cos2 (problem 3.4.1)

Average Error: 31.6 → 0.1

Time: 11.9s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

def code(x):
	return (math.tan((x * 0.5)) / x) / (x / math.sin(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(x / sin(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) / (x / sin(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[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6

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

2. Applied egg-rr 16.1

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

3. Simplified 16.0

   \[\leadsto \frac{\color{blue}{\sin x \cdot \tan \left(\frac{x}{2}\right)}}{x \cdot x} \]
   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)): 5 points increase in error, 0 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)): 0 points increase in error, 5 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)): 5 points increase in error, 0 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, 5 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. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13376

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

Alternative 2
Error	0.3
Cost	7240

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x}\\ \mathbf{if}\;x \leq -0.0055:\\ \;\;\;\;\frac{t_0}{x}\\ \mathbf{elif}\;x \leq 0.0055:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{x}\\ \end{array} \]

Alternative 3
Error	0.6
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0055 \lor \neg \left(x \leq 0.0055\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 4
Error	0.4
Cost	7112

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.0055:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0055:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \end{array} \]

Alternative 5
Error	15.3
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+76}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	46.3
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022340 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))