Average Error: 60.0 → 0.4
Time: 10.0s
Precision: binary64
Cost: 704
\[-0.026 < x \land x < 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[x \cdot 0.3333333333333333 + x \cdot \left(0.022222222222222223 \cdot \left(x \cdot x\right)\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
x \cdot 0.3333333333333333 + x \cdot \left(0.022222222222222223 \cdot \left(x \cdot x\right)\right)
(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 (tan x))))
↓
(FPCore (x)
 :precision binary64
 (+ (* x 0.3333333333333333) (* x (* 0.022222222222222223 (* x x)))))
double code(double x) {
	return (1.0 / x) - (1.0 / tan(x));
}
↓
double code(double x) {
	return (x * 0.3333333333333333) + (x * (0.022222222222222223 * (x * x)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Alternatives

Alternative 1
Error0.6
Cost192
\[x \cdot 0.3333333333333333\]
Alternative 2
Error60.5
Cost64
\[0\]
Alternative 3
Error61.6
Cost64
\[1\]

Error

Derivation

  1. Initial program 60.0

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

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

    \[\leadsto \color{blue}{x \cdot 0.3333333333333333 + 0.022222222222222223 \cdot {x}^{3}}\]
  4. Using strategy rm

  5. Applied unpow3_binary64_8260.4

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

  6. Applied associate-*r*_binary64_7000.4

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

  7. Simplified0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2021044 
(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))))