Average Error: 59.9 → 0.3
Time: 15.7s
Precision: binary64
Cost: 20352
\[-0.026 < x \land x < 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x} \]
↓
\[0.3333333333333333 \cdot x + \left(0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + 0.00021164021164021165 \cdot {x}^{7}\right)\right) \]
(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 (tan x))))
↓
(FPCore (x)
 :precision binary64
 (+
  (* 0.3333333333333333 x)
  (+
   (* 0.0021164021164021165 (pow x 5.0))
   (+
    (* 0.022222222222222223 (pow x 3.0))
    (* 0.00021164021164021165 (pow x 7.0))))))
double code(double x) {
	return (1.0 / x) - (1.0 / tan(x));
}
↓
double code(double x) {
	return (0.3333333333333333 * x) + ((0.0021164021164021165 * pow(x, 5.0)) + ((0.022222222222222223 * pow(x, 3.0)) + (0.00021164021164021165 * pow(x, 7.0))));
}

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) + ((0.0021164021164021165d0 * (x ** 5.0d0)) + ((0.022222222222222223d0 * (x ** 3.0d0)) + (0.00021164021164021165d0 * (x ** 7.0d0))))
end function

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

def code(x):
	return (1.0 / x) - (1.0 / math.tan(x))
↓
def code(x):
	return (0.3333333333333333 * x) + ((0.0021164021164021165 * math.pow(x, 5.0)) + ((0.022222222222222223 * math.pow(x, 3.0)) + (0.00021164021164021165 * math.pow(x, 7.0))))

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

function tmp = code(x)
	tmp = (1.0 / x) - (1.0 / tan(x));
end
↓
function tmp = code(x)
	tmp = (0.3333333333333333 * x) + ((0.0021164021164021165 * (x ^ 5.0)) + ((0.022222222222222223 * (x ^ 3.0)) + (0.00021164021164021165 * (x ^ 7.0))));
end

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[(0.0021164021164021165 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.022222222222222223 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.00021164021164021165 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{1}{x} - \frac{1}{\tan x}
↓
0.3333333333333333 \cdot x + \left(0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + 0.00021164021164021165 \cdot {x}^{7}\right)\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.9
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|x\right| < 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array} \]

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. Final simplification0.3

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

Alternatives

Alternative 1
Error0.4
Cost13824
\[\mathsf{fma}\left(0.0021164021164021165, {x}^{5}, x \cdot \left(0.3333333333333333 + \left(\left(1 + x \cdot \left(x \cdot 0.022222222222222223\right)\right) + -1\right)\right)\right) \]
Alternative 2
Error0.3
Cost13632
\[0.3333333333333333 \cdot x + \left(0.0021164021164021165 \cdot {x}^{5} + 0.022222222222222223 \cdot {x}^{3}\right) \]
Alternative 3
Error0.3
Cost13568
\[\mathsf{fma}\left(0.0021164021164021165, {x}^{5}, x \cdot \left(0.3333333333333333 + x \cdot \left(x \cdot 0.022222222222222223\right)\right)\right) \]
Alternative 4
Error0.4
Cost2624
\[\begin{array}{l} t_0 := 0.022222222222222223 \cdot \left(x \cdot x\right)\\ x \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot 0.022222222222222223\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0004938271604938272\right)\right) + 0.037037037037037035}{t_0 \cdot t_0 + \left(0.1111111111111111 + 0.3333333333333333 \cdot \left(\left(x \cdot x\right) \cdot -0.022222222222222223\right)\right)} \end{array} \]
Alternative 5
Error0.4
Cost832
\[x \cdot \left(0.3333333333333333 + \left(\left(1 + x \cdot \left(x \cdot 0.022222222222222223\right)\right) + -1\right)\right) \]
Alternative 6
Error0.4
Cost704
\[0.3333333333333333 \cdot x + x \cdot \left(0.022222222222222223 \cdot \left(x \cdot x\right)\right) \]
Alternative 7
Error0.4
Cost576
\[x \cdot \left(0.3333333333333333 + 0.022222222222222223 \cdot \left(x \cdot x\right)\right) \]
Alternative 8
Error0.7
Cost192
\[0.3333333333333333 \cdot x \]

Error

Reproduce

herbie shell --seed 2022291 
(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))))