Asymptote A

Percentage Accurate: 77.9% → 99.9%

Time: 5.9s

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: 77.9% 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

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

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 78.0%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
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. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lift-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. frac-sub N/A

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

   9. associate-/r* N/A

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

   10. neg-mul-1 N/A

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

   11. remove-double-neg N/A

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

   12. *-lft-identity N/A

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

   16. frac-2neg N/A

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

   17. div-inv N/A

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

   18. metadata-eval N/A

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

4. Applied rewrites 81.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.9%

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

Alternative 2: 98.6% 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 54.3%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.4%

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

Alternative 3: 52.8% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (+ 1.0 x)) (/ 1.0 (- x 1.0))) 0.0)
   (/ 2.0 (- x))
   (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;
	} 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));
	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 / (-x)), $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 54.3%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-sub N/A

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

      9. associate-/r* N/A

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

      10. neg-mul-1 N/A

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

      11. remove-double-neg N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

      15. metadata-eval N/A

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

      16. frac-2neg N/A

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

      17. div-inv N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 5.7%

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

   8. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 5.7

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

   10. Applied rewrites 5.7%

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

   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 54.7%

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

Alternative 4: 99.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.0%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
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 81.8

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

4. Applied rewrites 81.8%

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

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 99.5%

   \[\leadsto \frac{\color{blue}{-2}}{\mathsf{fma}\left(x, x, -1\right)} \]
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. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-in N/A

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

   6. distribute-rgt-out N/A

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

   7. neg-mul-1 N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. metadata-eval N/A

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

   11. sub-neg N/A

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

   12. lift--.f64 N/A

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

   13. lower-+.f64 N/A

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

   14. *-lft-identity N/A

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

   15. lower-fma.f64 99.5

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

9. Applied rewrites 99.5%

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

Alternative 5: 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 78.0%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
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 81.8

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

4. Applied rewrites 81.8%

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

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 99.5%

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

Alternative 6: 52.0% accurate, 2.1× 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 78.0%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
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. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lift-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. frac-sub N/A

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

   9. associate-/r* N/A

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

   10. neg-mul-1 N/A

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

   11. remove-double-neg N/A

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

   12. *-lft-identity N/A

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

   16. frac-2neg N/A

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

   17. div-inv N/A

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

   18. metadata-eval N/A

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

4. Applied rewrites 81.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 54.3%

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

Alternative 7: 50.9% 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 78.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 53.2%

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

Reproduce

herbie shell --seed 2024254 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))