Asymptote A

Percentage Accurate: 77.9% → 99.9%

Time: 5.6s

Alternatives: 6

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

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

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 72.8%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
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 77.6

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

4. Applied rewrites 77.6%

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

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 99.8%

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

Alternative 2: 98.4% accurate, 0.1× speedup

Math

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

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) x 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), x, 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), x, 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] * x + 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 51.0%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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 97.2

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

   5. Applied rewrites 97.2%

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

   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 99.3

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.3%

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

4. Final simplification 98.2%

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

Alternative 3: 99.3% 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 72.8%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
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 77.6

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

4. Applied rewrites 77.6%

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

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 99.8%

   \[\leadsto \frac{\frac{\color{blue}{-2}}{x - 1}}{x + 1} \]
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. *-commutative 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. lift--.f64 N/A

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

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

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

   8. lift-fma.f64 N/A

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

   9. lower-/.f64 99.1

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

9. Applied rewrites 99.1%

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

Alternative 4: 75.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
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. fp-cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f64 45.6

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

5. Applied rewrites 45.6%

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

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. lower--.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 70.7

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

8. Applied rewrites 70.7%

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

Alternative 5: 51.0% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fma (* x -2.0) x 2.0))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

3. Taylor expanded in x around 0

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

   1. +-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 45.1

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

5. Applied rewrites 45.1%

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

7. Applied rewrites 45.5%

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

9. Applied rewrites 45.7%

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

Alternative 6: 51.1% 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 72.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 45.7%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))