Asymptote A

Percentage Accurate: 76.5% → 99.3%

Time: 3.4s

Alternatives: 5

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(1 + x\right) \cdot \left(1 - x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (* (+ 1.0 x) (- 1.0 x))))

C

double code(double x) {
	return 2.0 / ((1.0 + x) * (1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / ((1.0d0 + x) * (1.0d0 - x))
end function

Java

public static double code(double x) {
	return 2.0 / ((1.0 + x) * (1.0 - x));
}

Python

def code(x):
	return 2.0 / ((1.0 + x) * (1.0 - x))

Julia

function code(x)
	return Float64(2.0 / Float64(Float64(1.0 + x) * Float64(1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / ((1.0 + x) * (1.0 - x));
end

Wolfram

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

Derivation

1. Initial program 73.7%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
2. Step-by-step derivation

   1. frac-2neg 73.7%

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

   2. metadata-eval 73.7%

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

   3. frac-sub 74.9%

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

   4. *-un-lft-identity 74.9%

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

   5. sub-neg 74.9%

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

   6. metadata-eval 74.9%

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

   7. distribute-neg-in 74.9%

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

   8. metadata-eval 74.9%

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

   9. +-commutative 74.9%

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

   10. +-commutative 74.9%

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

   11. sub-neg 74.9%

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

   12. metadata-eval 74.9%

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

   13. distribute-neg-in 74.9%

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

   14. metadata-eval 74.9%

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

3. Applied egg-rr 74.9%

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

4. Taylor expanded in x around 0 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55) (- (- 1.0 x) (/ 1.0 (+ x -1.0))) (/ (/ (- 2.0) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = (1.0 - x) - (1.0 / (x + -1.0));
	} else {
		tmp = (-2.0 / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.55d0) then
        tmp = (1.0d0 - x) - (1.0d0 / (x + (-1.0d0)))
    else
        tmp = (-2.0d0 / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = (1.0 - x) - (1.0 / (x + -1.0));
	} else {
		tmp = (-2.0 / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = (1.0 - x) - (1.0 / (x + -1.0))
	else:
		tmp = (-2.0 / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = Float64(Float64(1.0 - x) - Float64(1.0 / Float64(x + -1.0)));
	else
		tmp = Float64(Float64(Float64(-2.0) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.55)
		tmp = (1.0 - x) - (1.0 / (x + -1.0));
	else
		tmp = (-2.0 / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], N[(N[(1.0 - x), $MachinePrecision] - N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-2.0) / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 79.9%

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

   2. Taylor expanded in x around 0 60.9%

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

      1. neg-mul-1 60.9%

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

      2. unsub-neg 60.9%

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

   4. Simplified 60.9%

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

   if 1.55000000000000004 < x

   1. Initial program 55.5%

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

   2. Taylor expanded in x around inf 99.5%

      \[\leadsto \color{blue}{\frac{-2}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 99.5%

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

   4. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{-2}{x \cdot x}} \]
   5. Step-by-step derivation

      1. associate-/r* 99.4%

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

      2. div-inv 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. un-div-inv 99.4%

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

      2. frac-2neg 99.4%

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

      3. frac-2neg 99.4%

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

      4. metadata-eval 99.4%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 52.8%

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

      7. sqr-neg 52.8%

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

      8. sqrt-unprod 52.8%

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

      9. add-sqr-sqrt 52.8%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 99.4%

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

      12. sqr-neg 99.4%

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

      13. sqrt-unprod 99.2%

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

      14. add-sqr-sqrt 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \end{array} \]

Alternative 3: 74.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 79.8%

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

   2. Taylor expanded in x around 0 61.4%

      \[\leadsto \color{blue}{2} \]

   if 1 < x

   1. Initial program 56.2%

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

   2. Taylor expanded in x around inf 98.2%

      \[\leadsto \color{blue}{\frac{-2}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 98.2%

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

   4. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{-2}{x \cdot x}} \]
   5. Step-by-step derivation

      1. associate-/r* 98.2%

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

      2. div-inv 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. un-div-inv 98.2%

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

      2. frac-2neg 98.2%

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

      3. frac-2neg 98.2%

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

      4. metadata-eval 98.2%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 52.0%

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

      7. sqr-neg 52.0%

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

      8. sqrt-unprod 52.0%

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

      9. add-sqr-sqrt 52.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 98.1%

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

      12. sqr-neg 98.1%

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

      13. sqrt-unprod 97.9%

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

      14. add-sqr-sqrt 98.2%

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

   8. Applied egg-rr 98.2%

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

4. Final simplification 70.9%

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

Alternative 4: 74.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 79.8%

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

   2. Taylor expanded in x around 0 61.4%

      \[\leadsto \color{blue}{2} \]

   if 1 < x

   1. Initial program 56.2%

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

   2. Taylor expanded in x around inf 98.2%

      \[\leadsto \color{blue}{\frac{-2}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 98.2%

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

   4. Simplified 98.2%

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

4. Final simplification 70.9%

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

Alternative 5: 49.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 73.7%

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

2. Taylor expanded in x around 0 46.2%

   \[\leadsto \color{blue}{2} \]

3. Final simplification 46.2%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023230 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))