?

Average Error: 93.51% → 0.14%
Time: 15.5s
Precision: binary64
Cost: 6976

?

\[-0.026 < x \land x < 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x} \]
↓
\[\frac{x}{\frac{1}{\mathsf{fma}\left(0.022222222222222223, x \cdot x, 0.3333333333333333\right)}} \]
(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 (tan x))))
↓
(FPCore (x)
 :precision binary64
 (/ x (/ 1.0 (fma 0.022222222222222223 (* x x) 0.3333333333333333))))
double code(double x) {
	return (1.0 / x) - (1.0 / tan(x));
}
↓
double code(double x) {
	return x / (1.0 / fma(0.022222222222222223, (x * x), 0.3333333333333333));
}

function code(x)
	return Float64(Float64(1.0 / x) - Float64(1.0 / tan(x)))
end
↓
function code(x)
	return Float64(x / Float64(1.0 / fma(0.022222222222222223, Float64(x * x), 0.3333333333333333)))
end

code[x_] := N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(x / N[(1.0 / N[(0.022222222222222223 * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error?

Target

Original93.51%
Target0.14%
Herbie0.14%
\[\begin{array}{l} \mathbf{if}\;\left|x\right| < 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array} \]

Derivation?

  1. Initial program 93.51

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

  2. Taylor expanded in x around 0 0.64

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

  3. Applied egg-rr0.64

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

  4. Applied egg-rr0.14

    \[\leadsto \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(0.022222222222222223, x \cdot x, 0.3333333333333333\right)}}} \]

  5. Final simplification0.14

    \[\leadsto \frac{x}{\frac{1}{\mathsf{fma}\left(0.022222222222222223, x \cdot x, 0.3333333333333333\right)}} \]

Alternatives

Alternative 1
Error0.64%
Cost576
\[x \cdot \left(0.3333333333333333 + 0.022222222222222223 \cdot \left(x \cdot x\right)\right) \]
Alternative 2
Error0.99%
Cost320
\[\frac{0.037037037037037035}{\frac{0.1111111111111111}{x}} \]
Alternative 3
Error1.11%
Cost192
\[x \cdot 0.3333333333333333 \]

Error

Reproduce?

herbie shell --seed 2023089 
(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))))