invcot (example 3.9)

Percentage Accurate: 6.2% → 99.9%

Time: 14.7s

Alternatives: 3

Speedup: 21.0×

Specification

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

Math

\[\begin{array}{l} \\ \frac{1}{x} - \frac{1}{\tan x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} - \frac{1}{\tan x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{1}{\mathsf{fma}\left(x, x \cdot 0.022222222222222223, 0.3333333333333333\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ x (/ 1.0 (fma x (* x 0.022222222222222223) 0.3333333333333333))))

C

double code(double x) {
	return x / (1.0 / fma(x, (x * 0.022222222222222223), 0.3333333333333333));
}

Julia

function code(x)
	return Float64(x / Float64(1.0 / fma(x, Float64(x * 0.022222222222222223), 0.3333333333333333)))
end

Wolfram

code[x_] := N[(x / N[(1.0 / N[(x * N[(x * 0.022222222222222223), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.9%

   \[\frac{1}{x} - \frac{1}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {x}^{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{45} \cdot {x}^{2} + \frac{1}{3}\right)} \]

   3. unpow2 N/A

      \[\leadsto x \cdot \left(\frac{1}{45} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{3}\right) \]

   4. associate-*r* N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(\frac{1}{45} \cdot x\right) \cdot x} + \frac{1}{3}\right) \]

   5. *-commutative N/A

      \[\leadsto x \cdot \left(\color{blue}{x \cdot \left(\frac{1}{45} \cdot x\right)} + \frac{1}{3}\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{45} \cdot x, \frac{1}{3}\right)} \]

   7. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{45}}, \frac{1}{3}\right) \]

   8. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot 0.022222222222222223, 0.3333333333333333\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{45}\right)} + \frac{1}{3}\right) \]

   2. flip-+ N/A

      \[\leadsto x \cdot \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) - \frac{1}{3} \cdot \frac{1}{3}}{x \cdot \left(x \cdot \frac{1}{45}\right) - \frac{1}{3}}} \]

   3. clear-num N/A

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot \frac{1}{45}\right) - \frac{1}{3}}{\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) - \frac{1}{3} \cdot \frac{1}{3}}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{x}{\frac{x \cdot \left(x \cdot \frac{1}{45}\right) - \frac{1}{3}}{\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) - \frac{1}{3} \cdot \frac{1}{3}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{\frac{x \cdot \left(x \cdot \frac{1}{45}\right) - \frac{1}{3}}{\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) - \frac{1}{3} \cdot \frac{1}{3}}}} \]

   6. clear-num N/A

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) - \frac{1}{3} \cdot \frac{1}{3}}{x \cdot \left(x \cdot \frac{1}{45}\right) - \frac{1}{3}}}}} \]

   7. flip-+ N/A

      \[\leadsto \frac{x}{\frac{1}{\color{blue}{x \cdot \left(x \cdot \frac{1}{45}\right) + \frac{1}{3}}}} \]

   8. lift-fma.f64 N/A

      \[\leadsto \frac{x}{\frac{1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{45}, \frac{1}{3}\right)}}} \]

   9. lower-/.f64 100.0

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

7. Applied rewrites 100.0%

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

Alternative 2: 99.5% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \frac{x}{3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x 3.0))

C

double code(double x) {
	return x / 3.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / 3.0d0
end function

Java

public static double code(double x) {
	return x / 3.0;
}

Python

def code(x):
	return x / 3.0

Julia

function code(x)
	return Float64(x / 3.0)
end

MATLAB

function tmp = code(x)
	tmp = x / 3.0;
end

Wolfram

code[x_] := N[(x / 3.0), $MachinePrecision]

Derivation

1. Initial program 5.9%

   \[\frac{1}{x} - \frac{1}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot {x}^{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{45} \cdot {x}^{2} + \frac{1}{3}\right)} \]

   3. unpow2 N/A

      \[\leadsto x \cdot \left(\frac{1}{45} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{3}\right) \]

   4. associate-*r* N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(\frac{1}{45} \cdot x\right) \cdot x} + \frac{1}{3}\right) \]

   5. *-commutative N/A

      \[\leadsto x \cdot \left(\color{blue}{x \cdot \left(\frac{1}{45} \cdot x\right)} + \frac{1}{3}\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{45} \cdot x, \frac{1}{3}\right)} \]

   7. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{45}}, \frac{1}{3}\right) \]

   8. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot 0.022222222222222223, 0.3333333333333333\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{45}\right)} + \frac{1}{3}\right) \]

   2. flip-+ N/A

      \[\leadsto x \cdot \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) - \frac{1}{3} \cdot \frac{1}{3}}{x \cdot \left(x \cdot \frac{1}{45}\right) - \frac{1}{3}}} \]

   3. clear-num N/A

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot \frac{1}{45}\right) - \frac{1}{3}}{\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) - \frac{1}{3} \cdot \frac{1}{3}}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{x}{\frac{x \cdot \left(x \cdot \frac{1}{45}\right) - \frac{1}{3}}{\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) - \frac{1}{3} \cdot \frac{1}{3}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{\frac{x \cdot \left(x \cdot \frac{1}{45}\right) - \frac{1}{3}}{\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) - \frac{1}{3} \cdot \frac{1}{3}}}} \]

   6. clear-num N/A

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) - \frac{1}{3} \cdot \frac{1}{3}}{x \cdot \left(x \cdot \frac{1}{45}\right) - \frac{1}{3}}}}} \]

   7. flip-+ N/A

      \[\leadsto \frac{x}{\frac{1}{\color{blue}{x \cdot \left(x \cdot \frac{1}{45}\right) + \frac{1}{3}}}} \]

   8. lift-fma.f64 N/A

      \[\leadsto \frac{x}{\frac{1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{45}, \frac{1}{3}\right)}}} \]

   9. lower-/.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{x}{\color{blue}{3}} \]
9. Step-by-step derivation

10. Applied rewrites 99.7%

    \[\leadsto \frac{x}{\color{blue}{3}} \]
11. Add Preprocessing

Alternative 3: 99.0% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.3333333333333333 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * 0.3333333333333333

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.9%

   \[\frac{1}{x} - \frac{1}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot x} \]
4. Step-by-step derivation

   1. lower-*.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Final simplification 99.3%

   \[\leadsto x \cdot 0.3333333333333333 \]
7. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (< (fabs x) 0.026)
   (* (/ x 3.0) (+ 1.0 (/ (* x x) 15.0)))
   (- (/ 1.0 x) (/ 1.0 (tan x)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) < 0.026) {
		tmp = (x / 3.0) * (1.0 + ((x * x) / 15.0));
	} else {
		tmp = (1.0 / x) - (1.0 / tan(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) < 0.026d0) then
        tmp = (x / 3.0d0) * (1.0d0 + ((x * x) / 15.0d0))
    else
        tmp = (1.0d0 / x) - (1.0d0 / tan(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) < 0.026) {
		tmp = (x / 3.0) * (1.0 + ((x * x) / 15.0));
	} else {
		tmp = (1.0 / x) - (1.0 / Math.tan(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) < 0.026:
		tmp = (x / 3.0) * (1.0 + ((x * x) / 15.0))
	else:
		tmp = (1.0 / x) - (1.0 / math.tan(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) < 0.026)
		tmp = Float64(Float64(x / 3.0) * Float64(1.0 + Float64(Float64(x * x) / 15.0)));
	else
		tmp = Float64(Float64(1.0 / x) - Float64(1.0 / tan(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) < 0.026)
		tmp = (x / 3.0) * (1.0 + ((x * x) / 15.0));
	else
		tmp = (1.0 / x) - (1.0 / tan(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Less[N[Abs[x], $MachinePrecision], 0.026], N[(N[(x / 3.0), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] / 15.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :pre (and (< -0.026 x) (< x 0.026))

  :alt
  (! :herbie-platform default (if (< (fabs x) 13/500) (* (/ x 3) (+ 1 (/ (* x x) 15))) (- (/ 1 x) (/ 1 (tan x)))))

  (- (/ 1.0 x) (/ 1.0 (tan x))))