Asymptote A

Percentage Accurate: 77.9% → 99.9%

Time: 5.0s

Alternatives: 6

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return (-2.0 / (x + -1.0)) / (x + 1.0);
}

Python

def code(x):
	return (-2.0 / (x + -1.0)) / (x + 1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. sub-neg 74.0%

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

   2. +-commutative 74.0%

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

   3. distribute-neg-frac2 74.0%

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

   4. neg-sub0 74.0%

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

   5. associate-+l- 74.0%

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

   6. neg-sub0 74.0%

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

   7. remove-double-neg 74.0%

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

   8. distribute-neg-in 74.0%

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

   9. sub-neg 74.0%

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

   10. distribute-neg-frac2 74.0%

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

   11. sub-neg 74.0%

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

   12. +-commutative 74.0%

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

   13. unsub-neg 74.0%

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

   14. sub-neg 74.0%

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

   15. +-commutative 74.0%

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

   16. unsub-neg 74.0%

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

   17. metadata-eval 74.0%

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

3. Simplified 74.0%

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

   1. sub-neg 74.0%

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

   2. distribute-neg-frac 74.0%

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

   3. metadata-eval 74.0%

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

6. Applied egg-rr 74.0%

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

8. Simplified 99.0%

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

   1. associate-/r* 99.9%

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

   2. div-inv 99.8%

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

   3. metadata-eval 99.8%

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

   4. +-commutative 99.8%

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

   5. flip-+ 99.0%

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

   6. metadata-eval 99.0%

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

   7. metadata-eval 99.0%

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

   8. sub-neg 99.0%

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

   9. metadata-eval 99.0%

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

   10. neg-mul-1 99.0%

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

   11. distribute-lft-in 99.0%

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

   12. +-commutative 99.0%

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

   13. mul-1-neg 99.0%

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

   14. distribute-neg-frac2 99.0%

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

   15. +-commutative 99.0%

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

   16. flip-- 99.8%

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

   17. frac-2neg 99.8%

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

10. Applied egg-rr 99.8%

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

    1. associate-*l/ 99.9%

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

    2. frac-2neg 99.9%

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

    3. metadata-eval 99.9%

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

    4. un-div-inv 99.9%

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

    5. flip-- 99.1%

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

    6. metadata-eval 99.1%

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

    7. metadata-eval 99.1%

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

    8. +-commutative 99.1%

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

    9. distribute-neg-frac2 99.1%

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

    10. +-commutative 99.1%

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

    11. distribute-neg-in 99.1%

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

    12. metadata-eval 99.1%

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

    13. sub-neg 99.1%

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

    14. flip-+ 99.9%

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

    15. +-commutative 99.9%

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

12. Applied egg-rr 99.9%

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

Alternative 2: 75.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (+ 2.0 (* 2.0 (* x x))) (/ (/ 2.0 x) (- -1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0 + (2.0 * (x * x));
	} else {
		tmp = (2.0 / x) / (-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 = 2.0d0 + (2.0d0 * (x * x))
    else
        tmp = (2.0d0 / x) / ((-1.0d0) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0 + (2.0 * (x * x));
	} else {
		tmp = (2.0 / x) / (-1.0 - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 2.0 + (2.0 * (x * x))
	else:
		tmp = (2.0 / x) / (-1.0 - x)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 2.0 + (2.0 * (x * x));
	else
		tmp = (2.0 / x) / (-1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 81.2%

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

      1. sub-neg 81.2%

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

      2. +-commutative 81.2%

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

      3. distribute-neg-frac2 81.2%

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

      4. neg-sub0 81.2%

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

      5. associate-+l- 81.2%

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

      6. neg-sub0 81.2%

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

      7. remove-double-neg 81.2%

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

      8. distribute-neg-in 81.2%

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

      9. sub-neg 81.2%

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

      10. distribute-neg-frac2 81.2%

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

      11. sub-neg 81.2%

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

      12. +-commutative 81.2%

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

      13. unsub-neg 81.2%

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

      14. sub-neg 81.2%

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

      15. +-commutative 81.2%

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

      16. unsub-neg 81.2%

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

      17. metadata-eval 81.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in x around 0 63.0%

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

      1. unpow2 63.0%

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

   7. Applied egg-rr 63.0%

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

   if 1 < x

   1. Initial program 55.0%

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

      1. sub-neg 55.0%

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

      2. +-commutative 55.0%

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

      3. distribute-neg-frac2 55.0%

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

      4. neg-sub0 55.0%

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

      5. associate-+l- 55.0%

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

      6. neg-sub0 55.0%

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

      7. remove-double-neg 55.0%

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

      8. distribute-neg-in 55.0%

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

      9. sub-neg 55.0%

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

      10. distribute-neg-frac2 55.0%

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

      11. sub-neg 55.0%

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

      12. +-commutative 55.0%

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

      13. unsub-neg 55.0%

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

      14. sub-neg 55.0%

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

      15. +-commutative 55.0%

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

      16. unsub-neg 55.0%

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

      17. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

      1. frac-sub 57.5%

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

      2. *-rgt-identity 57.5%

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

      3. metadata-eval 57.5%

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

      4. div-inv 57.5%

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

      5. associate-/r* 57.6%

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

      6. metadata-eval 57.6%

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

      7. div-inv 57.6%

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

      8. *-un-lft-identity 57.6%

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

      9. associate--l- 63.1%

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

      10. div-inv 63.1%

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

      11. metadata-eval 63.1%

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

      12. *-rgt-identity 63.1%

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

      13. div-inv 63.1%

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

      14. metadata-eval 63.1%

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

      15. *-rgt-identity 63.1%

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

   6. Applied egg-rr 63.1%

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

   7. Taylor expanded in x around inf 95.6%

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

Alternative 3: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.85:\\ \;\;\;\;2 + 2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x \cdot \left(x + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.85) (+ 2.0 (* 2.0 (* x x))) (/ -2.0 (* x (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.85) {
		tmp = 2.0 + (2.0 * (x * x));
	} else {
		tmp = -2.0 / (x * (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.85d0) then
        tmp = 2.0d0 + (2.0d0 * (x * x))
    else
        tmp = (-2.0d0) / (x * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.85) {
		tmp = 2.0 + (2.0 * (x * x));
	} else {
		tmp = -2.0 / (x * (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.85:
		tmp = 2.0 + (2.0 * (x * x))
	else:
		tmp = -2.0 / (x * (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.85)
		tmp = Float64(2.0 + Float64(2.0 * Float64(x * x)));
	else
		tmp = Float64(-2.0 / Float64(x * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.85)
		tmp = 2.0 + (2.0 * (x * x));
	else
		tmp = -2.0 / (x * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.85], N[(2.0 + N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.849999999999999978

   1. Initial program 81.2%

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

      1. sub-neg 81.2%

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

      2. +-commutative 81.2%

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

      3. distribute-neg-frac2 81.2%

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

      4. neg-sub0 81.2%

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

      5. associate-+l- 81.2%

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

      6. neg-sub0 81.2%

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

      7. remove-double-neg 81.2%

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

      8. distribute-neg-in 81.2%

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

      9. sub-neg 81.2%

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

      10. distribute-neg-frac2 81.2%

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

      11. sub-neg 81.2%

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

      12. +-commutative 81.2%

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

      13. unsub-neg 81.2%

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

      14. sub-neg 81.2%

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

      15. +-commutative 81.2%

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

      16. unsub-neg 81.2%

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

      17. metadata-eval 81.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in x around 0 63.0%

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

      1. unpow2 63.0%

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

   7. Applied egg-rr 63.0%

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

   if 0.849999999999999978 < x

   1. Initial program 55.0%

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

      1. sub-neg 55.0%

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

      2. +-commutative 55.0%

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

      3. distribute-neg-frac2 55.0%

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

      4. neg-sub0 55.0%

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

      5. associate-+l- 55.0%

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

      6. neg-sub0 55.0%

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

      7. remove-double-neg 55.0%

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

      8. distribute-neg-in 55.0%

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

      9. sub-neg 55.0%

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

      10. distribute-neg-frac2 55.0%

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

      11. sub-neg 55.0%

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

      12. +-commutative 55.0%

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

      13. unsub-neg 55.0%

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

      14. sub-neg 55.0%

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

      15. +-commutative 55.0%

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

      16. unsub-neg 55.0%

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

      17. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

      1. sub-neg 55.0%

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

      2. distribute-neg-frac 55.0%

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

      3. metadata-eval 55.0%

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

   6. Applied egg-rr 55.0%

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

   8. Simplified 97.5%

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

   9. Taylor expanded in x around inf 93.4%

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

Alternative 4: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return -2.0 / ((x + -1.0) * (x + 1.0));
}

Python

def code(x):
	return -2.0 / ((x + -1.0) * (x + 1.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. sub-neg 74.0%

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

   2. +-commutative 74.0%

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

   3. distribute-neg-frac2 74.0%

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

   4. neg-sub0 74.0%

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

   5. associate-+l- 74.0%

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

   6. neg-sub0 74.0%

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

   7. remove-double-neg 74.0%

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

   8. distribute-neg-in 74.0%

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

   9. sub-neg 74.0%

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

   10. distribute-neg-frac2 74.0%

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

   11. sub-neg 74.0%

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

   12. +-commutative 74.0%

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

   13. unsub-neg 74.0%

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

   14. sub-neg 74.0%

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

   15. +-commutative 74.0%

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

   16. unsub-neg 74.0%

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

   17. metadata-eval 74.0%

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

3. Simplified 74.0%

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

   1. sub-neg 74.0%

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

   2. distribute-neg-frac 74.0%

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

   3. metadata-eval 74.0%

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

6. Applied egg-rr 74.0%

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

8. Simplified 99.0%

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

9. Final simplification 99.0%

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

Alternative 5: 52.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return -2.0 / (-1.0 - x);
}

Python

def code(x):
	return -2.0 / (-1.0 - x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. sub-neg 74.0%

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

   2. +-commutative 74.0%

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

   3. distribute-neg-frac2 74.0%

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

   4. neg-sub0 74.0%

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

   5. associate-+l- 74.0%

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

   6. neg-sub0 74.0%

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

   7. remove-double-neg 74.0%

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

   8. distribute-neg-in 74.0%

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

   9. sub-neg 74.0%

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

   10. distribute-neg-frac2 74.0%

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

   11. sub-neg 74.0%

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

   12. +-commutative 74.0%

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

   13. unsub-neg 74.0%

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

   14. sub-neg 74.0%

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

   15. +-commutative 74.0%

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

   16. unsub-neg 74.0%

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

   17. metadata-eval 74.0%

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

3. Simplified 74.0%

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

   1. frac-sub 75.0%

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

   2. *-rgt-identity 75.0%

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

   3. metadata-eval 75.0%

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

   4. div-inv 75.0%

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

   5. associate-/r* 75.0%

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

   6. metadata-eval 75.0%

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

   7. div-inv 75.0%

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

   8. *-un-lft-identity 75.0%

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

   9. associate--l- 78.3%

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

   10. div-inv 78.3%

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

   11. metadata-eval 78.3%

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

   12. *-rgt-identity 78.3%

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

   13. div-inv 78.3%

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

   14. metadata-eval 78.3%

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

   15. *-rgt-identity 78.3%

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

6. Applied egg-rr 78.3%

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

7. Taylor expanded in x around 0 47.5%

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

Alternative 6: 51.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

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

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

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

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 74.0%

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

   1. sub-neg 74.0%

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

   2. +-commutative 74.0%

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

   3. distribute-neg-frac2 74.0%

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

   4. neg-sub0 74.0%

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

   5. associate-+l- 74.0%

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

   6. neg-sub0 74.0%

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

   7. remove-double-neg 74.0%

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

   8. distribute-neg-in 74.0%

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

   9. sub-neg 74.0%

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

   10. distribute-neg-frac2 74.0%

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

   11. sub-neg 74.0%

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

   12. +-commutative 74.0%

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

   13. unsub-neg 74.0%

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

   14. sub-neg 74.0%

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

   15. +-commutative 74.0%

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

   16. unsub-neg 74.0%

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

   17. metadata-eval 74.0%

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

3. Simplified 74.0%

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

5. Taylor expanded in x around 0 46.5%

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

Reproduce

herbie shell --seed 2024182 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))