2cos (problem 3.3.5)

Average Error: 37.8% → 99.5%

Time: 16.2s

Precision: binary64

Cost: 32512.00

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 37.8

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

2. Applied egg-rr 61.6

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

3. Simplified 90.0

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

   [Start] 61.6	\[ \cos x \cdot \cos \varepsilon + \left(\sin \varepsilon \cdot \left(-\sin x\right) + \left(-\cos x\right)\right) \]
   +-commutative [=>] 61.6	\[ \cos x \cdot \cos \varepsilon + \color{blue}{\left(\left(-\cos x\right) + \sin \varepsilon \cdot \left(-\sin x\right)\right)} \]
   *-commutative [=>] 61.6	\[ \cos x \cdot \cos \varepsilon + \left(\left(-\cos x\right) + \color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon}\right) \]
   distribute-lft-neg-in [<=] 61.6	\[ \cos x \cdot \cos \varepsilon + \left(\left(-\cos x\right) + \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)}\right) \]
   associate-+r+ [=>] 90.0	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right) + \left(-\sin x \cdot \sin \varepsilon\right)} \]
   +-commutative [<=] 90.0	\[ \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right)} + \left(-\sin x \cdot \sin \varepsilon\right) \]
   +-commutative [=>] 90.0	\[ \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right) + \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right)} \]
   distribute-rgt-neg-in [=>] 90.0	\[ \color{blue}{\sin x \cdot \left(-\sin \varepsilon\right)} + \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) \]
   fma-def [=>] 90.0	\[ \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right)} \]
   +-commutative [=>] 90.0	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos x \cdot \cos \varepsilon + \left(-\cos x\right)}\right) \]
   *-commutative [=>] 90.0	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos \varepsilon \cdot \cos x} + \left(-\cos x\right)\right) \]
   neg-mul-1 [=>] 90.0	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos \varepsilon \cdot \cos x + \color{blue}{-1 \cdot \cos x}\right) \]
   distribute-rgt-out [=>] 90.0	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)}\right) \]

4. Applied egg-rr 99.1

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

5. Simplified 99.5

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

   [Start] 99.1	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\cos x \cdot {\sin \varepsilon}^{2}}{-1 - \cos \varepsilon}\right) \]
   sub-neg [=>] 99.1	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\cos x \cdot {\sin \varepsilon}^{2}}{\color{blue}{-1 + \left(-\cos \varepsilon\right)}}\right) \]
   +-commutative [=>] 99.1	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\cos x \cdot {\sin \varepsilon}^{2}}{\color{blue}{\left(-\cos \varepsilon\right) + -1}}\right) \]
   metadata-eval [<=] 99.1	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\cos x \cdot {\sin \varepsilon}^{2}}{\left(-\cos \varepsilon\right) + \color{blue}{\left(-1\right)}}\right) \]
   distribute-neg-in [<=] 99.1	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\cos x \cdot {\sin \varepsilon}^{2}}{\color{blue}{-\left(\cos \varepsilon + 1\right)}}\right) \]
   *-commutative [<=] 99.1	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\color{blue}{{\sin \varepsilon}^{2} \cdot \cos x}}{-\left(\cos \varepsilon + 1\right)}\right) \]
   associate-/l* [=>] 99.1	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\frac{{\sin \varepsilon}^{2}}{\frac{-\left(\cos \varepsilon + 1\right)}{\cos x}}}\right) \]
   associate-/r/ [=>] 99.1	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\frac{{\sin \varepsilon}^{2}}{-\left(\cos \varepsilon + 1\right)} \cdot \cos x}\right) \]
   unpow2 [=>] 99.1	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{-\left(\cos \varepsilon + 1\right)} \cdot \cos x\right) \]
   neg-mul-1 [=>] 99.1	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\sin \varepsilon \cdot \sin \varepsilon}{\color{blue}{-1 \cdot \left(\cos \varepsilon + 1\right)}} \cdot \cos x\right) \]
   times-frac [=>] 99.1	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \frac{\sin \varepsilon}{\cos \varepsilon + 1}\right)} \cdot \cos x\right) \]
   +-commutative [=>] 99.1	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(\frac{\sin \varepsilon}{-1} \cdot \frac{\sin \varepsilon}{\color{blue}{1 + \cos \varepsilon}}\right) \cdot \cos x\right) \]
   hang-0p-tan [=>] 99.5	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(\frac{\sin \varepsilon}{-1} \cdot \color{blue}{\tan \left(\frac{\varepsilon}{2}\right)}\right) \cdot \cos x\right) \]

6. Applied egg-rr 99.5

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

7. Simplified 99.5

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

   [Start] 99.5	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\sin \varepsilon}{\frac{-1}{\tan \left(\varepsilon \cdot 0.5\right)}} \cdot \cos x\right) \]
   metadata-eval [<=] 99.5	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\sin \varepsilon}{\frac{\color{blue}{\frac{1}{-1}}}{\tan \left(\varepsilon \cdot 0.5\right)}} \cdot \cos x\right) \]
   associate-/r* [<=] 99.5	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \frac{\sin \varepsilon}{\color{blue}{\frac{1}{-1 \cdot \tan \left(\varepsilon \cdot 0.5\right)}}} \cdot \cos x\right) \]
   associate-/r/ [=>] 99.5	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\left(\frac{\sin \varepsilon}{1} \cdot \left(-1 \cdot \tan \left(\varepsilon \cdot 0.5\right)\right)\right)} \cdot \cos x\right) \]
   mul-1-neg [=>] 99.5	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(\frac{\sin \varepsilon}{1} \cdot \color{blue}{\left(-\tan \left(\varepsilon \cdot 0.5\right)\right)}\right) \cdot \cos x\right) \]
   /-rgt-identity [=>] 99.5	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(\color{blue}{\sin \varepsilon} \cdot \left(-\tan \left(\varepsilon \cdot 0.5\right)\right)\right) \cdot \cos x\right) \]
   *-commutative [<=] 99.5	\[ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \left(\sin \varepsilon \cdot \left(-\tan \color{blue}{\left(0.5 \cdot \varepsilon\right)}\right)\right) \cdot \cos x\right) \]

8. Applied egg-rr 99.5

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

9. Simplified 99.5

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

   [Start] 99.5	\[ \sin \varepsilon \cdot \left(\sin x \cdot -1 - \tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x\right) \]
   sub-neg [=>] 99.5	\[ \sin \varepsilon \cdot \color{blue}{\left(\sin x \cdot -1 + \left(-\tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x\right)\right)} \]
   *-commutative [=>] 99.5	\[ \sin \varepsilon \cdot \left(\color{blue}{-1 \cdot \sin x} + \left(-\tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x\right)\right) \]
   mul-1-neg [=>] 99.5	\[ \sin \varepsilon \cdot \left(\color{blue}{\left(-\sin x\right)} + \left(-\tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x\right)\right) \]
   distribute-neg-in [<=] 99.5	\[ \sin \varepsilon \cdot \color{blue}{\left(-\left(\sin x + \tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x\right)\right)} \]
   +-commutative [<=] 99.5	\[ \sin \varepsilon \cdot \left(-\color{blue}{\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \cos x + \sin x\right)}\right) \]
   *-commutative [=>] 99.5	\[ \sin \varepsilon \cdot \left(-\left(\color{blue}{\cos x \cdot \tan \left(\varepsilon \cdot 0.5\right)} + \sin x\right)\right) \]
   fma-def [=>] 99.5	\[ \sin \varepsilon \cdot \left(-\color{blue}{\mathsf{fma}\left(\cos x, \tan \left(\varepsilon \cdot 0.5\right), \sin x\right)}\right) \]
   *-commutative [<=] 99.5	\[ \sin \varepsilon \cdot \left(-\mathsf{fma}\left(\cos x, \tan \color{blue}{\left(0.5 \cdot \varepsilon\right)}, \sin x\right)\right) \]

10. Final simplification 99.5

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

Alternatives

Alternative 1
Error	99.1%
Cost	26441.00

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-17} \lor \neg \left(x \leq 1.3 \cdot 10^{-42}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	99.5%
Cost	26240.00

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

Alternative 3
Error	76.6%
Cost	13768.00

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0069:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 4
Error	71.9%
Cost	13644.00

\[\begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_1 := \sin x \cdot \left(-2 \cdot t_0\right)\\ \mathbf{if}\;x \leq -510000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.4 \cdot 10^{-97}:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot x\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-23}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	76.4%
Cost	13632.00

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

Alternative 6
Error	70.3%
Cost	13513.00

\[\begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{-52} \lor \neg \left(x \leq 1.28 \cdot 10^{-23}\right):\\ \;\;\;\;\sin x \cdot \left(-2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \]

Alternative 7
Error	68.3%
Cost	13449.00

\[\begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-49} \lor \neg \left(x \leq 1.5 \cdot 10^{-22}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 8
Error	67.5%
Cost	13257.00

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-13} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 9
Error	66.7%
Cost	6984.00

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-13}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \left(\varepsilon + x\right)\\ \end{array} \]

Alternative 10
Error	66.7%
Cost	6921.00

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-13} \lor \neg \left(\varepsilon \leq 1.55 \cdot 10^{-5}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 11
Error	46.8%
Cost	6857.00

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00018 \lor \neg \left(\varepsilon \leq 0.00016\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \]

Alternative 12
Error	21.5%
Cost	320.00

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

Alternative 13
Error	13.0%
Cost	64.00

\[0 \]

Reproduce

herbie shell --seed 2023098 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))