Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 9

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, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / N[(1.0 + N[(1.0 / x), $MachinePrecision]), $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. 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. *-commutative 100.0%

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

   2. *-commutative 100.0%

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

   3. mul-1-neg 100.0%

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

   4. unsub-neg 100.0%

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

   5. associate--r+ 100.0%

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

8. Simplified 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. div-inv 99.8%

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

   3. associate--r+ 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-/r* 99.8%

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

   6. +-commutative 99.8%

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

   7. clear-num 99.7%

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

   8. +-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

    1. *-lft-identity 99.7%

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

    2. associate-*r/ 99.8%

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

    3. associate-*r/ 72.2%

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

    4. associate-*l/ 99.8%

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

    5. associate-/r/ 100.0%

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

    6. div-sub 100.0%

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

    7. sub-neg 100.0%

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

    8. *-lft-identity 100.0%

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

    9. associate-*l/ 99.8%

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

    10. distribute-lft-in 99.8%

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

    11. lft-mult-inverse 100.0%

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

    12. *-rgt-identity 100.0%

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

    13. *-inverses 100.0%

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

    14. metadata-eval 100.0%

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

12. Simplified 100.0%

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

    1. *-un-lft-identity 100.0%

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

14. Applied egg-rr 100.0%

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

    1. associate-*r/ 100.0%

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

    2. associate-*l/ 99.8%

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

    3. distribute-lft-in 99.8%

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

    4. associate-*r/ 99.8%

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

    5. lft-mult-inverse 100.0%

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

    6. associate-*l/ 100.0%

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

    7. metadata-eval 100.0%

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

16. Simplified 100.0%

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

17. Final simplification 100.0%

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

Alternative 2: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.54:\\ \;\;\;\;\frac{x}{1 + x} + \left(-1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.9)
   1.0
   (if (<= x 0.54) (+ (/ x (+ 1.0 x)) (- -1.0 x)) (+ 1.0 (/ -1.0 (- 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.9) {
		tmp = 1.0;
	} else if (x <= 0.54) {
		tmp = (x / (1.0 + x)) + (-1.0 - x);
	} else {
		tmp = 1.0 + (-1.0 / (1.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.9d0)) then
        tmp = 1.0d0
    else if (x <= 0.54d0) then
        tmp = (x / (1.0d0 + x)) + ((-1.0d0) - x)
    else
        tmp = 1.0d0 + ((-1.0d0) / (1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.9) {
		tmp = 1.0;
	} else if (x <= 0.54) {
		tmp = (x / (1.0 + x)) + (-1.0 - x);
	} else {
		tmp = 1.0 + (-1.0 / (1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.9:
		tmp = 1.0
	elif x <= 0.54:
		tmp = (x / (1.0 + x)) + (-1.0 - x)
	else:
		tmp = 1.0 + (-1.0 / (1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.9)
		tmp = 1.0;
	elseif (x <= 0.54)
		tmp = Float64(Float64(x / Float64(1.0 + x)) + Float64(-1.0 - x));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.9)
		tmp = 1.0;
	elseif (x <= 0.54)
		tmp = (x / (1.0 + x)) + (-1.0 - x);
	else
		tmp = 1.0 + (-1.0 / (1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.9], 1.0, If[LessEqual[x, 0.54], N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if 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 inf 98.9%

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

   if -1.8999999999999999 < x < 0.54000000000000004

   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 98.2%

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

      1. sub-neg 98.2%

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

      2. metadata-eval 98.2%

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

      3. neg-mul-1 98.2%

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

      4. +-commutative 98.2%

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

      5. unsub-neg 98.2%

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

   7. Simplified 98.2%

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

   if 0.54000000000000004 < 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. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. associate--r+ 100.0%

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

   8. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 99.9%

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

      3. associate--r+ 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/r* 99.9%

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

      6. +-commutative 99.9%

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

      7. clear-num 99.9%

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

      8. +-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. *-lft-identity 99.9%

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

       2. associate-*r/ 100.0%

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

       3. associate-*r/ 52.2%

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

       4. associate-*l/ 100.0%

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

       5. associate-/r/ 100.0%

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

       6. div-sub 100.0%

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

       7. sub-neg 100.0%

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

       8. *-lft-identity 100.0%

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

       9. associate-*l/ 99.8%

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

       10. distribute-lft-in 99.8%

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

       11. lft-mult-inverse 100.0%

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

       12. *-rgt-identity 100.0%

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

       13. *-inverses 100.0%

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

       14. metadata-eval 100.0%

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

   12. Simplified 100.0%

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

       1. *-un-lft-identity 100.0%

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

   14. Applied egg-rr 100.0%

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

       1. associate-*r/ 100.0%

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

       2. associate-*l/ 99.8%

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

       3. distribute-lft-in 99.8%

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

       4. associate-*r/ 99.9%

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

       5. lft-mult-inverse 100.0%

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

       6. associate-*l/ 100.0%

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

       7. metadata-eval 100.0%

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

   16. Simplified 100.0%

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

   17. Taylor expanded in x around inf 98.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.54:\\ \;\;\;\;\frac{x}{1 + x} + \left(-1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 + x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} + t\_0\\ \mathbf{elif}\;x \leq 0.54:\\ \;\;\;\;t\_0 + \left(-1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ 1.0 x))))
   (if (<= x -1.0)
     (+ (/ 1.0 x) t_0)
     (if (<= x 0.54) (+ t_0 (- -1.0 x)) (+ 1.0 (/ -1.0 (- 1.0 x)))))))

C

double code(double x) {
	double t_0 = x / (1.0 + x);
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) + t_0;
	} else if (x <= 0.54) {
		tmp = t_0 + (-1.0 - x);
	} else {
		tmp = 1.0 + (-1.0 / (1.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (1.0d0 + x)
    if (x <= (-1.0d0)) then
        tmp = (1.0d0 / x) + t_0
    else if (x <= 0.54d0) then
        tmp = t_0 + ((-1.0d0) - x)
    else
        tmp = 1.0d0 + ((-1.0d0) / (1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x / (1.0 + x);
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) + t_0;
	} else if (x <= 0.54) {
		tmp = t_0 + (-1.0 - x);
	} else {
		tmp = 1.0 + (-1.0 / (1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (1.0 + x)
	tmp = 0
	if x <= -1.0:
		tmp = (1.0 / x) + t_0
	elif x <= 0.54:
		tmp = t_0 + (-1.0 - x)
	else:
		tmp = 1.0 + (-1.0 / (1.0 - x))
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(1.0 + x))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(1.0 / x) + t_0);
	elseif (x <= 0.54)
		tmp = Float64(t_0 + Float64(-1.0 - x));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (1.0 + x);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (1.0 / x) + t_0;
	elseif (x <= 0.54)
		tmp = t_0 + (-1.0 - x);
	else
		tmp = 1.0 + (-1.0 / (1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(N[(1.0 / x), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[x, 0.54], N[(t$95$0 + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if 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 inf 98.9%

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

   if -1 < x < 0.54000000000000004

   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 98.2%

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

      1. sub-neg 98.2%

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

      2. metadata-eval 98.2%

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

      3. neg-mul-1 98.2%

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

      4. +-commutative 98.2%

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

      5. unsub-neg 98.2%

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

   7. Simplified 98.2%

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

   if 0.54000000000000004 < 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. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. associate--r+ 100.0%

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

   8. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 99.9%

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

      3. associate--r+ 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/r* 99.9%

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

      6. +-commutative 99.9%

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

      7. clear-num 99.9%

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

      8. +-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. *-lft-identity 99.9%

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

       2. associate-*r/ 100.0%

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

       3. associate-*r/ 52.2%

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

       4. associate-*l/ 100.0%

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

       5. associate-/r/ 100.0%

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

       6. div-sub 100.0%

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

       7. sub-neg 100.0%

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

       8. *-lft-identity 100.0%

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

       9. associate-*l/ 99.8%

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

       10. distribute-lft-in 99.8%

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

       11. lft-mult-inverse 100.0%

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

       12. *-rgt-identity 100.0%

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

       13. *-inverses 100.0%

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

       14. metadata-eval 100.0%

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

   12. Simplified 100.0%

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

       1. *-un-lft-identity 100.0%

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

   14. Applied egg-rr 100.0%

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

       1. associate-*r/ 100.0%

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

       2. associate-*l/ 99.8%

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

       3. distribute-lft-in 99.8%

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

       4. associate-*r/ 99.9%

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

       5. lft-mult-inverse 100.0%

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

       6. associate-*l/ 100.0%

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

       7. metadata-eval 100.0%

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

   16. Simplified 100.0%

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

   17. Taylor expanded in x around inf 98.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} + \frac{x}{1 + x}\\ \mathbf{elif}\;x \leq 0.54:\\ \;\;\;\;\frac{x}{1 + x} + \left(-1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 + x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} + t\_0\\ \mathbf{elif}\;x \leq 0.54:\\ \;\;\;\;-1 + \left(t\_0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ 1.0 x))))
   (if (<= x -1.0)
     (+ (/ 1.0 x) t_0)
     (if (<= x 0.54) (+ -1.0 (- t_0 x)) (+ 1.0 (/ -1.0 (- 1.0 x)))))))

C

double code(double x) {
	double t_0 = x / (1.0 + x);
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) + t_0;
	} else if (x <= 0.54) {
		tmp = -1.0 + (t_0 - x);
	} else {
		tmp = 1.0 + (-1.0 / (1.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (1.0d0 + x)
    if (x <= (-1.0d0)) then
        tmp = (1.0d0 / x) + t_0
    else if (x <= 0.54d0) then
        tmp = (-1.0d0) + (t_0 - x)
    else
        tmp = 1.0d0 + ((-1.0d0) / (1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x / (1.0 + x);
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) + t_0;
	} else if (x <= 0.54) {
		tmp = -1.0 + (t_0 - x);
	} else {
		tmp = 1.0 + (-1.0 / (1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (1.0 + x)
	tmp = 0
	if x <= -1.0:
		tmp = (1.0 / x) + t_0
	elif x <= 0.54:
		tmp = -1.0 + (t_0 - x)
	else:
		tmp = 1.0 + (-1.0 / (1.0 - x))
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(1.0 + x))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(1.0 / x) + t_0);
	elseif (x <= 0.54)
		tmp = Float64(-1.0 + Float64(t_0 - x));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (1.0 + x);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (1.0 / x) + t_0;
	elseif (x <= 0.54)
		tmp = -1.0 + (t_0 - x);
	else
		tmp = 1.0 + (-1.0 / (1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(N[(1.0 / x), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[x, 0.54], N[(-1.0 + N[(t$95$0 - x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if 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 inf 98.9%

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

   if -1 < x < 0.54000000000000004

   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 98.2%

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

      1. sub-neg 98.2%

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

      2. metadata-eval 98.2%

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

      3. neg-mul-1 98.2%

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

      4. +-commutative 98.2%

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

      5. unsub-neg 98.2%

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

   7. Simplified 98.2%

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

      1. +-commutative 98.2%

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

      2. associate-+l- 98.3%

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

   9. Applied egg-rr 98.3%

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

   if 0.54000000000000004 < 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. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. associate--r+ 100.0%

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

   8. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 99.9%

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

      3. associate--r+ 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/r* 99.9%

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

      6. +-commutative 99.9%

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

      7. clear-num 99.9%

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

      8. +-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. *-lft-identity 99.9%

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

       2. associate-*r/ 100.0%

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

       3. associate-*r/ 52.2%

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

       4. associate-*l/ 100.0%

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

       5. associate-/r/ 100.0%

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

       6. div-sub 100.0%

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

       7. sub-neg 100.0%

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

       8. *-lft-identity 100.0%

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

       9. associate-*l/ 99.8%

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

       10. distribute-lft-in 99.8%

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

       11. lft-mult-inverse 100.0%

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

       12. *-rgt-identity 100.0%

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

       13. *-inverses 100.0%

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

       14. metadata-eval 100.0%

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

   12. Simplified 100.0%

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

       1. *-un-lft-identity 100.0%

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

   14. Applied egg-rr 100.0%

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

       1. associate-*r/ 100.0%

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

       2. associate-*l/ 99.8%

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

       3. distribute-lft-in 99.8%

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

       4. associate-*r/ 99.9%

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

       5. lft-mult-inverse 100.0%

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

       6. associate-*l/ 100.0%

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

       7. metadata-eval 100.0%

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

   16. Simplified 100.0%

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

   17. Taylor expanded in x around inf 98.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} + \frac{x}{1 + x}\\ \mathbf{elif}\;x \leq 0.54:\\ \;\;\;\;-1 + \left(\frac{x}{1 + x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 0.5) -1.0 (+ 1.0 (/ -1.0 (- 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 0.5) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-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)) then
        tmp = 1.0d0
    else if (x <= 0.5d0) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 + ((-1.0d0) / (1.0d0 - x))
    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 <= 0.5) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-1.0 / (1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 0.5:
		tmp = -1.0
	else:
		tmp = 1.0 + (-1.0 / (1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 0.5)
		tmp = -1.0;
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 0.5)
		tmp = -1.0;
	else
		tmp = 1.0 + (-1.0 / (1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 0.5], -1.0, N[(1.0 + N[(-1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if 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 inf 98.9%

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

   if -1 < x < 0.5

   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 98.2%

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

   if 0.5 < 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. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. associate--r+ 100.0%

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

   8. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 99.9%

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

      3. associate--r+ 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/r* 99.9%

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

      6. +-commutative 99.9%

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

      7. clear-num 99.9%

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

      8. +-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. *-lft-identity 99.9%

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

       2. associate-*r/ 100.0%

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

       3. associate-*r/ 52.2%

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

       4. associate-*l/ 100.0%

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

       5. associate-/r/ 100.0%

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

       6. div-sub 100.0%

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

       7. sub-neg 100.0%

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

       8. *-lft-identity 100.0%

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

       9. associate-*l/ 99.8%

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

       10. distribute-lft-in 99.8%

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

       11. lft-mult-inverse 100.0%

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

       12. *-rgt-identity 100.0%

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

       13. *-inverses 100.0%

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

       14. metadata-eval 100.0%

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

   12. Simplified 100.0%

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

       1. *-un-lft-identity 100.0%

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

   14. Applied egg-rr 100.0%

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

       1. associate-*r/ 100.0%

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

       2. associate-*l/ 99.8%

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

       3. distribute-lft-in 99.8%

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

       4. associate-*r/ 99.9%

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

       5. lft-mult-inverse 100.0%

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

       6. associate-*l/ 100.0%

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

       7. metadata-eval 100.0%

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

   16. Simplified 100.0%

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

   17. Taylor expanded in x around inf 98.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = Float64(x + Float64(1.0 / Float64(x + -1.0)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(1.0 - x)));
	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 = x + (1.0 / (x + -1.0));
	else
		tmp = 1.0 + (-1.0 / (1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if 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 inf 98.9%

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

   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 98.2%

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

   if 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. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. associate--r+ 100.0%

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

   8. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 99.9%

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

      3. associate--r+ 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/r* 99.9%

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

      6. +-commutative 99.9%

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

      7. clear-num 99.9%

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

      8. +-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. *-lft-identity 99.9%

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

       2. associate-*r/ 100.0%

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

       3. associate-*r/ 52.2%

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

       4. associate-*l/ 100.0%

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

       5. associate-/r/ 100.0%

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

       6. div-sub 100.0%

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

       7. sub-neg 100.0%

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

       8. *-lft-identity 100.0%

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

       9. associate-*l/ 99.8%

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

       10. distribute-lft-in 99.8%

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

       11. lft-mult-inverse 100.0%

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

       12. *-rgt-identity 100.0%

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

       13. *-inverses 100.0%

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

       14. metadata-eval 100.0%

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

   12. Simplified 100.0%

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

       1. *-un-lft-identity 100.0%

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

   14. Applied egg-rr 100.0%

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

       1. associate-*r/ 100.0%

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

       2. associate-*l/ 99.8%

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

       3. distribute-lft-in 99.8%

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

       4. associate-*r/ 99.9%

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

       5. lft-mult-inverse 100.0%

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

       6. associate-*l/ 100.0%

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

       7. metadata-eval 100.0%

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

   16. Simplified 100.0%

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

   17. Taylor expanded in x around inf 98.8%

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

4. Final simplification 98.6%

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

Alternative 7: 98.7% 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 98.8%

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

   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 98.2%

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

4. Final simplification 98.5%

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

Alternative 8: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 9: 50.0% 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 43.8%

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

6. Final simplification 43.8%

   \[\leadsto -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024030 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))