2cos (problem 3.3.5)

Average Error: 39.8 → 0.4

Time: 13.5s

Precision: binary64

Cost: 46144

Math

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

↓

\[\begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_1 := t_0 \cdot -2\\ \mathsf{fma}\left(t_1, \cos \left(\varepsilon \cdot 0.5\right) \cdot \sin x, t_1 \cdot \left(t_0 \cdot \cos x\right)\right) \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))) (t_1 (* t_0 -2.0)))
   (fma t_1 (* (cos (* eps 0.5)) (sin x)) (* t_1 (* t_0 (cos x))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double t_1 = t_0 * -2.0;
	return fma(t_1, (cos((eps * 0.5)) * sin(x)), (t_1 * (t_0 * cos(x))));
}

Julia

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

↓

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	t_1 = Float64(t_0 * -2.0)
	return fma(t_1, Float64(cos(Float64(eps * 0.5)) * sin(x)), Float64(t_1 * Float64(t_0 * cos(x))))
end

Wolfram

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

↓

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * -2.0), $MachinePrecision]}, N[(t$95$1 * N[(N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 39.8

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

2. Applied egg-rr 15.4

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

3. Simplified 15.4

   \[\leadsto \color{blue}{\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)\right)} \]
   Proof
   (*.f64 (sin.f64 (*.f64 eps 1/2)) (*.f64 -2 (sin.f64 (*.f64 1/2 (+.f64 x (+.f64 x eps)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 (*.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 eps 0)) 1/2)) (*.f64 -2 (sin.f64 (*.f64 1/2 (+.f64 x (+.f64 x eps)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 (*.f64 (+.f64 eps (Rewrite<= +-inverses_binary64 (-.f64 x x))) 1/2)) (*.f64 -2 (sin.f64 (*.f64 1/2 (+.f64 x (+.f64 x eps)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 (*.f64 (+.f64 eps (-.f64 x x)) 1/2)) (*.f64 -2 (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 x (+.f64 x eps)) 1/2))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 (*.f64 (+.f64 eps (-.f64 x x)) 1/2)) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 (*.f64 (+.f64 x (+.f64 x eps)) 1/2)) -2))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sin.f64 (*.f64 (+.f64 eps (-.f64 x x)) 1/2)) (sin.f64 (*.f64 (+.f64 x (+.f64 x eps)) 1/2))) -2)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 -2 (*.f64 (sin.f64 (*.f64 (+.f64 eps (-.f64 x x)) 1/2)) (sin.f64 (*.f64 (+.f64 x (+.f64 x eps)) 1/2))))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.4

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

5. Simplified 0.3

   \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \varepsilon\right), \sin x, \sin \left(0.5 \cdot \varepsilon\right) \cdot \cos x\right)}\right) \]
   Proof
   (fma.f64 (cos.f64 (*.f64 1/2 eps)) (sin.f64 x) (*.f64 (sin.f64 (*.f64 1/2 eps)) (cos.f64 x))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 1/2 eps)) (sin.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1))) (*.f64 (sin.f64 (*.f64 1/2 eps)) (cos.f64 x))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 1/2 eps)) (sin.f64 (*.f64 x (Rewrite<= metadata-eval (+.f64 1/2 1/2)))) (*.f64 (sin.f64 (*.f64 1/2 eps)) (cos.f64 x))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 1/2 eps)) (sin.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 1/2 x) (*.f64 1/2 x)))) (*.f64 (sin.f64 (*.f64 1/2 eps)) (cos.f64 x))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 1/2 eps)) (sin.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 1/2 (+.f64 x x)))) (*.f64 (sin.f64 (*.f64 1/2 eps)) (cos.f64 x))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 1/2 eps)) (sin.f64 (*.f64 1/2 (+.f64 x x))) (*.f64 (sin.f64 (*.f64 1/2 eps)) (cos.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 1/2 eps)) (sin.f64 (*.f64 1/2 (+.f64 x x))) (*.f64 (sin.f64 (*.f64 1/2 eps)) (cos.f64 (*.f64 x (Rewrite<= metadata-eval (+.f64 1/2 1/2)))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 1/2 eps)) (sin.f64 (*.f64 1/2 (+.f64 x x))) (*.f64 (sin.f64 (*.f64 1/2 eps)) (cos.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 1/2 x) (*.f64 1/2 x)))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 1/2 eps)) (sin.f64 (*.f64 1/2 (+.f64 x x))) (*.f64 (sin.f64 (*.f64 1/2 eps)) (cos.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 1/2 (+.f64 x x)))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 1/2 eps)) (sin.f64 (*.f64 1/2 (+.f64 x x))) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 (*.f64 1/2 (+.f64 x x))) (sin.f64 (*.f64 1/2 eps))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 (*.f64 1/2 eps)) (sin.f64 (*.f64 1/2 (+.f64 x x)))) (*.f64 (cos.f64 (*.f64 1/2 (+.f64 x x))) (sin.f64 (*.f64 1/2 eps))))): 9 points increase in error, 6 points decrease in error
   (+.f64 (*.f64 (cos.f64 (*.f64 1/2 eps)) (sin.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 x x) 1/2)))) (*.f64 (cos.f64 (*.f64 1/2 (+.f64 x x))) (sin.f64 (*.f64 1/2 eps)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 (*.f64 (+.f64 x x) 1/2)) (cos.f64 (*.f64 1/2 eps)))) (*.f64 (cos.f64 (*.f64 1/2 (+.f64 x x))) (sin.f64 (*.f64 1/2 eps)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (sin.f64 (*.f64 (+.f64 x x) 1/2)) (cos.f64 (*.f64 1/2 eps))) (*.f64 (cos.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 x x) 1/2))) (sin.f64 (*.f64 1/2 eps)))): 0 points increase in error, 0 points decrease in error

6. Applied egg-rr 0.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2, \cos \left(\varepsilon \cdot 0.5\right) \cdot \sin x, \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right) \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \cos x\right)\right)} \]

7. Final simplification 0.4

   \[\leadsto \mathsf{fma}\left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2, \cos \left(\varepsilon \cdot 0.5\right) \cdot \sin x, \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right) \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \cos x\right)\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	39360

\[\begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_0 \cdot \left(-2 \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot 0.5\right), \sin x, t_0 \cdot \cos x\right)\right) \end{array} \]

Alternative 2
Error	0.4
Cost	33088

\[\begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_0 \cdot \left(-2 \cdot \left(\cos \left(\varepsilon \cdot 0.5\right) \cdot \sin x + t_0 \cdot \cos x\right)\right) \end{array} \]

Alternative 3
Error	0.7
Cost	32840

\[\begin{array}{l} t_0 := \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.00039:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.95 \cdot 10^{-5}:\\ \;\;\;\;\left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right) \cdot \sin \left(x + \varepsilon \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.7
Cost	32840

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.8 \cdot 10^{-5}:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.000104:\\ \;\;\;\;\left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right) \cdot \sin \left(x + \varepsilon \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \end{array} \]

Alternative 5
Error	15.4
Cost	13632

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

Alternative 6
Error	15.3
Cost	13504

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

Alternative 7
Error	39.8
Cost	13120

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

Reproduce

herbie shell --seed 2022334 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))