Average Error: 59.9 → 0.3

Time: 10.2s

Precision: binary64

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

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

↓

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

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

↓

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

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

↓

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

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

↓

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

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

↓

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

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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `x`

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

Initial program 59.9
\[\frac{1}{x} - \frac{1}{\tan x} \]
Taylor expanded in x around 0 0.3
\[\leadsto \color{blue}{0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + 0.3333333333333333 \cdot x\right)} \]
Applied egg-rr0.3
\[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \color{blue}{\mathsf{fma}\left({x}^{3}, 0.022222222222222223, 0.3333333333333333 \cdot x\right)} \]
Final simplification0.3
\[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \mathsf{fma}\left({x}^{3}, 0.022222222222222223, x \cdot 0.3333333333333333\right) \]

Reproduce

herbie shell --seed 2022153 
(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))))