invcot (example 3.9)

Percentage Accurate: 6.3% → 99.6%

Time: 17.1s

Alternatives: 5

Speedup: 35.7×

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: 6.3% 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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \left(0.1111111111111111 - {x}^{4} \cdot {\left(\mathsf{fma}\left({x}^{2}, 0.0021164021164021165, 0.022222222222222223\right)\right)}^{2}\right) \cdot \frac{x}{0.3333333333333333 - \mathsf{fma}\left(0.022222222222222223, {x}^{2}, {x}^{4} \cdot 0.0021164021164021165\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (-
   0.1111111111111111
   (*
    (pow x 4.0)
    (pow (fma (pow x 2.0) 0.0021164021164021165 0.022222222222222223) 2.0)))
  (/
   x
   (-
    0.3333333333333333
    (fma
     0.022222222222222223
     (pow x 2.0)
     (* (pow x 4.0) 0.0021164021164021165))))))

C

double code(double x) {
	return (0.1111111111111111 - (pow(x, 4.0) * pow(fma(pow(x, 2.0), 0.0021164021164021165, 0.022222222222222223), 2.0))) * (x / (0.3333333333333333 - fma(0.022222222222222223, pow(x, 2.0), (pow(x, 4.0) * 0.0021164021164021165))));
}

Julia

function code(x)
	return Float64(Float64(0.1111111111111111 - Float64((x ^ 4.0) * (fma((x ^ 2.0), 0.0021164021164021165, 0.022222222222222223) ^ 2.0))) * Float64(x / Float64(0.3333333333333333 - fma(0.022222222222222223, (x ^ 2.0), Float64((x ^ 4.0) * 0.0021164021164021165)))))
end

Wolfram

code[x_] := N[(N[(0.1111111111111111 - N[(N[Power[x, 4.0], $MachinePrecision] * N[Power[N[(N[Power[x, 2.0], $MachinePrecision] * 0.0021164021164021165 + 0.022222222222222223), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(0.3333333333333333 - N[(0.022222222222222223 * N[Power[x, 2.0], $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * 0.0021164021164021165), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in x around 0 99.4%

   \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + 0.0021164021164021165 \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. flip-+ 99.4%

      \[\leadsto x \cdot \color{blue}{\frac{0.3333333333333333 \cdot 0.3333333333333333 - \left({x}^{2} \cdot \left(0.022222222222222223 + 0.0021164021164021165 \cdot {x}^{2}\right)\right) \cdot \left({x}^{2} \cdot \left(0.022222222222222223 + 0.0021164021164021165 \cdot {x}^{2}\right)\right)}{0.3333333333333333 - {x}^{2} \cdot \left(0.022222222222222223 + 0.0021164021164021165 \cdot {x}^{2}\right)}} \]

   2. associate-*r/ 99.5%

      \[\leadsto \color{blue}{\frac{x \cdot \left(0.3333333333333333 \cdot 0.3333333333333333 - \left({x}^{2} \cdot \left(0.022222222222222223 + 0.0021164021164021165 \cdot {x}^{2}\right)\right) \cdot \left({x}^{2} \cdot \left(0.022222222222222223 + 0.0021164021164021165 \cdot {x}^{2}\right)\right)\right)}{0.3333333333333333 - {x}^{2} \cdot \left(0.022222222222222223 + 0.0021164021164021165 \cdot {x}^{2}\right)}} \]

5. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{x \cdot \left(0.1111111111111111 - {x}^{4} \cdot {\left(\mathsf{fma}\left({x}^{2}, 0.0021164021164021165, 0.022222222222222223\right)\right)}^{2}\right)}{0.3333333333333333 - \mathsf{fma}\left(0.022222222222222223, {x}^{2}, 0.0021164021164021165 \cdot {x}^{4}\right)}} \]
6. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \frac{\color{blue}{\left(0.1111111111111111 - {x}^{4} \cdot {\left(\mathsf{fma}\left({x}^{2}, 0.0021164021164021165, 0.022222222222222223\right)\right)}^{2}\right) \cdot x}}{0.3333333333333333 - \mathsf{fma}\left(0.022222222222222223, {x}^{2}, 0.0021164021164021165 \cdot {x}^{4}\right)} \]

   2. associate-/l* 99.6%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - {x}^{4} \cdot {\left(\mathsf{fma}\left({x}^{2}, 0.0021164021164021165, 0.022222222222222223\right)\right)}^{2}\right) \cdot \frac{x}{0.3333333333333333 - \mathsf{fma}\left(0.022222222222222223, {x}^{2}, 0.0021164021164021165 \cdot {x}^{4}\right)}} \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{\left(0.1111111111111111 - {x}^{4} \cdot {\left(\mathsf{fma}\left({x}^{2}, 0.0021164021164021165, 0.022222222222222223\right)\right)}^{2}\right) \cdot \frac{x}{0.3333333333333333 - \mathsf{fma}\left(0.022222222222222223, {x}^{2}, 0.0021164021164021165 \cdot {x}^{4}\right)}} \]

8. Final simplification 99.6%

   \[\leadsto \left(0.1111111111111111 - {x}^{4} \cdot {\left(\mathsf{fma}\left({x}^{2}, 0.0021164021164021165, 0.022222222222222223\right)\right)}^{2}\right) \cdot \frac{x}{0.3333333333333333 - \mathsf{fma}\left(0.022222222222222223, {x}^{2}, {x}^{4} \cdot 0.0021164021164021165\right)} \]
9. Add Preprocessing

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + {x}^{2} \cdot \left(0.0021164021164021165 + x \cdot \left(x \cdot 0.00021164021164021165\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   0.3333333333333333
   (*
    (pow x 2.0)
    (+
     0.022222222222222223
     (*
      (pow x 2.0)
      (+ 0.0021164021164021165 (* x (* x 0.00021164021164021165)))))))))

C

double code(double x) {
	return x * (0.3333333333333333 + (pow(x, 2.0) * (0.022222222222222223 + (pow(x, 2.0) * (0.0021164021164021165 + (x * (x * 0.00021164021164021165)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.3333333333333333d0 + ((x ** 2.0d0) * (0.022222222222222223d0 + ((x ** 2.0d0) * (0.0021164021164021165d0 + (x * (x * 0.00021164021164021165d0)))))))
end function

Java

public static double code(double x) {
	return x * (0.3333333333333333 + (Math.pow(x, 2.0) * (0.022222222222222223 + (Math.pow(x, 2.0) * (0.0021164021164021165 + (x * (x * 0.00021164021164021165)))))));
}

Python

def code(x):
	return x * (0.3333333333333333 + (math.pow(x, 2.0) * (0.022222222222222223 + (math.pow(x, 2.0) * (0.0021164021164021165 + (x * (x * 0.00021164021164021165)))))))

Julia

function code(x)
	return Float64(x * Float64(0.3333333333333333 + Float64((x ^ 2.0) * Float64(0.022222222222222223 + Float64((x ^ 2.0) * Float64(0.0021164021164021165 + Float64(x * Float64(x * 0.00021164021164021165))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.3333333333333333 + ((x ^ 2.0) * (0.022222222222222223 + ((x ^ 2.0) * (0.0021164021164021165 + (x * (x * 0.00021164021164021165)))))));
end

Wolfram

code[x_] := N[(x * N[(0.3333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.022222222222222223 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.0021164021164021165 + N[(x * N[(x * 0.00021164021164021165), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in x around 0 99.5%

   \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + {x}^{2} \cdot \left(0.0021164021164021165 + 0.00021164021164021165 \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 99.5%

      \[\leadsto x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + {x}^{2} \cdot \left(0.0021164021164021165 + 0.00021164021164021165 \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

   2. associate-*r* 99.5%

      \[\leadsto x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + {x}^{2} \cdot \left(0.0021164021164021165 + \color{blue}{\left(0.00021164021164021165 \cdot x\right) \cdot x}\right)\right)\right) \]

5. Applied egg-rr 99.5%

   \[\leadsto x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + {x}^{2} \cdot \left(0.0021164021164021165 + \color{blue}{\left(0.00021164021164021165 \cdot x\right) \cdot x}\right)\right)\right) \]

6. Final simplification 99.5%

   \[\leadsto x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + {x}^{2} \cdot \left(0.0021164021164021165 + x \cdot \left(x \cdot 0.00021164021164021165\right)\right)\right)\right) \]
7. Add Preprocessing

Alternative 3: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + x \cdot \left(x \cdot 0.0021164021164021165\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   0.3333333333333333
   (*
    (pow x 2.0)
    (+ 0.022222222222222223 (* x (* x 0.0021164021164021165)))))))

C

double code(double x) {
	return x * (0.3333333333333333 + (pow(x, 2.0) * (0.022222222222222223 + (x * (x * 0.0021164021164021165)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.3333333333333333d0 + ((x ** 2.0d0) * (0.022222222222222223d0 + (x * (x * 0.0021164021164021165d0)))))
end function

Java

public static double code(double x) {
	return x * (0.3333333333333333 + (Math.pow(x, 2.0) * (0.022222222222222223 + (x * (x * 0.0021164021164021165)))));
}

Python

def code(x):
	return x * (0.3333333333333333 + (math.pow(x, 2.0) * (0.022222222222222223 + (x * (x * 0.0021164021164021165)))))

Julia

function code(x)
	return Float64(x * Float64(0.3333333333333333 + Float64((x ^ 2.0) * Float64(0.022222222222222223 + Float64(x * Float64(x * 0.0021164021164021165))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.3333333333333333 + ((x ^ 2.0) * (0.022222222222222223 + (x * (x * 0.0021164021164021165)))));
end

Wolfram

code[x_] := N[(x * N[(0.3333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.022222222222222223 + N[(x * N[(x * 0.0021164021164021165), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in x around 0 99.4%

   \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + 0.0021164021164021165 \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 99.4%

      \[\leadsto x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + 0.0021164021164021165 \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

   2. associate-*r* 99.4%

      \[\leadsto x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + \color{blue}{\left(0.0021164021164021165 \cdot x\right) \cdot x}\right)\right) \]

5. Applied egg-rr 99.4%

   \[\leadsto x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + \color{blue}{\left(0.0021164021164021165 \cdot x\right) \cdot x}\right)\right) \]

6. Final simplification 99.4%

   \[\leadsto x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + x \cdot \left(x \cdot 0.0021164021164021165\right)\right)\right) \]
7. Add Preprocessing

Alternative 4: 99.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0.3333333333333333 + x \cdot \left(x \cdot 0.022222222222222223\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 0.3333333333333333 (* x (* x 0.022222222222222223)))))

C

double code(double x) {
	return x * (0.3333333333333333 + (x * (x * 0.022222222222222223)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.3333333333333333d0 + (x * (x * 0.022222222222222223d0)))
end function

Java

public static double code(double x) {
	return x * (0.3333333333333333 + (x * (x * 0.022222222222222223)));
}

Python

def code(x):
	return x * (0.3333333333333333 + (x * (x * 0.022222222222222223)))

Julia

function code(x)
	return Float64(x * Float64(0.3333333333333333 + Float64(x * Float64(x * 0.022222222222222223))))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.3333333333333333 + (x * (x * 0.022222222222222223)));
end

Wolfram

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

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in x around 0 99.3%

   \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 + 0.022222222222222223 \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 99.3%

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

   2. associate-*r* 99.3%

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

5. Applied egg-rr 99.3%

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

6. Final simplification 99.3%

   \[\leadsto x \cdot \left(0.3333333333333333 + x \cdot \left(x \cdot 0.022222222222222223\right)\right) \]
7. Add Preprocessing

Alternative 5: 99.0% accurate, 35.7× 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 6.7%

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

3. Taylor expanded in x around 0 98.7%

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

4. Final simplification 98.7%

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

Developer target: 99.9% accurate, 0.5× 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 2024095 
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :pre (and (< -0.026 x) (< x 0.026))

  :alt
  (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))))