invcot (example 3.9)

Average Error: 59.9 → 0.3

Time: 16.4s

Precision: binary64

Cost: 20352

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

Math

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

↓

\[0.3333333333333333 \cdot x + \left(0.00021164021164021165 \cdot {x}^{7} - \left({x}^{5} \cdot -0.0021164021164021165 + {x}^{3} \cdot -0.022222222222222223\right)\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (+
  (* 0.3333333333333333 x)
  (-
   (* 0.00021164021164021165 (pow x 7.0))
   (+
    (* (pow x 5.0) -0.0021164021164021165)
    (* (pow x 3.0) -0.022222222222222223)))))

C

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

↓

double code(double x) {
	return (0.3333333333333333 * x) + ((0.00021164021164021165 * pow(x, 7.0)) - ((pow(x, 5.0) * -0.0021164021164021165) + (pow(x, 3.0) * -0.022222222222222223)));
}

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) + ((0.00021164021164021165d0 * (x ** 7.0d0)) - (((x ** 5.0d0) * (-0.0021164021164021165d0)) + ((x ** 3.0d0) * (-0.022222222222222223d0))))
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) + ((0.00021164021164021165 * Math.pow(x, 7.0)) - ((Math.pow(x, 5.0) * -0.0021164021164021165) + (Math.pow(x, 3.0) * -0.022222222222222223)));
}

Python

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

↓

def code(x):
	return (0.3333333333333333 * x) + ((0.00021164021164021165 * math.pow(x, 7.0)) - ((math.pow(x, 5.0) * -0.0021164021164021165) + (math.pow(x, 3.0) * -0.022222222222222223)))

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(Float64(0.00021164021164021165 * (x ^ 7.0)) - Float64(Float64((x ^ 5.0) * -0.0021164021164021165) + Float64((x ^ 3.0) * -0.022222222222222223))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (0.3333333333333333 * x) + ((0.00021164021164021165 * (x ^ 7.0)) - (((x ^ 5.0) * -0.0021164021164021165) + ((x ^ 3.0) * -0.022222222222222223)));
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[(N[(0.00021164021164021165 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[x, 5.0], $MachinePrecision] * -0.0021164021164021165), $MachinePrecision] + N[(N[Power[x, 3.0], $MachinePrecision] * -0.022222222222222223), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	59.9
Target	0.1
Herbie	0.3

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

Derivation

1. Initial program 59.9

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

2. Taylor expanded in x around 0 0.3

   \[\leadsto \color{blue}{0.3333333333333333 \cdot x + \left(0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + 0.00021164021164021165 \cdot {x}^{7}\right)\right)} \]

3. Simplified 0.3

   \[\leadsto \color{blue}{0.3333333333333333 \cdot x + \left(0.00021164021164021165 \cdot {x}^{7} - \left({x}^{5} \cdot -0.0021164021164021165 + {x}^{3} \cdot -0.022222222222222223\right)\right)} \]
   Proof

   [Start] 0.3	\[ 0.3333333333333333 \cdot x + \left(0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + 0.00021164021164021165 \cdot {x}^{7}\right)\right) \]
   rational_best-simplify-116 [=>] 0.3	\[ 0.3333333333333333 \cdot x + \color{blue}{\left(0.00021164021164021165 \cdot {x}^{7} + \left(0.0021164021164021165 \cdot {x}^{5} + 0.022222222222222223 \cdot {x}^{3}\right)\right)} \]
   rational_best-simplify-62 [=>] 0.3	\[ 0.3333333333333333 \cdot x + \left(0.00021164021164021165 \cdot {x}^{7} + \color{blue}{\left(0.022222222222222223 \cdot {x}^{3} - \left(-0.0021164021164021165 \cdot {x}^{5}\right)\right)}\right) \]
   rational_best-simplify-65 [=>] 0.3	\[ 0.3333333333333333 \cdot x + \left(0.00021164021164021165 \cdot {x}^{7} + \color{blue}{\left(-\left(\left(-0.0021164021164021165 \cdot {x}^{5}\right) - 0.022222222222222223 \cdot {x}^{3}\right)\right)}\right) \]
   rational_best-simplify-60 [=>] 0.3	\[ 0.3333333333333333 \cdot x + \color{blue}{\left(0.00021164021164021165 \cdot {x}^{7} - \left(\left(-0.0021164021164021165 \cdot {x}^{5}\right) - 0.022222222222222223 \cdot {x}^{3}\right)\right)} \]
   rational_best-simplify-61 [=>] 0.3	\[ 0.3333333333333333 \cdot x + \left(0.00021164021164021165 \cdot {x}^{7} - \color{blue}{\left(\left(-0.0021164021164021165 \cdot {x}^{5}\right) + \left(-0.022222222222222223 \cdot {x}^{3}\right)\right)}\right) \]
   rational_best-simplify-52 [=>] 0.3	\[ 0.3333333333333333 \cdot x + \left(0.00021164021164021165 \cdot {x}^{7} - \left(\color{blue}{{x}^{5} \cdot \left(-0.0021164021164021165\right)} + \left(-0.022222222222222223 \cdot {x}^{3}\right)\right)\right) \]
   metadata-eval [=>] 0.3	\[ 0.3333333333333333 \cdot x + \left(0.00021164021164021165 \cdot {x}^{7} - \left({x}^{5} \cdot \color{blue}{-0.0021164021164021165} + \left(-0.022222222222222223 \cdot {x}^{3}\right)\right)\right) \]
   rational_best-simplify-52 [=>] 0.3	\[ 0.3333333333333333 \cdot x + \left(0.00021164021164021165 \cdot {x}^{7} - \left({x}^{5} \cdot -0.0021164021164021165 + \color{blue}{{x}^{3} \cdot \left(-0.022222222222222223\right)}\right)\right) \]
   metadata-eval [=>] 0.3	\[ 0.3333333333333333 \cdot x + \left(0.00021164021164021165 \cdot {x}^{7} - \left({x}^{5} \cdot -0.0021164021164021165 + {x}^{3} \cdot \color{blue}{-0.022222222222222223}\right)\right) \]

4. Final simplification 0.3

   \[\leadsto 0.3333333333333333 \cdot x + \left(0.00021164021164021165 \cdot {x}^{7} - \left({x}^{5} \cdot -0.0021164021164021165 + {x}^{3} \cdot -0.022222222222222223\right)\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	13632

\[0.3333333333333333 \cdot x - \left({x}^{5} \cdot -0.0021164021164021165 + {x}^{3} \cdot -0.022222222222222223\right) \]

Alternative 2
Error	0.4
Cost	6912

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

Alternative 3
Error	0.7
Cost	192

\[0.3333333333333333 \cdot x \]

Reproduce

herbie shell --seed 2023104 
(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))))