Prelude:atanh from fay-base-0.20.0.1

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 7

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: 99.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ 1.0 x) (- 1.0 x)) -0.5)
   (- -1.0 (/ 2.0 x))
   (fma (fma x 2.0 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(x, 2.0, 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(x, 2.0, 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[(x * 2.0 + 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. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \left(1 + 2 \cdot \frac{1}{x}\right)} \]

      3. associate--r+ N/A

         \[\leadsto \color{blue}{\left(0 - 1\right) - 2 \cdot \frac{1}{x}} \]

      4. metadata-eval 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 99.2

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

   5. Applied rewrites 99.2%

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

   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. *-commutative N/A

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

      6. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 99.0%

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

Alternative 3: 98.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ 1.0 x) (- 1.0 x)) -0.5) -1.0 (fma (fma x 2.0 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(x, 2.0, 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(x, 2.0, 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[(x * 2.0 + 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 98.2%

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

   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. *-commutative N/A

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

      6. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 98.5%

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

Alternative 4: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 + x) / (1.0d0 - x)) <= (-0.5d0)) then
        tmp = -1.0d0
    else
        tmp = ((-1.0d0) - x) * ((-1.0d0) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 + x) / (1.0 - x)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = (-1.0 - x) * (-1.0 - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 + x) / (1.0 - x)) <= -0.5:
		tmp = -1.0
	else:
		tmp = (-1.0 - x) * (-1.0 - x)
	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 = Float64(Float64(-1.0 - x) * Float64(-1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 + x) / (1.0 - x)) <= -0.5)
		tmp = -1.0;
	else
		tmp = (-1.0 - x) * (-1.0 - x);
	end
	tmp_2 = 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[(-1.0 - x), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $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 98.2%

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

   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

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 - {x}^{4}\right) \cdot {\left(1 - x\right)}^{-1}}{\mathsf{fma}\left(x, x, 1\right)}}}{1 - x} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(1 - {x}^{4}\right) \cdot {\left(1 - x\right)}^{-1}}{\mathsf{fma}\left(x, x, 1\right)}}{1 - x}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{\left(1 - {x}^{4}\right) \cdot {\left(1 - x\right)}^{-1}}{\mathsf{fma}\left(x, x, 1\right)}\right)}{\mathsf{neg}\left(\left(1 - x\right)\right)}} \]

      3. distribute-frac-neg2 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\frac{\left(1 - {x}^{4}\right) \cdot {\left(1 - x\right)}^{-1}}{\mathsf{fma}\left(x, x, 1\right)}\right)}{1 - x}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\frac{\mathsf{neg}\left(\frac{\left(1 - {x}^{4}\right) \cdot {\left(1 - x\right)}^{-1}}{\mathsf{fma}\left(x, x, 1\right)}\right)}{1 - x}} \]

      5. distribute-frac-neg N/A

         \[\leadsto -\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{\left(1 - {x}^{4}\right) \cdot {\left(1 - x\right)}^{-1}}{\mathsf{fma}\left(x, x, 1\right)}}{1 - x}\right)\right)} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto -\color{blue}{\frac{\frac{\left(1 - {x}^{4}\right) \cdot {\left(1 - x\right)}^{-1}}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{neg}\left(\left(1 - x\right)\right)}} \]

   6. Applied rewrites 100.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow-1 N/A

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

      12. div-inv N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. unsub-neg N/A

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

      20. lower--.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

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

       5. unsub-neg N/A

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

       6. lower--.f64 98.3

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

   11. Applied rewrites 98.3%

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

4. Final simplification 98.2%

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

Alternative 5: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((1.0 + x) / (1.0 - x)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = fma(x, 2.0, 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(x, 2.0, 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[(x * 2.0 + 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 98.2%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 98.2%

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

Alternative 6: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1 + x}{1 - x} \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ 1.0 x) (- 1.0 x)) -1e-309) -1.0 1.0))

C

double code(double x) {
	double tmp;
	if (((1.0 + x) / (1.0 - x)) <= -1e-309) {
		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)) <= (-1d-309)) 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)) <= -1e-309) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 + x) / (1.0 - x)) <= -1e-309:
		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)) <= -1e-309)
		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)) <= -1e-309)
		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], -1e-309], -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)) < -1.000000000000002e-309

   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 98.2%

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

   if -1.000000000000002e-309 < (/.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 97.3%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 + x}{1 - x} \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.0% 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 52.1%

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

Reproduce

herbie shell --seed 2024254 
(FPCore (x)
  :name "Prelude:atanh from fay-base-0.20.0.1"
  :precision binary64
  (/ (+ x 1.0) (- 1.0 x)))