invcot (example 3.9)

Average Accuracy: 6.4% → 99.4%

Time: 12.1s

Precision: binary64

Cost: 6912

\[-0.026 < x \land x < 0.026\]

Math

\[\frac{1}{x} - \frac{1}{\tan x} \]

↓

\[0.3333333333333333 \cdot x + 0.022222222222222223 \cdot {x}^{3} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 (tan x))))

↓

(FPCore (x)
 :precision binary64
 (+ (* 0.3333333333333333 x) (* 0.022222222222222223 (pow x 3.0))))

C

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

↓

double code(double x) {
	return (0.3333333333333333 * x) + (0.022222222222222223 * pow(x, 3.0));
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.3333333333333333d0 * x) + (0.022222222222222223d0 * (x ** 3.0d0))
end function

Java

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

↓

public static double code(double x) {
	return (0.3333333333333333 * x) + (0.022222222222222223 * Math.pow(x, 3.0));
}

Python

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

↓

def code(x):
	return (0.3333333333333333 * x) + (0.022222222222222223 * math.pow(x, 3.0))

Julia

function code(x)
	return Float64(Float64(1.0 / x) - Float64(1.0 / tan(x)))
end

↓

function code(x)
	return Float64(Float64(0.3333333333333333 * x) + Float64(0.022222222222222223 * (x ^ 3.0)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (0.3333333333333333 * x) + (0.022222222222222223 * (x ^ 3.0));
end

Wolfram

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

↓

code[x_] := N[(N[(0.3333333333333333 * x), $MachinePrecision] + N[(0.022222222222222223 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	6.4%
Target	99.9%
Herbie	99.4%

\[\begin{array}{l} \mathbf{if}\;\left|x\right| < 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array} \]

Derivation

1. Initial program 6.4%

   \[\frac{1}{x} - \frac{1}{\tan x} \]

2. Taylor expanded in x around 0 99.4%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot x + 0.022222222222222223 \cdot {x}^{3}} \]

3. Final simplification 99.4%

   \[\leadsto 0.3333333333333333 \cdot x + 0.022222222222222223 \cdot {x}^{3} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	576

\[\left(x \cdot 0.1111111111111111\right) \cdot \frac{x}{0.3333333333333333 \cdot x} \]

Alternative 2
Accuracy	99.1%
Cost	320

\[\frac{0.037037037037037035}{\frac{0.1111111111111111}{x}} \]

Alternative 3
Accuracy	98.9%
Cost	192

\[0.3333333333333333 \cdot x \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :pre (and (< -0.026 x) (< x 0.026))

  :herbie-target
  (if (< (fabs x) 0.026) (* (/ x 3.0) (+ 1.0 (/ (* x x) 15.0))) (- (/ 1.0 x) (/ 1.0 (tan x))))

  (- (/ 1.0 x) (/ 1.0 (tan x))))