Prelude:atanh from fay-base-0.20.0.1

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{-2}{x}\\ \mathbf{if}\;x \leq -0.48:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ -2.0 x))))
   (if (<= x -0.48) t_0 (if (<= x 1.0) (+ 1.0 (* x 2.0)) t_0))))

C

double code(double x) {
	double t_0 = -1.0 + (-2.0 / x);
	double tmp;
	if (x <= -0.48) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + ((-2.0d0) / x)
    if (x <= (-0.48d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 + (x * 2.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -1.0 + (-2.0 / x);
	double tmp;
	if (x <= -0.48) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = -1.0 + (-2.0 / x)
	tmp = 0
	if x <= -0.48:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0 + (x * 2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(-1.0 + Float64(-2.0 / x))
	tmp = 0.0
	if (x <= -0.48)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(1.0 + Float64(x * 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -1.0 + (-2.0 / x);
	tmp = 0.0;
	if (x <= -0.48)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = 1.0 + (x * 2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-1.0 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.48], t$95$0, If[LessEqual[x, 1.0], N[(1.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.47999999999999998 or 1 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(-1 \cdot \left(2 \cdot \frac{1}{x}\right)\right)}\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\mathsf{neg}\left(\frac{2 \cdot 1}{x}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{x}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \color{blue}{x}\right)\right) \]

      9. metadata-eval 98.9%

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(-2, x\right)\right) \]

   5. Simplified 98.9%

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

   if -0.47999999999999998 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(2 \cdot x\right)}\right) \]

      2. *-lowering-*.f64 98.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \color{blue}{x}\right)\right) \]

   5. Simplified 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.48:\\ \;\;\;\;-1 + \frac{-2}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{-2}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{-2}{x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ -2.0 x))))
   (if (<= x -1.0) t_0 (if (<= x 1.0) 1.0 t_0))))

C

double code(double x) {
	double t_0 = -1.0 + (-2.0 / x);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + ((-2.0d0) / x)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -1.0 + (-2.0 / x);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = -1.0 + (-2.0 / x)
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(-1.0 + Float64(-2.0 / x))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -1.0 + (-2.0 / x);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-1.0 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.0], 1.0, t$95$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

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(-1 \cdot \left(2 \cdot \frac{1}{x}\right)\right)}\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\mathsf{neg}\left(\frac{2 \cdot 1}{x}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{x}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \color{blue}{x}\right)\right) \]

      9. metadata-eval 98.9%

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(-2, x\right)\right) \]

   5. Simplified 98.9%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.3%

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

Alternative 4: 97.8% 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

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

   5. Simplified 98.2%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.3%

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

Alternative 5: 49.5% 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

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

5. Simplified 54.4%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x)
  :name "Prelude:atanh from fay-base-0.20.0.1"
  :precision binary64
  (/ (+ x 1.0) (- 1.0 x)))