invcot (example 3.9)

Average Error: 59.8 → 0.3

Time: 15.7s

Precision: binary64

Cost: 39168

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

Math

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

↓

\[\mathsf{fma}\left(0.3333333333333333, x, \mathsf{fma}\left(0.0021164021164021165, {x}^{5}, \mathsf{fma}\left(0.022222222222222223, {x}^{3}, 0.00021164021164021165 \cdot {x}^{7}\right)\right)\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (fma
  0.3333333333333333
  x
  (fma
   0.0021164021164021165
   (pow x 5.0)
   (fma
    0.022222222222222223
    (pow x 3.0)
    (* 0.00021164021164021165 (pow x 7.0))))))

C

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

↓

double code(double x) {
	return fma(0.3333333333333333, x, fma(0.0021164021164021165, pow(x, 5.0), fma(0.022222222222222223, pow(x, 3.0), (0.00021164021164021165 * pow(x, 7.0)))));
}

Julia

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

↓

function code(x)
	return fma(0.3333333333333333, x, fma(0.0021164021164021165, (x ^ 5.0), fma(0.022222222222222223, (x ^ 3.0), Float64(0.00021164021164021165 * (x ^ 7.0)))))
end

Wolfram

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

↓

code[x_] := N[(0.3333333333333333 * x + N[(0.0021164021164021165 * N[Power[x, 5.0], $MachinePrecision] + N[(0.022222222222222223 * N[Power[x, 3.0], $MachinePrecision] + N[(0.00021164021164021165 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	59.8
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 59.8

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

2. Taylor expanded in x around 0 0.3

   \[\leadsto \color{blue}{0.3333333333333333 \cdot x + \left(0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + 0.00021164021164021165 \cdot {x}^{7}\right)\right)} \]

3. Simplified 0.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, x, \mathsf{fma}\left(0.0021164021164021165, {x}^{5}, \mathsf{fma}\left(0.022222222222222223, {x}^{3}, 0.00021164021164021165 \cdot {x}^{7}\right)\right)\right)} \]
   Proof

   [Start] 0.3	\[ 0.3333333333333333 \cdot x + \left(0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + 0.00021164021164021165 \cdot {x}^{7}\right)\right) \]
   fma-def [=>] 0.3	\[ \color{blue}{\mathsf{fma}\left(0.3333333333333333, x, 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + 0.00021164021164021165 \cdot {x}^{7}\right)\right)} \]
   fma-def [=>] 0.3	\[ \mathsf{fma}\left(0.3333333333333333, x, \color{blue}{\mathsf{fma}\left(0.0021164021164021165, {x}^{5}, 0.022222222222222223 \cdot {x}^{3} + 0.00021164021164021165 \cdot {x}^{7}\right)}\right) \]
   fma-def [=>] 0.3	\[ \mathsf{fma}\left(0.3333333333333333, x, \mathsf{fma}\left(0.0021164021164021165, {x}^{5}, \color{blue}{\mathsf{fma}\left(0.022222222222222223, {x}^{3}, 0.00021164021164021165 \cdot {x}^{7}\right)}\right)\right) \]

4. Final simplification 0.3

   \[\leadsto \mathsf{fma}\left(0.3333333333333333, x, \mathsf{fma}\left(0.0021164021164021165, {x}^{5}, \mathsf{fma}\left(0.022222222222222223, {x}^{3}, 0.00021164021164021165 \cdot {x}^{7}\right)\right)\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	20352

\[0.3333333333333333 \cdot x + \left(0.0021164021164021165 \cdot {x}^{5} + \left(0.00021164021164021165 \cdot {x}^{7} + 0.022222222222222223 \cdot {x}^{3}\right)\right) \]

Alternative 2
Error	0.3
Cost	20352

\[0.00021164021164021165 \cdot {x}^{7} + \left(0.3333333333333333 \cdot x + \left(0.0021164021164021165 \cdot {x}^{5} + 0.022222222222222223 \cdot {x}^{3}\right)\right) \]

Alternative 3
Error	0.3
Cost	13632

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

Alternative 4
Error	0.4
Cost	576

\[x \cdot \left(0.3333333333333333 + 0.022222222222222223 \cdot \left(x \cdot x\right)\right) \]

Alternative 5
Error	0.4
Cost	576

\[\frac{1}{\frac{3}{x} + x \cdot -0.2} \]

Alternative 6
Error	0.6
Cost	320

\[\frac{0.037037037037037035}{\frac{0.1111111111111111}{x}} \]

Alternative 7
Error	0.7
Cost	320

\[\frac{1}{\frac{3}{x}} \]

Alternative 8
Error	0.7
Cost	192

\[0.3333333333333333 \cdot x \]

Reproduce

herbie shell --seed 2023057 
(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))))