Prelude:atanh from fay-base-0.20.0.1

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return (x + 1.0) / (1.0 - x);
}

Python

def code(x):
	return (x + 1.0) / (1.0 - x)

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return (x + 1.0) / (1.0 - x);
}

Python

def code(x):
	return (x + 1.0) / (1.0 - x)

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return (x + 1.0) / (1.0 - x);
}

Python

def code(x):
	return (x + 1.0) / (1.0 - x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;-1 + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.1) (not (<= x 1.0))) (+ -1.0 (/ -2.0 x)) (+ 1.0 (* x 2.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -2.1) || !(x <= 1.0)) {
		tmp = -1.0 + (-2.0 / x);
	} else {
		tmp = 1.0 + (x * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.1d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-1.0d0) + ((-2.0d0) / x)
    else
        tmp = 1.0d0 + (x * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -2.1) || !(x <= 1.0)) {
		tmp = -1.0 + (-2.0 / x);
	} else {
		tmp = 1.0 + (x * 2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -2.1) or not (x <= 1.0):
		tmp = -1.0 + (-2.0 / x)
	else:
		tmp = 1.0 + (x * 2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -2.1) || !(x <= 1.0))
		tmp = Float64(-1.0 + Float64(-2.0 / x));
	else
		tmp = Float64(1.0 + Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -2.1) || ~((x <= 1.0)))
		tmp = -1.0 + (-2.0 / x);
	else
		tmp = 1.0 + (x * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -2.1], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-1.0 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.10000000000000009 or 1 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 99.4%

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

      1. distribute-lft-in 99.4%

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

      2. metadata-eval 99.4%

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

      3. associate-*r/ 99.4%

         \[\leadsto -1 + -1 \cdot \color{blue}{\frac{2 \cdot 1}{x}} \]

      4. metadata-eval 99.4%

         \[\leadsto -1 + -1 \cdot \frac{\color{blue}{2}}{x} \]

      5. associate-*r/ 99.4%

         \[\leadsto -1 + \color{blue}{\frac{-1 \cdot 2}{x}} \]

      6. metadata-eval 99.4%

         \[\leadsto -1 + \frac{\color{blue}{-2}}{x} \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{-1 + \frac{-2}{x}} \]

   if -2.10000000000000009 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.7%

      \[\leadsto \color{blue}{1 + 2 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;-1 + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;-1 + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (+ -1.0 (/ -2.0 x)) 1.0))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -1.0 + (-2.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-1.0d0) + ((-2.0d0) / x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -1.0 + (-2.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = -1.0 + (-2.0 / x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(-1.0 + Float64(-2.0 / x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = -1.0 + (-2.0 / x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-1.0 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 98.7%

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

      1. distribute-lft-in 98.7%

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

      2. metadata-eval 98.7%

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

      3. associate-*r/ 98.7%

         \[\leadsto -1 + -1 \cdot \color{blue}{\frac{2 \cdot 1}{x}} \]

      4. metadata-eval 98.7%

         \[\leadsto -1 + -1 \cdot \frac{\color{blue}{2}}{x} \]

      5. associate-*r/ 98.7%

         \[\leadsto -1 + \color{blue}{\frac{-1 \cdot 2}{x}} \]

      6. metadata-eval 98.7%

         \[\leadsto -1 + \frac{\color{blue}{-2}}{x} \]

   5. Simplified 98.7%

      \[\leadsto \color{blue}{-1 + \frac{-2}{x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;-1 + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -1.0) -1.0 (if (<= x 1.0) 1.0 -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = -1.0d0
    else if (x <= 1.0d0) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0
	elif x <= 1.0:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 1.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 1.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], -1.0, If[LessEqual[x, 1.0], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 98.1%

      \[\leadsto \color{blue}{-1} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 49.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around inf 53.6%

   \[\leadsto \color{blue}{-1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024097 
(FPCore (x)
  :name "Prelude:atanh from fay-base-0.20.0.1"
  :precision binary64
  (/ (+ x 1.0) (- 1.0 x)))