Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. +-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 100.0%

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

   2. frac-2neg 100.0%

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

   3. metadata-eval 100.0%

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

   4. frac-add 100.0%

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

   5. *-un-lft-identity 100.0%

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

   6. +-commutative 100.0%

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

   7. distribute-neg-in 100.0%

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

   8. metadata-eval 100.0%

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

   9. sub-neg 100.0%

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

   10. +-commutative 100.0%

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

   11. +-commutative 100.0%

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

   12. +-commutative 100.0%

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

   13. distribute-neg-in 100.0%

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

   14. metadata-eval 100.0%

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

   15. sub-neg 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-+l- 100.0%

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

   2. sub-neg 100.0%

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

   3. *-commutative 100.0%

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

   4. mul-1-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. metadata-eval 100.0%

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

   7. sub-neg 100.0%

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

   8. div-sub 100.0%

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

   9. *-rgt-identity 100.0%

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

   10. associate-*r/ 100.0%

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

   11. rgt-mult-inverse 100.0%

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

   12. *-commutative 100.0%

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

   13. metadata-eval 100.0%

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

   14. sub-neg 100.0%

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

   15. div-sub 100.0%

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

   16. *-rgt-identity 100.0%

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

   17. associate-*r/ 99.9%

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

   18. rgt-mult-inverse 100.0%

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

8. Simplified 100.0%

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

   1. div-sub 100.0%

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

   2. sub-neg 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-/r* 100.0%

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

   5. div-inv 100.0%

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

   6. cancel-sign-sub-inv 100.0%

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

   7. metadata-eval 100.0%

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

   8. *-un-lft-identity 100.0%

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

   9. +-commutative 100.0%

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

   10. sub-neg 100.0%

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

   11. associate-+l+ 100.0%

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

   12. distribute-neg-frac 100.0%

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

   13. metadata-eval 100.0%

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

   14. div-inv 100.0%

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

10. Applied egg-rr 100.0%

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

    1. sub-neg 100.0%

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

    2. associate-/l/ 100.0%

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

    3. +-commutative 100.0%

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

    4. metadata-eval 100.0%

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

    5. distribute-neg-frac 100.0%

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

    6. sub-neg 100.0%

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

    7. associate--l+ 100.0%

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

    8. div-sub 100.0%

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

    9. associate-+l- 100.0%

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

    10. associate--r+ 100.0%

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

    11. metadata-eval 100.0%

        \[\leadsto \frac{\left(\color{blue}{0} - x\right) + \frac{-1}{x}}{\left(1 - x\right) \cdot \left(1 + \frac{1}{x}\right)} \]

    12. neg-sub0 100.0%

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

    13. +-commutative 100.0%

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

    14. sub-neg 100.0%

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

    15. sub-neg 100.0%

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

    16. +-commutative 100.0%

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

    17. distribute-rgt1-in 100.0%

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

    18. distribute-lft-neg-out 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around 0 76.1%

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

    1. mul-1-neg 76.1%

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

    2. sub-neg 76.1%

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

    3. div-sub 76.1%

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

    4. *-lft-identity 76.1%

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

    5. fma-neg 76.1%

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

    6. distribute-frac-neg 76.1%

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

    7. unpow2 76.1%

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

    8. distribute-lft-neg-out 76.1%

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

    9. associate-/l* 100.0%

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

    10. *-inverses 100.0%

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

    11. *-rgt-identity 100.0%

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

    12. fma-neg 100.0%

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

    13. *-lft-identity 100.0%

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

15. Simplified 100.0%

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

16. Final simplification 100.0%

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

Alternative 2: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.76 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x - \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(-1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.76) (not (<= x 1.0)))
   (/ (- x (/ -1.0 x)) x)
   (+ (/ x (+ x 1.0)) (- -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.76) || !(x <= 1.0)) {
		tmp = (x - (-1.0 / x)) / x;
	} else {
		tmp = (x / (x + 1.0)) + (-1.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.76d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (x - ((-1.0d0) / x)) / x
    else
        tmp = (x / (x + 1.0d0)) + ((-1.0d0) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.76) || !(x <= 1.0)) {
		tmp = (x - (-1.0 / x)) / x;
	} else {
		tmp = (x / (x + 1.0)) + (-1.0 - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.76) or not (x <= 1.0):
		tmp = (x - (-1.0 / x)) / x
	else:
		tmp = (x / (x + 1.0)) + (-1.0 - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.76) || !(x <= 1.0))
		tmp = Float64(Float64(x - Float64(-1.0 / x)) / x);
	else
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(-1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.76) || ~((x <= 1.0)))
		tmp = (x - (-1.0 / x)) / x;
	else
		tmp = (x / (x + 1.0)) + (-1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.76], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(x - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.76000000000000001 or 1 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 100.0%

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

      2. frac-2neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. frac-add 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. metadata-eval 100.0%

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

      15. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. mul-1-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. div-sub 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. associate-*r/ 100.0%

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

      11. rgt-mult-inverse 100.0%

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

      12. *-commutative 100.0%

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

      13. metadata-eval 100.0%

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

      14. sub-neg 100.0%

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

      15. div-sub 100.0%

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

      16. *-rgt-identity 100.0%

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

      17. associate-*r/ 99.8%

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

      18. rgt-mult-inverse 100.0%

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

   8. Simplified 100.0%

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

      1. div-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/r* 100.0%

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

      5. div-inv 100.0%

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

      6. cancel-sign-sub-inv 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-un-lft-identity 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. associate-+l+ 100.0%

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

      12. distribute-neg-frac 100.0%

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

      13. metadata-eval 100.0%

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

      14. div-inv 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. sub-neg 100.0%

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

       2. associate-/l/ 100.0%

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

       3. +-commutative 100.0%

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

       4. metadata-eval 100.0%

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

       5. distribute-neg-frac 100.0%

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

       6. sub-neg 100.0%

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

       7. associate--l+ 100.0%

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

       8. div-sub 100.0%

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

       9. associate-+l- 100.0%

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

       10. associate--r+ 100.0%

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

       11. metadata-eval 100.0%

           \[\leadsto \frac{\left(\color{blue}{0} - x\right) + \frac{-1}{x}}{\left(1 - x\right) \cdot \left(1 + \frac{1}{x}\right)} \]

       12. neg-sub0 100.0%

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

       13. +-commutative 100.0%

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

       14. sub-neg 100.0%

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

       15. sub-neg 100.0%

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

       16. +-commutative 100.0%

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

       17. distribute-rgt1-in 100.0%

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

       18. distribute-lft-neg-out 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around inf 98.8%

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

       1. neg-mul-1 98.8%

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

   15. Simplified 98.8%

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

   if -1.76000000000000001 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. sub-neg 99.4%

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

      2. neg-mul-1 99.4%

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

      3. metadata-eval 99.4%

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

      4. +-commutative 99.4%

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

      5. unsub-neg 99.4%

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

   7. Simplified 99.4%

      \[\leadsto \frac{x}{1 + x} + \color{blue}{\left(-1 - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.76 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x - \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(-1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.8\right):\\ \;\;\;\;\frac{x - \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.8)))
   (/ (- x (/ -1.0 x)) x)
   (+ x (/ 1.0 (+ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.8)) {
		tmp = (x - (-1.0 / x)) / x;
	} else {
		tmp = x + (1.0 / (-1.0 + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.8d0))) then
        tmp = (x - ((-1.0d0) / x)) / x
    else
        tmp = x + (1.0d0 / ((-1.0d0) + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.8)) {
		tmp = (x - (-1.0 / x)) / x;
	} else {
		tmp = x + (1.0 / (-1.0 + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.8):
		tmp = (x - (-1.0 / x)) / x
	else:
		tmp = x + (1.0 / (-1.0 + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.8))
		tmp = Float64(Float64(x - Float64(-1.0 / x)) / x);
	else
		tmp = Float64(x + Float64(1.0 / Float64(-1.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.8)))
		tmp = (x - (-1.0 / x)) / x;
	else
		tmp = x + (1.0 / (-1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.8]], $MachinePrecision]], N[(N[(x - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.80000000000000004 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 100.0%

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

      2. frac-2neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. frac-add 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. metadata-eval 100.0%

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

      15. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. mul-1-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. div-sub 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. associate-*r/ 100.0%

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

      11. rgt-mult-inverse 100.0%

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

      12. *-commutative 100.0%

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

      13. metadata-eval 100.0%

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

      14. sub-neg 100.0%

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

      15. div-sub 100.0%

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

      16. *-rgt-identity 100.0%

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

      17. associate-*r/ 99.8%

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

      18. rgt-mult-inverse 100.0%

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

   8. Simplified 100.0%

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

      1. div-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/r* 100.0%

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

      5. div-inv 100.0%

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

      6. cancel-sign-sub-inv 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-un-lft-identity 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. associate-+l+ 100.0%

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

      12. distribute-neg-frac 100.0%

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

      13. metadata-eval 100.0%

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

      14. div-inv 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. sub-neg 100.0%

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

       2. associate-/l/ 100.0%

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

       3. +-commutative 100.0%

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

       4. metadata-eval 100.0%

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

       5. distribute-neg-frac 100.0%

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

       6. sub-neg 100.0%

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

       7. associate--l+ 100.0%

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

       8. div-sub 100.0%

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

       9. associate-+l- 100.0%

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

       10. associate--r+ 100.0%

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

       11. metadata-eval 100.0%

           \[\leadsto \frac{\left(\color{blue}{0} - x\right) + \frac{-1}{x}}{\left(1 - x\right) \cdot \left(1 + \frac{1}{x}\right)} \]

       12. neg-sub0 100.0%

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

       13. +-commutative 100.0%

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

       14. sub-neg 100.0%

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

       15. sub-neg 100.0%

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

       16. +-commutative 100.0%

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

       17. distribute-rgt1-in 100.0%

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

       18. distribute-lft-neg-out 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around inf 98.8%

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

       1. neg-mul-1 98.8%

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

   15. Simplified 98.8%

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

   if -1 < x < 1.80000000000000004

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.4%

      \[\leadsto \color{blue}{x} + \frac{1}{x + -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.8\right):\\ \;\;\;\;\frac{x - \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.9:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 1.9) (+ x (/ 1.0 (+ -1.0 x))) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.9) {
		tmp = x + (1.0 / (-1.0 + x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.9d0) then
        tmp = x + (1.0d0 / ((-1.0d0) + x))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.9) {
		tmp = x + (1.0 / (-1.0 + x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.9:
		tmp = x + (1.0 / (-1.0 + x))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.9)
		tmp = Float64(x + Float64(1.0 / Float64(-1.0 + x)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.9)
		tmp = x + (1.0 / (-1.0 + x));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.9], N[(x + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.8999999999999999 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{1} \]

   if -1 < x < 1.8999999999999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.4%

      \[\leadsto \color{blue}{x} + \frac{1}{x + -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.9:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.0d0) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.0], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{1} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.4%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 7: 50.6% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. +-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 51.6%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))