Asymptote A

Percentage Accurate: 77.7% → 99.9%

Time: 8.6s

Alternatives: 6

Speedup: 1.6×

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: 77.7% 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 75.7%

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

   1. frac-sub N/A

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

   2. *-lft-identity N/A

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

   3. *-rgt-identity N/A

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

   4. associate--r+ N/A

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

   5. sub-neg N/A

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

   6. div-sub N/A

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

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(x - 1\right) + \left(\mathsf{neg}\left(x\right)\right)}{\left(x + 1\right) \cdot \left(x - 1\right)}\right), \color{blue}{\left(\frac{1}{\left(x + 1\right) \cdot \left(x - 1\right)}\right)}\right) \]

4. Applied egg-rr 79.9%

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

   1. associate-/l/ N/A

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

   2. sub-div N/A

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

   3. remove-double-neg N/A

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

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

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

   5. distribute-lft-neg-in N/A

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

   6. associate-/r* N/A

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

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(x + \left(-1 - x\right)\right) - 1}{\mathsf{neg}\left(\left(x + -1\right)\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\left(-1 - x\right) + x\right) - 1}{\mathsf{neg}\left(\left(x + -1\right)\right)}\right), \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)\right) \]

   9. associate-+l- N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 - \left(x - x\right)\right) - 1}{\mathsf{neg}\left(\left(x + -1\right)\right)}\right), \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)\right) \]

   10. +-inverses N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 - 0\right) - 1}{\mathsf{neg}\left(\left(x + -1\right)\right)}\right), \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 - 1}{\mathsf{neg}\left(\left(x + -1\right)\right)}\right), \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{\mathsf{neg}\left(\left(x + -1\right)\right)}\right), \left(\mathsf{neg}\left(\left(\color{blue}{1} + x\right)\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(\left(x + -1\right)\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(1 + x\right)}\right)\right)\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \left(\mathsf{neg}\left(\left(-1 + x\right)\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)\right) \]

   15. distribute-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + \color{blue}{x}\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)\right) \]

   17. unsub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \left(1 - x\right)\right), \left(\mathsf{neg}\left(\left(1 + \color{blue}{x}\right)\right)\right)\right) \]

   18. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{\_.f64}\left(1, x\right)\right), \left(\mathsf{neg}\left(\left(1 + \color{blue}{x}\right)\right)\right)\right) \]

   19. distribute-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{\_.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{\_.f64}\left(1, x\right)\right), \left(-1 + \left(\mathsf{neg}\left(\color{blue}{x}\right)\right)\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{\frac{-2}{1 - x}}{-1 - x}} \]
7. Add Preprocessing

Alternative 2: 74.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(2.0 + Float64(Float64(x * x) * 2.0));
	else
		tmp = 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 + ((x * x) * 2.0);
	else
		tmp = (-2.0 / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 82.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 + 2 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(2, \color{blue}{\left(2 \cdot {x}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      4. *-lowering-*.f64 68.1%

         \[\leadsto \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 68.1%

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

   if 1 < x

   1. Initial program 52.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 98.5%

         \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 98.5%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{x}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, x\right), x\right) \]

   7. Applied egg-rr 99.8%

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

4. Final simplification 75.4%

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

Alternative 3: 74.6% accurate, 1.1× 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(-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 82.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2} \]
   4. Step-by-step derivation

   5. Simplified 68.2%

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

   if 1 < x

   1. Initial program 52.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 98.5%

         \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 98.5%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{x}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, x\right), x\right) \]

   7. Applied egg-rr 99.8%

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

Alternative 4: 74.3% accurate, 1.1× 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 82.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2} \]
   4. Step-by-step derivation

   5. Simplified 68.2%

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

   if 1 < x

   1. Initial program 52.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 98.5%

         \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 98.5%

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

Alternative 5: 99.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. frac-sub N/A

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

   2. *-lft-identity N/A

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

   3. *-rgt-identity N/A

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

   4. associate--r+ N/A

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

   5. sub-neg N/A

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

   6. div-sub N/A

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

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(x - 1\right) + \left(\mathsf{neg}\left(x\right)\right)}{\left(x + 1\right) \cdot \left(x - 1\right)}\right), \color{blue}{\left(\frac{1}{\left(x + 1\right) \cdot \left(x - 1\right)}\right)}\right) \]

4. Applied egg-rr 79.9%

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

   1. div-inv N/A

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

   2. associate-/l/ N/A

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

   3. fmsub-define N/A

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

   4. +-commutative N/A

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

   5. associate-+l- N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. fmsub-define N/A

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

   9. neg-mul-1 N/A

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

   10. associate-/l/ N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. associate-/l/ N/A

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

   14. sub-div N/A

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

   15. metadata-eval N/A

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

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left(\left(x + -1\right) \cdot \left(1 + x\right)\right)}\right) \]

6. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{-2}{-1 + x \cdot x}} \]
7. Add Preprocessing

Alternative 6: 50.5% 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 75.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{2} \]
4. Step-by-step derivation

5. Simplified 53.1%

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

Reproduce

herbie shell --seed 2024164 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))