invcot (example 3.9)

Average Error: 60.0 → 0.3

Time: 10.0s

Precision: binary64

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

Math

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

↓

\[\mathsf{fma}\left(0.0021164021164021165, {x}^{5}, \mathsf{fma}\left(0.022222222222222223, {x}^{3}, x \cdot 0.3333333333333333\right)\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (fma
  0.0021164021164021165
  (pow x 5.0)
  (fma 0.022222222222222223 (pow x 3.0) (* x 0.3333333333333333))))

C

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

↓

double code(double x) {
	return fma(0.0021164021164021165, pow(x, 5.0), fma(0.022222222222222223, pow(x, 3.0), (x * 0.3333333333333333)));
}

Julia

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

↓

function code(x)
	return fma(0.0021164021164021165, (x ^ 5.0), fma(0.022222222222222223, (x ^ 3.0), Float64(x * 0.3333333333333333)))
end

Wolfram

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

↓

code[x_] := N[(0.0021164021164021165 * N[Power[x, 5.0], $MachinePrecision] + N[(0.022222222222222223 * N[Power[x, 3.0], $MachinePrecision] + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

Target

Original	60.0
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 60.0

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

2. Taylor expanded in x around 0 0.3

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

3. Simplified 0.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.0021164021164021165, {x}^{5}, \mathsf{fma}\left(0.022222222222222223, {x}^{3}, 0.3333333333333333 \cdot x\right)\right)} \]

4. Final simplification 0.3

   \[\leadsto \mathsf{fma}\left(0.0021164021164021165, {x}^{5}, \mathsf{fma}\left(0.022222222222222223, {x}^{3}, x \cdot 0.3333333333333333\right)\right) \]

Reproduce

herbie shell --seed 2022165 
(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))))