cos2 (problem 3.4.1)

Average Error: 31.8 → 0.1

Time: 10.9s

Precision: binary64

Cost: 13376

Math

\[\frac{1 - \cos x}{x \cdot x} \]

↓

\[\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

↓

(FPCore (x) :precision binary64 (* (/ (tan (* x 0.5)) x) (/ (sin x) x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

↓

double code(double x) {
	return (tan((x * 0.5)) / x) * (sin(x) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (tan((x * 0.5d0)) / x) * (sin(x) / x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

↓

public static double code(double x) {
	return (Math.tan((x * 0.5)) / x) * (Math.sin(x) / x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

↓

def code(x):
	return (math.tan((x * 0.5)) / x) * (math.sin(x) / x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

↓

function code(x)
	return Float64(Float64(tan(Float64(x * 0.5)) / x) * Float64(sin(x) / x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

↓

function tmp = code(x)
	tmp = (tan((x * 0.5)) / x) * (sin(x) / x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.8

   \[\frac{1 - \cos x}{x \cdot x} \]

2. Applied egg-rr 16.8

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

3. Simplified 16.6

   \[\leadsto \frac{\color{blue}{\sin x \cdot \tan \left(\frac{x}{2}\right)}}{x \cdot x} \]
   Proof
   (/.f64 (*.f64 (sin.f64 x) (tan.f64 (/.f64 x 2))) (*.f64 x x)): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (sin.f64 x) (Rewrite<= hang-0p-tan_binary64 (/.f64 (sin.f64 x) (+.f64 1 (cos.f64 x))))) (*.f64 x x)): 0 points increase in error, 5 points decrease in error
   (/.f64 (*.f64 (sin.f64 x) (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (sin.f64 x) 1)) (+.f64 1 (cos.f64 x)))) (*.f64 x x)): 5 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (sin.f64 x) (Rewrite<= associate-*r/_binary64 (*.f64 (sin.f64 x) (/.f64 1 (+.f64 1 (cos.f64 x)))))) (*.f64 x x)): 0 points increase in error, 5 points decrease in error
   (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sin.f64 x) (sin.f64 x)) (/.f64 1 (+.f64 1 (cos.f64 x))))) (*.f64 x x)): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x}} \]

5. Final simplification 0.1

   \[\leadsto \frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x} \]

Alternatives

Alternative 1
Error	0.4
Cost	13508

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)\\ \mathbf{elif}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(x \cdot \left(x \cdot -0.041666666666666664\right) + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	13448

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{\frac{1}{\frac{1}{t_0}}}}\\ \mathbf{elif}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(x \cdot \left(x \cdot -0.041666666666666664\right) + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot t_0\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7432

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x}\\ \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{1}{x} \cdot t_0\\ \mathbf{elif}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(x \cdot \left(x \cdot -0.041666666666666664\right) + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	7432

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{\frac{1}{\frac{1}{t_0}}}}\\ \mathbf{elif}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(x \cdot \left(x \cdot -0.041666666666666664\right) + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \end{array} \]

Alternative 5
Error	0.6
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0048 \lor \neg \left(x \leq 0.0052\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 6
Error	0.3
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0048 \lor \neg \left(x \leq 0.0052\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 7
Error	0.3
Cost	7112

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x}\\ \mathbf{if}\;x \leq -0.0048:\\ \;\;\;\;\frac{1}{x} \cdot t_0\\ \mathbf{elif}\;x \leq 0.0052:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x}\\ \end{array} \]

Alternative 8
Error	15.7
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+75} \lor \neg \left(x \leq 1.15 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} + \frac{-1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9
Error	14.1
Cost	832

\[\frac{\frac{1}{x}}{x \cdot 0.16666666666666666 + 2 \cdot \frac{1}{x}} \]

Alternative 10
Error	30.7
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))