Average Error: 59.8 → 0.3
Time: 10.4s
Precision: binary64
\[-0.026 < x \land x < 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x} \]
↓
\[\mathsf{fma}\left(0.3333333333333333, x, 0.0021164021164021165 \cdot {x}^{5}\right) + 0.022222222222222223 \cdot {x}^{3} \]
(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 (tan x))))
↓
(FPCore (x)
 :precision binary64
 (+
  (fma 0.3333333333333333 x (* 0.0021164021164021165 (pow x 5.0)))
  (* 0.022222222222222223 (pow x 3.0))))
double code(double x) {
	return (1.0 / x) - (1.0 / tan(x));
}
↓
double code(double x) {
	return fma(0.3333333333333333, x, (0.0021164021164021165 * pow(x, 5.0))) + (0.022222222222222223 * pow(x, 3.0));
}

function code(x)
	return Float64(Float64(1.0 / x) - Float64(1.0 / tan(x)))
end
↓
function code(x)
	return Float64(fma(0.3333333333333333, x, Float64(0.0021164021164021165 * (x ^ 5.0))) + Float64(0.022222222222222223 * (x ^ 3.0)))
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 + N[(0.0021164021164021165 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.022222222222222223 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Target

Original59.8
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 59.8

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

  2. Taylor expanded in x around 0 0.3

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

  3. Applied egg-rr1.5

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

  4. Applied egg-rr0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2022170 
(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))))