Asymptote A

Percentage Accurate: 77.5% → 99.9%

Time: 6.9s

Alternatives: 7

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: 77.5% 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: 99.9% 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 78.8%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
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 82.3

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

   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 82.3

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

4. Applied rewrites 82.3%

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

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: 98.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((pow((x - -1.0), -1.0) - pow((x - 1.0), -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(x - -1.0) ^ -1.0) - (Float64(x - 1.0) ^ -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[Power[N[(x - -1.0), $MachinePrecision], -1.0], $MachinePrecision] - N[Power[N[(x - 1.0), $MachinePrecision], -1.0], $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 54.0%

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

   3. Taylor expanded in x around inf

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

      1. 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 96.8

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

   5. Applied rewrites 96.8%

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

   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 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 98.2%

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

Alternative 3: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
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 82.3

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

   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 82.3

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

4. Applied rewrites 82.3%

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

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

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

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

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

   14. lift-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 99.0

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

9. Applied rewrites 99.0%

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

    1. lift-fma.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. metadata-eval N/A

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

    4. sub-neg N/A

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

    5. lower--.f64 99.0

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

11. Applied rewrites 99.0%

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

Alternative 4: 99.5% 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 78.8%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
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 82.3

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

   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 82.3

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

4. Applied rewrites 82.3%

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

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

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

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

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

   14. lift-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 99.0

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

9. Applied rewrites 99.0%

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

Alternative 5: 63.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = -x - (-1.0 - 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 = -x - ((-1.0d0) - 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 = -x - (-1.0 - x);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 87.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 70.5%

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

   if 1 < x

   1. Initial program 49.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 47.4

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

   8. Applied rewrites 47.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 47.4%

       \[\leadsto \left(-x\right) - \left(-1 - x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 54.5

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

5. Applied rewrites 54.5%

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

6. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f64 76.7

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

8. Applied rewrites 76.7%

   \[\leadsto \left(1 - x\right) - \color{blue}{\left(-1 - x\right)} \]
9. Add Preprocessing

Alternative 7: 51.3% 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 78.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 54.6%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))