Prelude:atanh from fay-base-0.20.0.1

Percentage Accurate: 100.0% → 100.0%

Time: 1.3s

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. Final simplification 100.0%

   \[\leadsto \frac{x + 1}{1 - x} \]

Alternative 2: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1:\\ \;\;\;\;-1 + \frac{-2}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.1) (+ -1.0 (/ -2.0 x)) (if (<= x 1.0) (+ 1.0 (* x 2.0)) -1.0)))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.1d0)) then
        tmp = (-1.0d0) + ((-2.0d0) / x)
    else if (x <= 1.0d0) then
        tmp = 1.0d0 + (x * 2.0d0)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= -2.1:
		tmp = -1.0 + (-2.0 / x)
	elif x <= 1.0:
		tmp = 1.0 + (x * 2.0)
	else:
		tmp = -1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -2.1], N[(-1.0 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.10000000000000009

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 99.1%

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

      1. distribute-neg-in 99.1%

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

      2. metadata-eval 99.1%

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

      3. associate-*r/ 99.1%

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

      4. metadata-eval 99.1%

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

      5. distribute-neg-frac 99.1%

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

      6. metadata-eval 99.1%

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

   4. Simplified 99.1%

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

   if -2.10000000000000009 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.1%

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

   if 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1:\\ \;\;\;\;-1 + \frac{-2}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1 + \frac{-2}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) (+ -1.0 (/ -2.0 x)) (if (<= x 1.0) 1.0 -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0 + (-2.0 / x);
	} 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) + ((-2.0d0) / x)
    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 + (-2.0 / x);
	} 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 + (-2.0 / x)
	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 = Float64(-1.0 + Float64(-2.0 / x));
	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 + (-2.0 / x);
	elseif (x <= 1.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(-1.0 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], 1.0, -1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 99.1%

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

      1. distribute-neg-in 99.1%

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

      2. metadata-eval 99.1%

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

      3. associate-*r/ 99.1%

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

      4. metadata-eval 99.1%

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

      5. distribute-neg-frac 99.1%

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

      6. metadata-eval 99.1%

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

   4. Simplified 99.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.8%

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

   if 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1 + \frac{-2}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 97.7% accurate, 1.4× 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. Taylor expanded in x around inf 99.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.8%

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

4. Final simplification 99.0%

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

Alternative 5: 50.2% 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. Taylor expanded in x around inf 54.5%

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

3. Final simplification 54.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023196 
(FPCore (x)
  :name "Prelude:atanh from fay-base-0.20.0.1"
  :precision binary64
  (/ (+ x 1.0) (- 1.0 x)))