invcot (example 3.9)

Average Error: 59.8 → 0.2

Time: 15.8s

Precision: binary64

Cost: 704

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (/ 1.0 (+ (/ 1.0 (/ x 3.0)) (* x -0.2))))

C

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

↓

double code(double x) {
	return 1.0 / ((1.0 / (x / 3.0)) + (x * -0.2));
}

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 = 1.0d0 / ((1.0d0 / (x / 3.0d0)) + (x * (-0.2d0)))
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 1.0 / ((1.0 / (x / 3.0)) + (x * -0.2));
}

Python

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

↓

def code(x):
	return 1.0 / ((1.0 / (x / 3.0)) + (x * -0.2))

Julia

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

↓

function code(x)
	return Float64(1.0 / Float64(Float64(1.0 / Float64(x / 3.0)) + Float64(x * -0.2)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = 1.0 / ((1.0 / (x / 3.0)) + (x * -0.2));
end

Wolfram

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

↓

code[x_] := N[(1.0 / N[(N[(1.0 / N[(x / 3.0), $MachinePrecision]), $MachinePrecision] + N[(x * -0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	59.8
Target	0.1
Herbie	0.2

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

Derivation

1. Initial program 59.8

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

2. Taylor expanded in x around 0 0.4

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

3. Applied egg-rr 29.4

   \[\leadsto \color{blue}{\frac{1}{\frac{0.3333333333333333 \cdot x + -0.022222222222222223 \cdot {x}^{3}}{{\left(0.3333333333333333 \cdot x\right)}^{2} - 0.0004938271604938272 \cdot {x}^{6}}}} \]

4. Taylor expanded in x around 0 0.5

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

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.4
Cost	576

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

Alternative 2
Error	0.7
Cost	320

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

Alternative 3
Error	0.7
Cost	192

\[x \cdot 0.3333333333333333 \]

Reproduce

herbie shell --seed 2022330 
(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))))