2cos (problem 3.3.5)

Average Error: 39.5 → 0.3

Time: 13.9s

Precision: binary64

Cost: 32512

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

function code(x, eps)
	return Float64(sin(eps) * Float64(-fma(cos(x), tan(Float64(eps * 0.5)), sin(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[eps], $MachinePrecision] * (-N[(N[Cos[x], $MachinePrecision] * N[Tan[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 39.5

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

2. Applied egg-rr 24.5

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

3. Taylor expanded in x around inf 24.5

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

4. Simplified 6.4

   \[\leadsto \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x} \]
   Proof
   (-.f64 (*.f64 (cos.f64 x) (+.f64 (cos.f64 eps) -1)) (*.f64 (sin.f64 eps) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (cos.f64 x) (+.f64 (cos.f64 eps) (Rewrite<= metadata-eval (neg.f64 1)))) (*.f64 (sin.f64 eps) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (cos.f64 x) (Rewrite<= sub-neg_binary64 (-.f64 (cos.f64 eps) 1))) (*.f64 (sin.f64 eps) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (cos.f64 x) (cos.f64 eps)) (*.f64 (cos.f64 x) 1))) (*.f64 (sin.f64 eps) (sin.f64 x))): 14 points increase in error, 13 points decrease in error
   (-.f64 (-.f64 (*.f64 (cos.f64 x) (cos.f64 eps)) (Rewrite=> *-rgt-identity_binary64 (cos.f64 x))) (*.f64 (sin.f64 eps) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 eps) (cos.f64 x))) (cos.f64 x)) (*.f64 (sin.f64 eps) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (*.f64 (cos.f64 eps) (cos.f64 x)) (cos.f64 x)) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 x) (sin.f64 eps)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 (cos.f64 eps) (cos.f64 x)) (+.f64 (cos.f64 x) (*.f64 (sin.f64 x) (sin.f64 eps))))): 108 points increase in error, 14 points decrease in error

5. Applied egg-rr 0.6

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

6. Simplified 0.3

   \[\leadsto \color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \cdot \cos x} - \sin \varepsilon \cdot \sin x \]
   Proof
   (*.f64 (*.f64 (/.f64 (sin.f64 eps) -1) (tan.f64 (/.f64 eps 2))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 (sin.f64 eps) -1) (Rewrite<= hang-0p-tan_binary64 (/.f64 (sin.f64 eps) (+.f64 1 (cos.f64 eps))))) (cos.f64 x)): 42 points increase in error, 13 points decrease in error
   (*.f64 (*.f64 (/.f64 (sin.f64 eps) -1) (/.f64 (sin.f64 eps) (Rewrite<= +-commutative_binary64 (+.f64 (cos.f64 eps) 1)))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (sin.f64 eps) (sin.f64 eps)) (*.f64 -1 (+.f64 (cos.f64 eps) 1)))) (cos.f64 x)): 25 points increase in error, 25 points decrease in error
   (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 eps) 2)) (*.f64 -1 (+.f64 (cos.f64 eps) 1))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)))) (*.f64 -1 (+.f64 (cos.f64 eps) 1))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (neg.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (pow.f64 (sin.f64 eps) 2)))) (*.f64 -1 (+.f64 (cos.f64 eps) 1))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 -1) (pow.f64 (sin.f64 eps) 2))) (*.f64 -1 (+.f64 (cos.f64 eps) 1))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (*.f64 (Rewrite=> metadata-eval 1) (pow.f64 (sin.f64 eps) 2)) (*.f64 -1 (+.f64 (cos.f64 eps) 1))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 1 -1) (/.f64 (pow.f64 (sin.f64 eps) 2) (+.f64 (cos.f64 eps) 1)))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (Rewrite=> metadata-eval -1) (/.f64 (pow.f64 (sin.f64 eps) 2) (+.f64 (cos.f64 eps) 1))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.f64 (pow.f64 (sin.f64 eps) 2) (+.f64 (cos.f64 eps) 1)))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= distribute-frac-neg_binary64 (/.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (+.f64 (cos.f64 eps) 1))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r/_binary64 (/.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (/.f64 (+.f64 (cos.f64 eps) 1) (cos.f64 x)))): 20 points increase in error, 14 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (cos.f64 x)) (+.f64 (cos.f64 eps) 1))): 15 points increase in error, 12 points decrease in error

7. Applied egg-rr 0.3

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

8. Simplified 0.3

   \[\leadsto \color{blue}{\left(-\sin \varepsilon\right) \cdot \mathsf{fma}\left(\cos x, \tan \left(\varepsilon \cdot 0.5\right), \sin x\right)} \]
   Proof
   (*.f64 (neg.f64 (sin.f64 eps)) (fma.f64 (cos.f64 x) (tan.f64 (*.f64 eps 1/2)) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
   (*.f64 (neg.f64 (sin.f64 eps)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 x) (tan.f64 (*.f64 eps 1/2))) (sin.f64 x)))): 8 points increase in error, 2 points decrease in error
   (*.f64 (neg.f64 (sin.f64 eps)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (tan.f64 (*.f64 eps 1/2)) (cos.f64 x))) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (neg.f64 (sin.f64 eps)) (*.f64 (tan.f64 (*.f64 eps 1/2)) (cos.f64 x))) (*.f64 (neg.f64 (sin.f64 eps)) (sin.f64 x)))): 13 points increase in error, 7 points decrease in error

9. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.8
Cost	26440

\[\begin{array}{l} t_0 := \cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;x \leq -1.06 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-88}:\\ \;\;\;\;\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.3
Cost	26240

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

Alternative 3
Error	14.7
Cost	13768

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.29:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 4
Error	15.0
Cost	13632

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

Alternative 5
Error	14.8
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0036:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 6
Error	14.5
Cost	13384

\[\begin{array}{l} t_0 := \cos x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{if}\;\varepsilon \leq -0.00192:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.011:\\ \;\;\;\;\varepsilon \cdot \left(-\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	14.8
Cost	13256

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.001:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0135:\\ \;\;\;\;\varepsilon \cdot \left(-\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	15.2
Cost	7304

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.00105:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.004:\\ \;\;\;\;\varepsilon \cdot \left(-\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	21.1
Cost	6920

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -5.1 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	33.8
Cost	6856

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -1.4 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.000145:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	50.1
Cost	320

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

Reproduce

herbie shell --seed 2022325 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))