invcot (example 3.9)

Average Error: 93.77% → 0.75%

Time: 13.9s

Precision: binary64

Cost: 576

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

Math

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

↓

\[\left(0.1111111111111111 \cdot x\right) \cdot \frac{x}{x \cdot 0.3333333333333333} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (* (* 0.1111111111111111 x) (/ x (* x 0.3333333333333333))))

C

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

↓

double code(double x) {
	return (0.1111111111111111 * x) * (x / (x * 0.3333333333333333));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / x) - (1.0d0 / tan(x))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.1111111111111111d0 * x) * (x / (x * 0.3333333333333333d0))
end function

Java

public static double code(double x) {
	return (1.0 / x) - (1.0 / Math.tan(x));
}

↓

public static double code(double x) {
	return (0.1111111111111111 * x) * (x / (x * 0.3333333333333333));
}

Python

def code(x):
	return (1.0 / x) - (1.0 / math.tan(x))

↓

def code(x):
	return (0.1111111111111111 * x) * (x / (x * 0.3333333333333333))

Julia

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

↓

function code(x)
	return Float64(Float64(0.1111111111111111 * x) * Float64(x / Float64(x * 0.3333333333333333)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / x) - (1.0 / tan(x));
end

↓

function tmp = code(x)
	tmp = (0.1111111111111111 * x) * (x / (x * 0.3333333333333333));
end

Wolfram

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

↓

code[x_] := N[(N[(0.1111111111111111 * x), $MachinePrecision] * N[(x / N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	93.77%
Target	0.09%
Herbie	0.75%

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

Derivation

1. Initial program 93.77

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

2. Taylor expanded in x around 0 0.96

   \[\leadsto \color{blue}{0.3333333333333333 \cdot x} \]

3. Applied egg-rr 92.59

   \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot x + 1\right) - 1} \]

4. Applied egg-rr 45.52

   \[\leadsto \color{blue}{\frac{0.1111111111111111 \cdot \left(x \cdot x\right) - 0}{0.3333333333333333 \cdot x - 0}} \]

5. Applied egg-rr 0.75

   \[\leadsto \color{blue}{\frac{0.1111111111111111 \cdot x}{1} \cdot \frac{x}{x \cdot 0.3333333333333333}} \]

6. Final simplification 0.75

   \[\leadsto \left(0.1111111111111111 \cdot x\right) \cdot \frac{x}{x \cdot 0.3333333333333333} \]

Alternatives

Alternative 1
Error	0.84%
Cost	320

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

Alternative 2
Error	0.96%
Cost	192

\[x \cdot 0.3333333333333333 \]

Reproduce

herbie shell --seed 2023121 
(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))))