invcot (example 3.9)

Average Error: 59.9 → 0.2

Time: 16.9s

Precision: binary64

Cost: 1344

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

Math

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

↓

\[\frac{1}{\frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.005714285714285714\right)}{-1} + \left(\frac{1}{\frac{x}{3}} + x \cdot -0.2\right)} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (/
  1.0
  (+
   (/ (* x (* (* x x) 0.005714285714285714)) -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 / (((x * ((x * x) * 0.005714285714285714)) / -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 / (((x * ((x * x) * 0.005714285714285714d0)) / (-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 / (((x * ((x * x) * 0.005714285714285714)) / -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 / (((x * ((x * x) * 0.005714285714285714)) / -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(Float64(x * Float64(Float64(x * x) * 0.005714285714285714)) / -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 / (((x * ((x * x) * 0.005714285714285714)) / -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[(N[(x * N[(N[(x * x), $MachinePrecision] * 0.005714285714285714), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision] + N[(N[(1.0 / N[(x / 3.0), $MachinePrecision]), $MachinePrecision] + N[(x * -0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	59.9
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 59.9

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

2. Applied egg-rr 59.9

   \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\tan x - x}{\tan x}}}} \]

3. Taylor expanded in x around 0 0.4

   \[\leadsto \frac{1}{\color{blue}{-0.005714285714285714 \cdot {x}^{3} + \left(3 \cdot \frac{1}{x} + -0.2 \cdot x\right)}} \]

4. Applied egg-rr 0.2

   \[\leadsto \frac{1}{-0.005714285714285714 \cdot {x}^{3} + \left(\color{blue}{\frac{1}{\frac{x}{3}}} + -0.2 \cdot x\right)} \]

5. Applied egg-rr 0.2

   \[\leadsto \frac{1}{\color{blue}{\frac{\left(-x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.005714285714285714\right)}{-1}} + \left(\frac{1}{\frac{x}{3}} + -0.2 \cdot x\right)} \]

6. Final simplification 0.2

   \[\leadsto \frac{1}{\frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.005714285714285714\right)}{-1} + \left(\frac{1}{\frac{x}{3}} + x \cdot -0.2\right)} \]

Alternatives

Alternative 1
Error	0.4
Cost	576

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

Alternative 2
Error	0.4
Cost	576

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

Alternative 3
Error	0.5
Cost	448

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

Alternative 4
Error	0.7
Cost	320

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

Alternative 5
Error	0.7
Cost	192

\[x \cdot 0.3333333333333333 \]

Reproduce

herbie shell --seed 2023187 
(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))))