2cos (problem 3.3.5)

Average Error: 62.07% → 0.48%

Time: 16.4s

Precision: binary64

Cost: 39168

Math

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

↓

\[\mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \cdot \cos x\right) \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (fma
  (sin x)
  (- (sin eps))
  (* (* (/ (sin eps) -1.0) (tan (/ eps 2.0))) (cos x))))

C

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

↓

double code(double x, double eps) {
	return fma(sin(x), -sin(eps), (((sin(eps) / -1.0) * tan((eps / 2.0))) * cos(x)));
}

Julia

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end

↓

function code(x, eps)
	return fma(sin(x), Float64(-sin(eps)), Float64(Float64(Float64(sin(eps) / -1.0) * tan(Float64(eps / 2.0))) * cos(x)))
end

Wolfram

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

↓

code[x_, eps_] := N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision]) + N[(N[(N[(N[Sin[eps], $MachinePrecision] / -1.0), $MachinePrecision] * N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.07

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

2. Applied egg-rr 38.43

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

3. Applied egg-rr 38.41

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

4. Taylor expanded in x around inf 38.43

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

5. Simplified 10.53

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

   [Start] 38.43	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x \]
   *-commutative [<=] 38.43	\[ \left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \color{blue}{\cos x \cdot \cos \varepsilon}\right) - \cos x \]
   associate--l+ [=>] 10.55	\[ \color{blue}{-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \left(\cos x \cdot \cos \varepsilon - \cos x\right)} \]
   mul-1-neg [=>] 10.55	\[ \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)} + \left(\cos x \cdot \cos \varepsilon - \cos x\right) \]
   distribute-rgt-neg-in [=>] 10.55	\[ \color{blue}{\sin x \cdot \left(-\sin \varepsilon\right)} + \left(\cos x \cdot \cos \varepsilon - \cos x\right) \]
   sub-neg [=>] 10.55	\[ \sin x \cdot \left(-\sin \varepsilon\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right)} \]
   fma-def [=>] 10.54	\[ \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right)} \]
   *-commutative [=>] 10.54	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos \varepsilon \cdot \cos x} + \left(-\cos x\right)\right) \]
   neg-mul-1 [=>] 10.54	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos \varepsilon \cdot \cos x + \color{blue}{-1 \cdot \cos x}\right) \]
   distribute-rgt-out [=>] 10.53	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)}\right) \]

6. Applied egg-rr 0.99

   \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\frac{\cos x \cdot {\sin \varepsilon}^{2}}{-1 - \cos \varepsilon}}\right) \]

7. Simplified 0.48

   \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \cdot \cos x}\right) \]
   Proof

   [Start] 0.99	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\cos x \cdot {\sin \varepsilon}^{2}}{-1 - \cos \varepsilon}\right) \]
   sub-neg [=>] 0.99	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\cos x \cdot {\sin \varepsilon}^{2}}{\color{blue}{-1 + \left(-\cos \varepsilon\right)}}\right) \]
   +-commutative [=>] 0.99	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\cos x \cdot {\sin \varepsilon}^{2}}{\color{blue}{\left(-\cos \varepsilon\right) + -1}}\right) \]
   metadata-eval [<=] 0.99	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\cos x \cdot {\sin \varepsilon}^{2}}{\left(-\cos \varepsilon\right) + \color{blue}{\left(-1\right)}}\right) \]
   distribute-neg-in [<=] 0.99	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\cos x \cdot {\sin \varepsilon}^{2}}{\color{blue}{-\left(\cos \varepsilon + 1\right)}}\right) \]
   *-commutative [<=] 0.99	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\color{blue}{{\sin \varepsilon}^{2} \cdot \cos x}}{-\left(\cos \varepsilon + 1\right)}\right) \]
   associate-/l* [=>] 0.99	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\frac{{\sin \varepsilon}^{2}}{\frac{-\left(\cos \varepsilon + 1\right)}{\cos x}}}\right) \]
   associate-/r/ [=>] 0.99	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\frac{{\sin \varepsilon}^{2}}{-\left(\cos \varepsilon + 1\right)} \cdot \cos x}\right) \]
   unpow2 [=>] 0.99	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{-\left(\cos \varepsilon + 1\right)} \cdot \cos x\right) \]
   neg-mul-1 [=>] 0.99	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\sin \varepsilon \cdot \sin \varepsilon}{\color{blue}{-1 \cdot \left(\cos \varepsilon + 1\right)}} \cdot \cos x\right) \]
   times-frac [=>] 1	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \frac{\sin \varepsilon}{\cos \varepsilon + 1}\right)} \cdot \cos x\right) \]
   +-commutative [=>] 1	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(\frac{\sin \varepsilon}{-1} \cdot \frac{\sin \varepsilon}{\color{blue}{1 + \cos \varepsilon}}\right) \cdot \cos x\right) \]
   hang-0p-tan [=>] 0.48	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(\frac{\sin \varepsilon}{-1} \cdot \color{blue}{\tan \left(\frac{\varepsilon}{2}\right)}\right) \cdot \cos x\right) \]

8. Final simplification 0.48

   \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \cdot \cos x\right) \]

Alternatives

Alternative 1
Error	1.51%
Cost	32777

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-61} \lor \neg \left(x \leq 5.2 \cdot 10^{-84}\right):\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right) \cdot -2\\ \end{array} \]

Alternative 2
Error	1.52%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-61} \lor \neg \left(x \leq 4 \cdot 10^{-84}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right) \cdot -2\\ \end{array} \]

Alternative 3
Error	23.02%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.013 \lor \neg \left(\varepsilon \leq 0.024\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \sin x \cdot \varepsilon\\ \end{array} \]

Alternative 4
Error	23.51%
Cost	13632

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

Alternative 5
Error	29.59%
Cost	13580

\[\begin{array}{l} t_0 := \sin x \cdot \left(-\varepsilon\right)\\ \mathbf{if}\;x \leq -0.00013:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-112}:\\ \;\;\;\;\left(-1 + \cos \varepsilon\right) - x \cdot \sin \varepsilon\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	32.26%
Cost	13449

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-59} \lor \neg \left(x \leq 2.6 \cdot 10^{-24}\right):\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 7
Error	33.12%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.6 \cdot 10^{-15} \lor \neg \left(\varepsilon \leq 1.35 \cdot 10^{-5}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 8
Error	33.77%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.6 \cdot 10^{-15} \lor \neg \left(\varepsilon \leq 5.6 \cdot 10^{-6}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 9
Error	52.73%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000108 \lor \neg \left(\varepsilon \leq 1.45 \cdot 10^{-13}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \]

Alternative 10
Error	78.26%
Cost	320

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

Reproduce

herbie shell --seed 2023089 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))