Asymptote A

Percentage Accurate: 77.2% → 99.9%

Time: 6.3s

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: 77.2% 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, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{-2}{x\_m - 1}}{\left(1 + x\_m\right) \cdot \left(\frac{1}{x\_m - 1} \cdot \left(x\_m - 1\right)\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ (/ -2.0 (- x_m 1.0)) (* (+ 1.0 x_m) (* (/ 1.0 (- x_m 1.0)) (- x_m 1.0)))))

C

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

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (-2.0 / (x_m - 1.0)) / ((1.0 + x_m) * ((1.0 / (x_m - 1.0)) * (x_m - 1.0)));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (-2.0 / (x_m - 1.0)) / ((1.0 + x_m) * ((1.0 / (x_m - 1.0)) * (x_m - 1.0)))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(-2.0 / N[(x$95$m - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x$95$m), $MachinePrecision] * N[(N[(1.0 / N[(x$95$m - 1.0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
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 80.3

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

   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 80.3

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

4. Applied rewrites 80.3%

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. flip-+ N/A

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

   5. lift--.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. lift-fma.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift--.f64 N/A

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

   14. frac-2neg N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 N/A

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

   17. lift--.f64 N/A

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

   18. sub-neg N/A

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

   19. metadata-eval N/A

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

   20. distribute-neg-in N/A

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

   21. metadata-eval N/A

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

   22. neg-mul-1 N/A

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

   23. *-commutative N/A

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

   24. +-commutative N/A

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

   25. *-commutative N/A

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

   26. neg-mul-1 N/A

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

   27. *-rgt-identity N/A

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

   28. sub-neg N/A

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

   29. *-rgt-identity N/A

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

   30. lower--.f64 98.9

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

9. Applied rewrites 98.9%

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

    1. lift-*.f64 N/A

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

    2. lift-fma.f64 N/A

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

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

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

    4. lift--.f64 N/A

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

    5. associate-*l* N/A

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

    6. lower-*.f64 N/A

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

    7. +-commutative N/A

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

    8. lower-+.f64 N/A

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

    9. lower-*.f64 99.9

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

    10. lift-/.f64 N/A

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

    11. metadata-eval N/A

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

    12. lift--.f64 N/A

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

    13. sub-neg N/A

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

    14. lift-neg.f64 N/A

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

    15. +-commutative N/A

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

    16. lift-neg.f64 N/A

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

    17. metadata-eval N/A

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

    18. distribute-neg-in N/A

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

    19. metadata-eval N/A

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

    20. sub-neg N/A

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

    21. lift--.f64 N/A

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

    22. frac-2neg N/A

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

    23. lower-/.f64 99.9

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

11. Applied rewrites 99.9%

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

12. Final simplification 99.9%

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

Alternative 2: 98.5% accurate, 0.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (- (/ 1.0 (+ 1.0 x_m)) (/ 1.0 (- x_m 1.0))) 0.0)
   (/ -2.0 (* x_m x_m))
   (fma (* x_m x_m) 2.0 2.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(N[(1.0 / N[(1.0 + x$95$m), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(x$95$m - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(-2.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m), $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 55.0%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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 97.1

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

   5. Applied rewrites 97.1%

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

   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%

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

Alternative 3: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{-2}{x\_m - 1}}{x\_m - -1} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (/ -2.0 (- x_m 1.0)) (- x_m -1.0)))

C

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

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (-2.0 / (x_m - 1.0)) / (x_m - -1.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (-2.0 / (x_m - 1.0)) / (x_m - -1.0)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(-2.0 / N[(x$95$m - 1.0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m - -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
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 80.3

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

   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 80.3

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

4. Applied rewrites 80.3%

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

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 4: 99.4% accurate, 1.8× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ -2.0 (fma x_m x_m -1.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(-2.0 / N[(x$95$m * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
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. 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 80.2

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

4. Applied rewrites 80.2%

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

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 98.9%

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

Alternative 5: 50.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{2}{x\_m - -1} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ 2.0 (- x_m -1.0)))

C

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

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 2.0 / (x_m - -1.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 2.0 / (x_m - -1.0)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(2.0 / N[(x$95$m - -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
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 80.3

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

   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 80.3

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

4. Applied rewrites 80.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 50.8%

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

Alternative 6: 49.9% accurate, 32.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 2 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 2.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 2.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 2.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 2.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 2.0

Julia

x_m = abs(x)
function code(x_m)
	return 2.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 2.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 2.0

Derivation

1. Initial program 76.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 49.5%

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

Reproduce

herbie shell --seed 2024240 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))