Asymptote A

Percentage Accurate: 77.8% → 99.9%

Time: 5.1s

Alternatives: 7

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: 77.8% 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: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{-1 - x}}{-1 + x} \end{array} \]

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(Float64(2.0 / 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[(N[(2.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

   1. frac-sub 80.0%

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

   2. associate-/r* 80.0%

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

   3. *-rgt-identity 80.0%

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

   4. *-un-lft-identity 80.0%

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

   5. sub-neg 80.0%

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

   6. metadata-eval 80.0%

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

   7. +-commutative 80.0%

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

   8. +-commutative 80.0%

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

   9. sub-neg 80.0%

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

   10. metadata-eval 80.0%

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

4. Applied egg-rr 80.0%

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

   1. associate--r+ 80.0%

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

   2. div-sub 79.5%

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

   3. associate--l+ 79.5%

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

   4. metadata-eval 79.5%

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

   5. +-commutative 79.5%

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

   6. +-commutative 79.5%

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

6. Applied egg-rr 79.5%

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

   1. div-sub 80.0%

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

   2. associate-+r- 79.9%

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

   3. remove-double-neg 79.9%

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

   4. distribute-frac-neg 79.9%

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

   5. distribute-neg-frac2 79.9%

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

   6. distribute-neg-in 79.9%

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

   7. metadata-eval 79.9%

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

   8. +-commutative 79.9%

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

   9. +-commutative 79.9%

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

   10. associate-+l- 99.9%

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

   11. +-inverses 99.9%

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

   12. metadata-eval 99.9%

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

   13. metadata-eval 99.9%

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

   14. unsub-neg 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 75.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{-1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{-1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.5) (+ (- 1.0 x) (/ -1.0 (+ -1.0 x))) (/ (/ -2.0 x) (+ -1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = (1.0 - x) + (-1.0 / (-1.0 + x));
	} else {
		tmp = (-2.0 / x) / (-1.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.5d0) then
        tmp = (1.0d0 - x) + ((-1.0d0) / ((-1.0d0) + x))
    else
        tmp = ((-2.0d0) / x) / ((-1.0d0) + x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = (1.0 - x) + (-1.0 / (-1.0 + x));
	} else {
		tmp = (-2.0 / x) / (-1.0 + x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.5:
		tmp = (1.0 - x) + (-1.0 / (-1.0 + x))
	else:
		tmp = (-2.0 / x) / (-1.0 + x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5)
		tmp = Float64(Float64(1.0 - x) + Float64(-1.0 / Float64(-1.0 + x)));
	else
		tmp = Float64(Float64(-2.0 / x) / Float64(-1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.5)
		tmp = (1.0 - x) + (-1.0 / (-1.0 + x));
	else
		tmp = (-2.0 / x) / (-1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.5], N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / x), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 86.0%

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

   3. Taylor expanded in x around 0 69.8%

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

      1. neg-mul-1 69.8%

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

      2. sub-neg 69.8%

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

   5. Simplified 69.8%

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

   if 1.5 < x

   1. Initial program 58.7%

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

      1. frac-sub 59.9%

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

      2. associate-/r* 59.9%

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

      3. *-rgt-identity 59.9%

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

      4. *-un-lft-identity 59.9%

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

      5. sub-neg 59.9%

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

      6. metadata-eval 59.9%

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

      7. +-commutative 59.9%

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

      8. +-commutative 59.9%

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

      9. sub-neg 59.9%

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

      10. metadata-eval 59.9%

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

   4. Applied egg-rr 59.9%

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

   5. Taylor expanded in x around inf 95.8%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{-1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{-1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.75:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{-1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.75) 2.0 (/ (/ -2.0 x) (+ -1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.75) {
		tmp = 2.0;
	} else {
		tmp = (-2.0 / x) / (-1.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.75d0) then
        tmp = 2.0d0
    else
        tmp = ((-2.0d0) / x) / ((-1.0d0) + x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.75) {
		tmp = 2.0;
	} else {
		tmp = (-2.0 / x) / (-1.0 + x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.75:
		tmp = 2.0
	else:
		tmp = (-2.0 / x) / (-1.0 + x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.75)
		tmp = 2.0;
	else
		tmp = Float64(Float64(-2.0 / x) / Float64(-1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.75)
		tmp = 2.0;
	else
		tmp = (-2.0 / x) / (-1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.75], 2.0, N[(N[(-2.0 / x), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.75

   1. Initial program 86.0%

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

   3. Taylor expanded in x around 0 70.0%

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

   if 0.75 < x

   1. Initial program 58.7%

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

      1. frac-sub 59.9%

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

      2. associate-/r* 59.9%

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

      3. *-rgt-identity 59.9%

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

      4. *-un-lft-identity 59.9%

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

      5. sub-neg 59.9%

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

      6. metadata-eval 59.9%

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

      7. +-commutative 59.9%

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

      8. +-commutative 59.9%

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

      9. sub-neg 59.9%

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

      10. metadata-eval 59.9%

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

   4. Applied egg-rr 59.9%

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

   5. Taylor expanded in x around inf 95.8%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.75:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{-1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.58:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.58) 2.0 (/ 1.0 (* x (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.58) {
		tmp = 2.0;
	} 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 (x <= 0.58d0) then
        tmp = 2.0d0
    else
        tmp = 1.0d0 / (x * (1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.58) {
		tmp = 2.0;
	} else {
		tmp = 1.0 / (x * (1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.58:
		tmp = 2.0
	else:
		tmp = 1.0 / (x * (1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.58)
		tmp = 2.0;
	else
		tmp = Float64(1.0 / Float64(x * Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.58)
		tmp = 2.0;
	else
		tmp = 1.0 / (x * (1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.58], 2.0, N[(1.0 / N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.57999999999999996

   1. Initial program 86.0%

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

   3. Taylor expanded in x around 0 70.0%

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

   if 0.57999999999999996 < x

   1. Initial program 58.7%

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

   3. Taylor expanded in x around inf 54.6%

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

      1. frac-sub 54.6%

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

      2. *-un-lft-identity 54.6%

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

      3. sub-neg 54.6%

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

      4. metadata-eval 54.6%

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

      5. div-sub 54.6%

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

      6. distribute-rgt-out-- 54.6%

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

      7. *-un-lft-identity 54.6%

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

      8. pow2 54.6%

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

      9. *-rgt-identity 54.6%

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

      10. distribute-rgt-out-- 54.6%

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

      11. *-un-lft-identity 54.6%

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

      12. pow2 54.6%

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

   5. Applied egg-rr 54.6%

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

   7. Simplified 60.1%

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

Alternative 5: 52.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{-1 + x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. frac-sub 80.0%

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

   2. associate-/r* 80.0%

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

   3. *-rgt-identity 80.0%

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

   4. *-un-lft-identity 80.0%

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

   5. sub-neg 80.0%

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

   6. metadata-eval 80.0%

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

   7. +-commutative 80.0%

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

   8. +-commutative 80.0%

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

   9. sub-neg 80.0%

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

   10. metadata-eval 80.0%

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

4. Applied egg-rr 80.0%

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

   1. associate--r+ 80.0%

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

   2. div-sub 79.5%

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

   3. associate--l+ 79.5%

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

   4. metadata-eval 79.5%

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

   5. +-commutative 79.5%

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

   6. +-commutative 79.5%

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

6. Applied egg-rr 79.5%

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

   1. div-sub 80.0%

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

   2. associate-+r- 79.9%

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

   3. remove-double-neg 79.9%

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

   4. distribute-frac-neg 79.9%

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

   5. distribute-neg-frac2 79.9%

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

   6. distribute-neg-in 79.9%

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

   7. metadata-eval 79.9%

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

   8. +-commutative 79.9%

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

   9. +-commutative 79.9%

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

   10. associate-+l- 99.9%

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

   11. +-inverses 99.9%

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

   12. metadata-eval 99.9%

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

   13. metadata-eval 99.9%

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

   14. unsub-neg 99.9%

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

8. Simplified 99.9%

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

9. Taylor expanded in x around 0 54.9%

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

10. Final simplification 54.9%

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

Alternative 6: 51.9% 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 79.5%

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

3. Taylor expanded in x around 0 54.0%

   \[\leadsto \color{blue}{2} \]
4. Add Preprocessing

Alternative 7: 10.9% accurate, 11.0× 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 79.5%

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

3. Taylor expanded in x around 0 53.5%

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

4. Taylor expanded in x around inf 11.2%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))