2cos (problem 3.3.5)

Average Error: 61.66% → 0.48%

Time: 15.4s

Precision: binary64

Cost: 32512

Math

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

↓

\[\mathsf{fma}\left(\cos x, \tan \left(\varepsilon \cdot 0.5\right), \sin x\right) \cdot \left(-\sin \varepsilon\right) \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (* (fma (cos x) (tan (* eps 0.5)) (sin x)) (- (sin eps))))

C

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

↓

double code(double x, double eps) {
	return fma(cos(x), tan((eps * 0.5)), sin(x)) * -sin(eps);
}

Julia

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

↓

function code(x, eps)
	return Float64(fma(cos(x), tan(Float64(eps * 0.5)), sin(x)) * Float64(-sin(eps)))
end

Wolfram

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

↓

code[x_, eps_] := N[(N[(N[Cos[x], $MachinePrecision] * N[Tan[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 61.66

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

2. Applied egg-rr 37.71

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

3. Simplified 9.82

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

   [Start] 37.71	\[ \cos x \cdot \cos \varepsilon + \left(\sin \varepsilon \cdot \left(-\sin x\right) + \left(-\cos x\right)\right) \]
   +-commutative [=>] 37.71	\[ \cos x \cdot \cos \varepsilon + \color{blue}{\left(\left(-\cos x\right) + \sin \varepsilon \cdot \left(-\sin x\right)\right)} \]
   *-commutative [=>] 37.71	\[ \cos x \cdot \cos \varepsilon + \left(\left(-\cos x\right) + \color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon}\right) \]
   distribute-lft-neg-in [<=] 37.71	\[ \cos x \cdot \cos \varepsilon + \left(\left(-\cos x\right) + \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)}\right) \]
   associate-+r+ [=>] 9.86	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right) + \left(-\sin x \cdot \sin \varepsilon\right)} \]
   +-commutative [<=] 9.86	\[ \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right)} + \left(-\sin x \cdot \sin \varepsilon\right) \]
   +-commutative [=>] 9.86	\[ \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right) + \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right)} \]
   distribute-rgt-neg-in [=>] 9.86	\[ \color{blue}{\sin x \cdot \left(-\sin \varepsilon\right)} + \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) \]
   fma-def [=>] 9.85	\[ \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right)} \]
   +-commutative [=>] 9.85	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos x \cdot \cos \varepsilon + \left(-\cos x\right)}\right) \]
   *-commutative [=>] 9.85	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos \varepsilon \cdot \cos x} + \left(-\cos x\right)\right) \]
   neg-mul-1 [=>] 9.85	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos \varepsilon \cdot \cos x + \color{blue}{-1 \cdot \cos x}\right) \]
   distribute-rgt-out [=>] 9.82	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)}\right) \]

4. Applied egg-rr 1.07

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

5. Simplified 1.04

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

   [Start] 1.07	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \frac{\sin \varepsilon}{\frac{-\left(\cos \varepsilon + 1\right)}{\sin \varepsilon}}\right) \]
   associate-/r/ [=>] 1.04	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\left(\frac{\sin \varepsilon}{-\left(\cos \varepsilon + 1\right)} \cdot \sin \varepsilon\right)}\right) \]
   distribute-neg-in [=>] 1.04	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\frac{\sin \varepsilon}{\color{blue}{\left(-\cos \varepsilon\right) + \left(-1\right)}} \cdot \sin \varepsilon\right)\right) \]
   metadata-eval [=>] 1.04	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\frac{\sin \varepsilon}{\left(-\cos \varepsilon\right) + \color{blue}{-1}} \cdot \sin \varepsilon\right)\right) \]

6. Taylor expanded in x around inf 1.03

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

7. Simplified 0.49

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

   [Start] 1.03	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, -1 \cdot \frac{\cos x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon}\right) \]
   +-commutative [=>] 1.03	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, -1 \cdot \frac{\cos x \cdot {\sin \varepsilon}^{2}}{\color{blue}{\cos \varepsilon + 1}}\right) \]
   associate-*r/ [=>] 1.03	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\frac{-1 \cdot \left(\cos x \cdot {\sin \varepsilon}^{2}\right)}{\cos \varepsilon + 1}}\right) \]
   *-commutative [=>] 1.03	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{-1 \cdot \color{blue}{\left({\sin \varepsilon}^{2} \cdot \cos x\right)}}{\cos \varepsilon + 1}\right) \]
   associate-*r* [=>] 1.03	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\color{blue}{\left(-1 \cdot {\sin \varepsilon}^{2}\right) \cdot \cos x}}{\cos \varepsilon + 1}\right) \]
   neg-mul-1 [<=] 1.03	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\color{blue}{\left(-{\sin \varepsilon}^{2}\right)} \cdot \cos x}{\cos \varepsilon + 1}\right) \]
   unpow2 [=>] 1.03	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\left(-\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}\right) \cdot \cos x}{\cos \varepsilon + 1}\right) \]
   distribute-lft-neg-out [<=] 1.03	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\color{blue}{\left(\left(-\sin \varepsilon\right) \cdot \sin \varepsilon\right)} \cdot \cos x}{\cos \varepsilon + 1}\right) \]
   associate-/l* [=>] 1.04	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\frac{\left(-\sin \varepsilon\right) \cdot \sin \varepsilon}{\frac{\cos \varepsilon + 1}{\cos x}}}\right) \]
   associate-/r/ [=>] 1.03	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\frac{\left(-\sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon + 1} \cdot \cos x}\right) \]
   associate-*r/ [<=] 1.04	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\left(\left(-\sin \varepsilon\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon + 1}\right)} \cdot \cos x\right) \]
   *-commutative [<=] 1.04	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\left(\frac{\sin \varepsilon}{\cos \varepsilon + 1} \cdot \left(-\sin \varepsilon\right)\right)} \cdot \cos x\right) \]
   +-commutative [<=] 1.04	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(\frac{\sin \varepsilon}{\color{blue}{1 + \cos \varepsilon}} \cdot \left(-\sin \varepsilon\right)\right) \cdot \cos x\right) \]
   hang-0p-tan [=>] 0.49	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(\color{blue}{\tan \left(\frac{\varepsilon}{2}\right)} \cdot \left(-\sin \varepsilon\right)\right) \cdot \cos x\right) \]

8. Applied egg-rr 0.5

   \[\leadsto \color{blue}{\sin \varepsilon \cdot \left(\sin x \cdot -1 - \tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x\right)} \]

9. Simplified 0.48

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \tan \left(\varepsilon \cdot 0.5\right), \sin x\right) \cdot \left(-\sin \varepsilon\right)} \]
   Proof

   [Start] 0.5	\[ \sin \varepsilon \cdot \left(\sin x \cdot -1 - \tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x\right) \]
   sub-neg [=>] 0.5	\[ \sin \varepsilon \cdot \color{blue}{\left(\sin x \cdot -1 + \left(-\tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x\right)\right)} \]
   +-commutative [=>] 0.5	\[ \sin \varepsilon \cdot \color{blue}{\left(\left(-\tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x\right) + \sin x \cdot -1\right)} \]
   *-commutative [=>] 0.5	\[ \sin \varepsilon \cdot \left(\left(-\tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x\right) + \color{blue}{-1 \cdot \sin x}\right) \]
   mul-1-neg [=>] 0.5	\[ \sin \varepsilon \cdot \left(\left(-\tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x\right) + \color{blue}{\left(-\sin x\right)}\right) \]
   distribute-neg-in [<=] 0.5	\[ \sin \varepsilon \cdot \color{blue}{\left(-\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x + \sin x\right)\right)} \]
   distribute-rgt-neg-in [<=] 0.5	\[ \color{blue}{-\sin \varepsilon \cdot \left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x + \sin x\right)} \]
   distribute-lft-neg-in [=>] 0.5	\[ \color{blue}{\left(-\sin \varepsilon\right) \cdot \left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x + \sin x\right)} \]
   *-commutative [=>] 0.5	\[ \color{blue}{\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x + \sin x\right) \cdot \left(-\sin \varepsilon\right)} \]
   *-commutative [=>] 0.5	\[ \left(\color{blue}{\cos x \cdot \tan \left(\varepsilon \cdot 0.5\right)} + \sin x\right) \cdot \left(-\sin \varepsilon\right) \]
   fma-def [=>] 0.48	\[ \color{blue}{\mathsf{fma}\left(\cos x, \tan \left(\varepsilon \cdot 0.5\right), \sin x\right)} \cdot \left(-\sin \varepsilon\right) \]

10. Final simplification 0.48

    \[\leadsto \mathsf{fma}\left(\cos x, \tan \left(\varepsilon \cdot 0.5\right), \sin x\right) \cdot \left(-\sin \varepsilon\right) \]

Alternatives

Alternative 1
Error	0.94%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-8} \lor \neg \left(x \leq 1.4 \cdot 10^{-37}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon \cdot \left(\left(-x\right) - \frac{\sin \left(\varepsilon \cdot 0.5\right)}{\cos \left(\varepsilon \cdot 0.5\right)}\right)\\ \end{array} \]

Alternative 2
Error	0.5%
Cost	26240

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

Alternative 3
Error	23.86%
Cost	13888

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

Alternative 4
Error	23.24%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.013 \lor \neg \left(\varepsilon \leq 0.033\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 5
Error	33.66%
Cost	13388

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -18000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 9 \cdot 10^{-84}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.005:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	34.23%
Cost	7436

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -1.4 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.1 \cdot 10^{-82}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0051:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	50.43%
Cost	7120

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{if}\;\varepsilon \leq -0.000165:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.18 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{-140}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	34.27%
Cost	6988

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -1.4 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.6 \cdot 10^{-86}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	76.96%
Cost	585

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.5 \cdot 10^{-155} \lor \neg \left(\varepsilon \leq 4.4 \cdot 10^{-140}\right):\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 10
Error	82.61%
Cost	256

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

Alternative 11
Error	87.54%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))