2cos (problem 3.3.5)

Average Accuracy: 38.3% → 97.9%

Time: 20.4s

Precision: binary64

Cost: 32777

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+30} \lor \neg \left(x \leq 1.85 \cdot 10^{-22}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.75e+30) (not (<= x 1.85e-22)))
   (fma (+ (cos eps) -1.0) (cos x) (* (sin eps) (- (sin x))))
   (* (sin (/ (+ eps (- x x)) 2.0)) (* -2.0 (sin (/ (+ eps (+ x x)) 2.0))))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.75e+30) || !(x <= 1.85e-22)) {
		tmp = fma((cos(eps) + -1.0), cos(x), (sin(eps) * -sin(x)));
	} else {
		tmp = sin(((eps + (x - x)) / 2.0)) * (-2.0 * sin(((eps + (x + x)) / 2.0)));
	}
	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 <= -1.75e+30) || !(x <= 1.85e-22))
		tmp = fma(Float64(cos(eps) + -1.0), cos(x), Float64(sin(eps) * Float64(-sin(x))));
	else
		tmp = Float64(sin(Float64(Float64(eps + Float64(x - x)) / 2.0)) * Float64(-2.0 * sin(Float64(Float64(eps + Float64(x + x)) / 2.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := If[Or[LessEqual[x, -1.75e+30], N[Not[LessEqual[x, 1.85e-22]], $MachinePrecision]], N[(N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75000000000000011e30 or 1.85e-22 < x

   1. Initial program 8.3%

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

   2. Applied egg-rr 98.8%

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

      [Start] 8.3	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      sub-neg [=>] 8.3	\[ \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]
      +-commutative [=>] 8.3	\[ \color{blue}{\left(-\cos x\right) + \cos \left(x + \varepsilon\right)} \]
      cos-sum [=>] 53.3	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} \]
      cancel-sign-sub-inv [=>] 53.3	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} \]
      associate-+r+ [=>] 98.8	\[ \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \left(-\sin x\right) \cdot \sin \varepsilon} \]
      *-commutative [=>] 98.8	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} \]

   3. Taylor expanded in x around inf 53.3%

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

   4. Simplified 98.8%

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

      [Start] 53.3	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x \]
      +-commutative [=>] 53.3	\[ \color{blue}{\left(\cos \varepsilon \cdot \cos x + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
      *-commutative [=>] 53.3	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} + -1 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right) - \cos x \]
      *-commutative [<=] 53.3	\[ \left(\cos x \cdot \cos \varepsilon + -1 \cdot \color{blue}{\left(\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      mul-1-neg [=>] 53.3	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(-\sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      sub0-neg [<=] 53.3	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\left(0 - \sin \varepsilon \cdot \sin x\right)}\right) - \cos x \]
      associate-+r- [=>] 53.3	\[ \color{blue}{\left(\left(\cos x \cdot \cos \varepsilon + 0\right) - \sin \varepsilon \cdot \sin x\right)} - \cos x \]
      +-rgt-identity [=>] 53.3	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin \varepsilon \cdot \sin x\right) - \cos x \]
      associate--r+ [<=] 53.3	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin \varepsilon \cdot \sin x + \cos x\right)} \]
      +-commutative [<=] 53.3	\[ \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin \varepsilon \cdot \sin x\right)} \]
      associate--r+ [=>] 98.8	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x} \]

   5. Applied egg-rr 98.8%

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

      [Start] 98.8	\[ \left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin x \cdot \sin \varepsilon \]
      *-un-lft-identity [=>] 98.8	\[ \left(\cos \varepsilon \cdot \cos x - \color{blue}{1 \cdot \cos x}\right) - \sin x \cdot \sin \varepsilon \]
      distribute-rgt-out-- [=>] 98.8	\[ \color{blue}{\cos x \cdot \left(\cos \varepsilon - 1\right)} - \sin x \cdot \sin \varepsilon \]

   6. Applied egg-rr 98.9%

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

      [Start] 98.8	\[ \cos x \cdot \left(\cos \varepsilon - 1\right) - \sin x \cdot \sin \varepsilon \]
      *-commutative [=>] 98.8	\[ \color{blue}{\left(\cos \varepsilon - 1\right) \cdot \cos x} - \sin x \cdot \sin \varepsilon \]
      fma-neg [=>] 98.9	\[ \color{blue}{\mathsf{fma}\left(\cos \varepsilon - 1, \cos x, -\sin x \cdot \sin \varepsilon\right)} \]
      sub-neg [=>] 98.9	\[ \mathsf{fma}\left(\color{blue}{\cos \varepsilon + \left(-1\right)}, \cos x, -\sin x \cdot \sin \varepsilon\right) \]
      metadata-eval [=>] 98.9	\[ \mathsf{fma}\left(\cos \varepsilon + \color{blue}{-1}, \cos x, -\sin x \cdot \sin \varepsilon\right) \]
      distribute-rgt-neg-in [=>] 98.9	\[ \mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \color{blue}{\sin x \cdot \left(-\sin \varepsilon\right)}\right) \]

   if -1.75000000000000011e30 < x < 1.85e-22

   1. Initial program 67.8%

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

   2. Applied egg-rr 67.6%

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

      [Start] 67.8	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      expm1-log1p-u [=>] 67.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x + \varepsilon\right)\right)\right)} - \cos x \]

   3. Applied egg-rr 85.4%

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

      [Start] 67.6	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x + \varepsilon\right)\right)\right) - \cos x \]
      expm1-log1p-u [<=] 67.8	\[ \color{blue}{\cos \left(x + \varepsilon\right)} - \cos x \]
      diff-cos [=>] 85.4	\[ \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]
      associate--l+ [=>] 85.4	\[ -2 \cdot \left(\sin \left(\frac{\color{blue}{x + \left(\varepsilon - x\right)}}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
      +-commutative [=>] 85.4	\[ -2 \cdot \left(\sin \left(\frac{x + \left(\varepsilon - x\right)}{2}\right) \cdot \sin \left(\frac{\color{blue}{x + \left(x + \varepsilon\right)}}{2}\right)\right) \]

   4. Simplified 97.0%

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

      [Start] 85.4	\[ -2 \cdot \left(\sin \left(\frac{x + \left(\varepsilon - x\right)}{2}\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right) \]
      *-commutative [=>] 85.4	\[ \color{blue}{\left(\sin \left(\frac{x + \left(\varepsilon - x\right)}{2}\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right) \cdot -2} \]
      associate-*l* [=>] 85.4	\[ \color{blue}{\sin \left(\frac{x + \left(\varepsilon - x\right)}{2}\right) \cdot \left(\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot -2\right)} \]
      associate-+r- [=>] 85.4	\[ \sin \left(\frac{\color{blue}{\left(x + \varepsilon\right) - x}}{2}\right) \cdot \left(\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot -2\right) \]
      +-commutative [=>] 85.4	\[ \sin \left(\frac{\color{blue}{\left(\varepsilon + x\right)} - x}{2}\right) \cdot \left(\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot -2\right) \]
      associate--l+ [=>] 97.0	\[ \sin \left(\frac{\color{blue}{\varepsilon + \left(x - x\right)}}{2}\right) \cdot \left(\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot -2\right) \]
      *-commutative [=>] 97.0	\[ \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \color{blue}{\left(-2 \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)} \]
      associate-+r+ [=>] 97.0	\[ \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\color{blue}{\left(x + x\right) + \varepsilon}}{2}\right)\right) \]
      +-commutative [=>] 97.0	\[ \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\color{blue}{\varepsilon + \left(x + x\right)}}{2}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+30} \lor \neg \left(x \leq 1.85 \cdot 10^{-22}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.9%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+30} \lor \neg \left(x \leq 1.2 \cdot 10^{-20}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	76.6%
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 3
Accuracy	77.2%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.017 \lor \neg \left(\varepsilon \leq 0.024\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 4
Accuracy	66.6%
Cost	13708

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -1.8 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{-78}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0052:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	66.6%
Cost	13388

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -1.8 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-78}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000125:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	65.8%
Cost	7116

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.8 \cdot 10^{-26}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-78}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000125:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \left(x + \varepsilon\right)\\ \end{array} \]

Alternative 7
Accuracy	65.9%
Cost	6988

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -1.8 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-78}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	46.8%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015 \lor \neg \left(\varepsilon \leq 2.2 \cdot 10^{-5}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \]

Alternative 9
Accuracy	20.7%
Cost	320

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

Reproduce

herbie shell --seed 2023126 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))