Prelude:atanh from fay-base-0.20.0.1

Percentage Accurate: 100.0% → 100.0%

Time: 2.9s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

      \[\leadsto \color{blue}{-1 \cdot \left(1 + 2 \cdot \frac{1}{x}\right)} \]
   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. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 + x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -0.5], -1.0, N[(N[(2.0 * x + 2.0), $MachinePrecision] * 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 98.9%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.3%

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

Alternative 4: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((1.0 + x) / (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(1.0 + x) / 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[(1.0 + x), $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 98.9%

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

   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 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 99.2%

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

Alternative 5: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1 + x}{1 - x} \leq -2 \cdot 10^{-312}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ 1.0 x) (- 1.0 x)) -2e-312) -1.0 1.0))

C

double code(double x) {
	double tmp;
	if (((1.0 + x) / (1.0 - x)) <= -2e-312) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 + x) / (1.0d0 - x)) <= (-2d-312)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 + x) / (1.0 - x)) <= -2e-312) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 + x) / (1.0 - x)) <= -2e-312:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 + x) / Float64(1.0 - x)) <= -2e-312)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 + x) / (1.0 - x)) <= -2e-312)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 + x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -2e-312], -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)) < -2.0000000000019e-312

   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 98.9%

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

   if -2.0000000000019e-312 < (/.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 98.6%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 + x}{1 - x} \leq -2 \cdot 10^{-312}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.7% 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 49.9%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x)
  :name "Prelude:atanh from fay-base-0.20.0.1"
  :precision binary64
  (/ (+ x 1.0) (- 1.0 x)))