Asymptote A

Percentage Accurate: 77.3% → 99.9%

Time: 7.0s

Alternatives: 8

Speedup: 1.8×

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: 77.3% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-lft-identity N/A

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

   10. *-rgt-identity N/A

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

   11. lift-+.f64 N/A

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

   12. associate--r+ N/A

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

   13. lower--.f64 N/A

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

   14. lower--.f64 77.7

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

   15. lift-+.f64 N/A

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

   16. metadata-eval N/A

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

   17. sub-neg N/A

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

   18. lower--.f64 77.7

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

4. Applied rewrites 77.7%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.9%

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

Alternative 2: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} - \frac{1}{x - 1} \leq 0:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 2, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (+ 1.0 x)) (/ 1.0 (- x 1.0))) 0.0)
   (/ -2.0 (* x x))
   (fma (* x x) 2.0 2.0)))

C

double code(double x) {
	double tmp;
	if (((1.0 / (1.0 + x)) - (1.0 / (x - 1.0))) <= 0.0) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = fma((x * x), 2.0, 2.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 2.0 + 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64)))) < 0.0

   1. Initial program 50.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} - \frac{1}{x - 1} \leq 0:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 2, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-lft-identity N/A

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

   10. *-rgt-identity N/A

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

   11. lift-+.f64 N/A

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

   12. associate--r+ N/A

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

   13. lower--.f64 N/A

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

   14. lower--.f64 77.7

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

   15. lift-+.f64 N/A

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

   16. metadata-eval N/A

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

   17. sub-neg N/A

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

   18. lower--.f64 77.7

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

4. Applied rewrites 77.7%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.9

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

9. Applied rewrites 99.9%

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

Alternative 4: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. lift-+.f64 N/A

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

   6. lift--.f64 N/A

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

   7. difference-of-sqr-1 N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. *-lft-identity N/A

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

   11. *-rgt-identity N/A

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

   12. lift-+.f64 N/A

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

   13. associate--r+ N/A

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

   14. lower--.f64 N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval N/A

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

   17. sub-neg N/A

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

   18. metadata-eval N/A

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

   19. lower-fma.f64 77.7

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

4. Applied rewrites 77.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{-2}}{\mathsf{fma}\left(x, x, -1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.2%

   \[\leadsto \frac{\color{blue}{-2}}{\mathsf{fma}\left(x, x, -1\right)} \]
8. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. difference-of-sqr--1 N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-in N/A

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

   9. neg-mul-1 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. metadata-eval N/A

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

   15. distribute-neg-in N/A

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

   16. neg-mul-1 N/A

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

   17. metadata-eval N/A

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

   18. lower-fma.f64 99.2

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

9. Applied rewrites 99.2%

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

10. Final simplification 99.2%

    \[\leadsto \frac{-2}{\mathsf{fma}\left(-1, x, -1\right) + \left(x - -1\right) \cdot x} \]
11. Add Preprocessing

Alternative 5: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. lift-+.f64 N/A

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

   6. lift--.f64 N/A

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

   7. difference-of-sqr-1 N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. *-lft-identity N/A

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

   11. *-rgt-identity N/A

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

   12. lift-+.f64 N/A

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

   13. associate--r+ N/A

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

   14. lower--.f64 N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval N/A

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

   17. sub-neg N/A

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

   18. metadata-eval N/A

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

   19. lower-fma.f64 77.7

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

4. Applied rewrites 77.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{-2}}{\mathsf{fma}\left(x, x, -1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.2%

   \[\leadsto \frac{\color{blue}{-2}}{\mathsf{fma}\left(x, x, -1\right)} \]
8. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. difference-of-sqr--1 N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. lift--.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 99.2

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

9. Applied rewrites 99.2%

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

10. Final simplification 99.2%

    \[\leadsto \frac{-2}{\left(x - -1\right) \cdot \left(x - 1\right)} \]
11. Add Preprocessing

Alternative 6: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{\mathsf{fma}\left(x, x, -1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. lift-+.f64 N/A

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

   6. lift--.f64 N/A

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

   7. difference-of-sqr-1 N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. *-lft-identity N/A

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

   11. *-rgt-identity N/A

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

   12. lift-+.f64 N/A

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

   13. associate--r+ N/A

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

   14. lower--.f64 N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval N/A

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

   17. sub-neg N/A

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

   18. metadata-eval N/A

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

   19. lower-fma.f64 77.7

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

4. Applied rewrites 77.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{-2}}{\mathsf{fma}\left(x, x, -1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.2%

   \[\leadsto \frac{\color{blue}{-2}}{\mathsf{fma}\left(x, x, -1\right)} \]
8. Add Preprocessing

Alternative 7: 51.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-lft-identity N/A

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

   10. *-rgt-identity N/A

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

   11. lift-+.f64 N/A

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

   12. associate--r+ N/A

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

   13. lower--.f64 N/A

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

   14. lower--.f64 77.7

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

   15. lift-+.f64 N/A

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

   16. metadata-eval N/A

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

   17. sub-neg N/A

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

   18. lower--.f64 77.7

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

4. Applied rewrites 77.7%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.9

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

9. Applied rewrites 99.9%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 48.0%

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

Alternative 8: 50.4% accurate, 32.0× 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 73.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 46.7%

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

Reproduce

herbie shell --seed 2024267 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))