invcot (example 3.9)

Average Error: 93.51% → 0.13%

Time: 16.2s

Precision: binary64

Cost: 576

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	93.51%
Target	0.15%
Herbie	0.13%

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

Derivation

1. Initial program 93.51

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

2. Taylor expanded in x around 0 0.67

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

3. Applied egg-rr 0.67

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

4. Applied egg-rr 0.15

   \[\leadsto \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(0.022222222222222223, x \cdot x, 0.3333333333333333\right)}}} \]

5. Taylor expanded in x around 0 0.13

   \[\leadsto \frac{x}{\color{blue}{3 + -0.2 \cdot {x}^{2}}} \]

6. Simplified 0.13

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

   [Start] 0.13	\[ \frac{x}{3 + -0.2 \cdot {x}^{2}} \]
   unpow2 [=>] 0.13	\[ \frac{x}{3 + -0.2 \cdot \color{blue}{\left(x \cdot x\right)}} \]

7. Final simplification 0.13

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

Alternatives

Alternative 1
Error	0.67%
Cost	576

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

Alternative 2
Error	0.56%
Cost	576

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

Alternative 3
Error	1.15%
Cost	192

\[x \cdot 0.3333333333333333 \]

Alternative 4
Error	0.63%
Cost	192

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

Reproduce

herbie shell --seed 2023088 
(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))))