Prelude:atanh from fay-base-0.20.0.1

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Alternatives: 6

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: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + 1}{1 - x} \leq -0.5:\\ \;\;\;\;-1 + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ x 1.0) (- 1.0 x)) -0.5)
   (+ -1.0 (/ -2.0 x))
   (fma 2.0 (fma x x x) 1.0)))

C

double code(double x) {
	double tmp;
	if (((x + 1.0) / (1.0 - x)) <= -0.5) {
		tmp = -1.0 + (-2.0 / x);
	} else {
		tmp = fma(2.0, fma(x, x, x), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x + 1.0) / Float64(1.0 - x)) <= -0.5)
		tmp = Float64(-1.0 + Float64(-2.0 / x));
	else
		tmp = fma(2.0, fma(x, x, x), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -0.5], N[(-1.0 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(x * x + x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) x)) < -0.5

   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

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 98.7

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

   5. Applied rewrites 98.7%

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

   if -0.5 < (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. rgt-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. *-lft-identity N/A

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

      14. lft-mult-inverse N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 99.5

          \[\leadsto \mathsf{fma}\left(2, \color{blue}{\mathsf{fma}\left(x, x, x\right)}, 1\right) \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + 1}{1 - x} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ x 1.0) (- 1.0 x)) -0.5) -1.0 (fma 2.0 (fma x x x) 1.0)))

C

double code(double x) {
	double tmp;
	if (((x + 1.0) / (1.0 - x)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = fma(2.0, fma(x, x, x), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x + 1.0) / Float64(1.0 - x)) <= -0.5)
		tmp = -1.0;
	else
		tmp = fma(2.0, fma(x, x, x), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -0.5], -1.0, N[(2.0 * N[(x * x + x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) x)) < -0.5

   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. Applied rewrites 96.8%

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

   if -0.5 < (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. rgt-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. *-lft-identity N/A

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

      14. lft-mult-inverse N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 99.5

          \[\leadsto \mathsf{fma}\left(2, \color{blue}{\mathsf{fma}\left(x, x, x\right)}, 1\right) \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + 1}{1 - x} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ x 1.0) (- 1.0 x)) -0.5) -1.0 (fma 2.0 x 1.0)))

C

double code(double x) {
	double tmp;
	if (((x + 1.0) / (1.0 - x)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = fma(2.0, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x + 1.0) / Float64(1.0 - x)) <= -0.5)
		tmp = -1.0;
	else
		tmp = fma(2.0, x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -0.5], -1.0, N[(2.0 * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) x)) < -0.5

   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. Applied rewrites 96.8%

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

   if -0.5 < (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) x))

   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. +-commutative N/A

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

      2. lower-fma.f64 99.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 1\right)} \]

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + 1}{1 - x} \leq -1 \cdot 10^{-313}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ x 1.0) (- 1.0 x)) -1e-313) -1.0 1.0))

C

double code(double x) {
	double tmp;
	if (((x + 1.0) / (1.0 - x)) <= -1e-313) {
		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) / (1.0d0 - x)) <= (-1d-313)) 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) / (1.0 - x)) <= -1e-313) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x + 1.0) / (1.0 - x)) <= -1e-313:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x + 1.0) / Float64(1.0 - x)) <= -1e-313)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x + 1.0) / (1.0 - x)) <= -1e-313)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -1e-313], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) x)) < -1.00000000001e-313

   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. Applied rewrites 96.8%

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

   if -1.00000000001e-313 < (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) x))

   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. Applied rewrites 97.9%

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

Alternative 6: 49.8% accurate, 18.0× 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. Applied rewrites 49.2%

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

Reproduce

herbie shell --seed 2024221 
(FPCore (x)
  :name "Prelude:atanh from fay-base-0.20.0.1"
  :precision binary64
  (/ (+ x 1.0) (- 1.0 x)))