Average Error: 60.0 → 0.4
Time: 13.0s
Precision: binary64
\[-0.026 < x \land x < 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x} \]
↓
\[\mathsf{fma}\left(0.3333333333333333, x, 0.022222222222222223 \cdot {x}^{3}\right) \]
(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 (tan x))))
↓
(FPCore (x)
 :precision binary64
 (fma 0.3333333333333333 x (* 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, (0.022222222222222223 * pow(x, 3.0)));
}

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

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

Error

Bits error versus x

Target

Original60.0
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|x\right| < 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array} \]

Derivation

  1. Initial program 60.0

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

  2. Taylor expanded in x around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2022131 
(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))))