2cos (problem 3.3.5)

Average Accuracy: 38.4% → 98.9%

Time: 19.9s

Precision: binary64

Cost: 32777

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0031 \lor \neg \left(\varepsilon \leq 0.00095\right):\\ \;\;\;\;\mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0031) (not (<= eps 0.00095)))
   (fma (- (sin x)) (sin eps) (* (cos x) (+ -1.0 (cos eps))))
   (* (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 ((eps <= -0.0031) || !(eps <= 0.00095)) {
		tmp = fma(-sin(x), sin(eps), (cos(x) * (-1.0 + cos(eps))));
	} 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 ((eps <= -0.0031) || !(eps <= 0.00095))
		tmp = fma(Float64(-sin(x)), sin(eps), Float64(cos(x) * Float64(-1.0 + cos(eps))));
	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[eps, -0.0031], N[Not[LessEqual[eps, 0.00095]], $MachinePrecision]], N[((-N[Sin[x], $MachinePrecision]) * N[Sin[eps], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $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 eps < -0.00309999999999999989 or 9.49999999999999998e-4 < eps

   1. Initial program 53.5%

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

   2. Applied egg-rr 98.7%

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

      [Start] 53.5	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      cos-sum [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

   3. Taylor expanded in x around inf 98.7%

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

   4. Simplified 98.7%

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

      [Start] 98.7	\[ \left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x \]
      *-commutative [<=] 98.7	\[ \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin x \cdot \sin \varepsilon\right) - \cos x \]
      sub-neg [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
      distribute-rgt-neg-out [<=] 98.7	\[ \left(\cos x \cdot \cos \varepsilon + \color{blue}{\sin x \cdot \left(-\sin \varepsilon\right)}\right) - \cos x \]
      +-commutative [<=] 98.7	\[ \color{blue}{\left(\sin x \cdot \left(-\sin \varepsilon\right) + \cos x \cdot \cos \varepsilon\right)} - \cos x \]
      fma-def [=>] 98.7	\[ \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon\right)} - \cos x \]
      *-commutative [=>] 98.7	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos \varepsilon \cdot \cos x}\right) - \cos x \]

   5. Taylor expanded in x around inf 98.7%

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

   6. Simplified 98.8%

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

      [Start] 98.7	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x \]
      associate--l+ [=>] 98.7	\[ \color{blue}{-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \left(\cos \varepsilon \cdot \cos x - \cos x\right)} \]
      associate-*r* [=>] 98.7	\[ \color{blue}{\left(-1 \cdot \sin x\right) \cdot \sin \varepsilon} + \left(\cos \varepsilon \cdot \cos x - \cos x\right) \]
      fma-def [=>] 98.7	\[ \color{blue}{\mathsf{fma}\left(-1 \cdot \sin x, \sin \varepsilon, \cos \varepsilon \cdot \cos x - \cos x\right)} \]
      mul-1-neg [=>] 98.7	\[ \mathsf{fma}\left(\color{blue}{-\sin x}, \sin \varepsilon, \cos \varepsilon \cdot \cos x - \cos x\right) \]
      sub-neg [=>] 98.7	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \color{blue}{\cos \varepsilon \cdot \cos x + \left(-\cos x\right)}\right) \]
      neg-mul-1 [=>] 98.7	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos \varepsilon \cdot \cos x + \color{blue}{-1 \cdot \cos x}\right) \]
      distribute-rgt-out [=>] 98.8	\[ \mathsf{fma}\left(-\sin x, \sin \varepsilon, \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)}\right) \]

   if -0.00309999999999999989 < eps < 9.49999999999999998e-4

   1. Initial program 23.4%

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

   2. Applied egg-rr 23.4%

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

      [Start] 23.4	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      add-cbrt-cube [=>] 23.4	\[ \color{blue}{\sqrt[3]{\left(\left(\cos \left(x + \varepsilon\right) - \cos x\right) \cdot \left(\cos \left(x + \varepsilon\right) - \cos x\right)\right) \cdot \left(\cos \left(x + \varepsilon\right) - \cos x\right)}} \]
      pow3 [=>] 23.4	\[ \sqrt[3]{\color{blue}{{\left(\cos \left(x + \varepsilon\right) - \cos x\right)}^{3}}} \]

   3. Applied egg-rr 41.2%

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

      [Start] 23.4	\[ \sqrt[3]{{\left(\cos \left(x + \varepsilon\right) - \cos x\right)}^{3}} \]
      rem-cbrt-cube [=>] 23.4	\[ \color{blue}{\cos \left(x + \varepsilon\right) - \cos x} \]
      diff-cos [=>] 41.2	\[ \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]
      div-inv [=>] 41.2	\[ -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
      associate--l+ [=>] 41.2	\[ -2 \cdot \left(\sin \left(\color{blue}{\left(x + \left(\varepsilon - x\right)\right)} \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
      metadata-eval [=>] 41.2	\[ -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
      div-inv [=>] 41.2	\[ -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]
      +-commutative [=>] 41.2	\[ -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]
      metadata-eval [=>] 41.2	\[ -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

   4. Simplified 99.1%

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

      [Start] 41.2	\[ -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
      *-commutative [=>] 41.2	\[ \color{blue}{\left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \cdot -2} \]
      associate-*l* [=>] 41.2	\[ \color{blue}{\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \left(\sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right) \cdot -2\right)} \]
      *-commutative [=>] 41.2	\[ \sin \color{blue}{\left(0.5 \cdot \left(x + \left(\varepsilon - x\right)\right)\right)} \cdot \left(\sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right) \cdot -2\right) \]
      associate-+r- [=>] 41.2	\[ \sin \left(0.5 \cdot \color{blue}{\left(\left(x + \varepsilon\right) - x\right)}\right) \cdot \left(\sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right) \cdot -2\right) \]
      +-commutative [=>] 41.2	\[ \sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \left(\sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right) \cdot -2\right) \]
      associate--l+ [=>] 99.1	\[ \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \left(\sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right) \cdot -2\right) \]
      *-commutative [=>] 99.1	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \color{blue}{\left(-2 \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
      *-commutative [=>] 99.1	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \left(-2 \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]
      associate-+r+ [=>] 99.1	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)}\right)\right) \]
      +-commutative [=>] 99.1	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x + x\right)\right)}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0031 \lor \neg \left(\varepsilon \leq 0.00095\right):\\ \;\;\;\;\mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	39168

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

Alternative 2
Accuracy	98.9%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0031 \lor \neg \left(\varepsilon \leq 1.32 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	77.0%
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 4
Accuracy	77.6%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00295 \lor \neg \left(\varepsilon \leq 0.011\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 5
Accuracy	68.1%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.6 \cdot 10^{-18} \lor \neg \left(\varepsilon \leq 2.32 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 6
Accuracy	67.4%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.6 \cdot 10^{-18} \lor \neg \left(\varepsilon \leq 8.1 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 7
Accuracy	47.2%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00014 \lor \neg \left(\varepsilon \leq 0.000112\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 8
Accuracy	21.4%
Cost	320

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

Reproduce

herbie shell --seed 2023135 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))