsilon \leq 0.0105\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

Alternative 14
Accuracy	64.8%
Cost	7184

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -4.9 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.7 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-103}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Accuracy	47.4%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000155 \lor \neg \left(\varepsilon \leq 0.000115\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 16
Accuracy	21.8%
Cost	320

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

Alternative 17
Accuracy	13.0%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023167 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))