Asymptote A

Percentage Accurate: 77.1% → 99.9%

Time: 6.6s

Alternatives: 9

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: 77.1% 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 80.1%

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

   1. sub-neg 80.1%

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

   2. +-commutative 80.1%

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

   3. distribute-neg-frac2 80.1%

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

   4. neg-sub0 80.1%

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

   5. associate-+l- 80.1%

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

   6. neg-sub0 80.1%

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

   7. remove-double-neg 80.1%

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

   8. distribute-neg-in 80.1%

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

   9. sub-neg 80.1%

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

   10. distribute-neg-frac2 80.1%

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

   11. sub-neg 80.1%

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

   12. +-commutative 80.1%

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

   13. unsub-neg 80.1%

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

   14. sub-neg 80.1%

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

   15. +-commutative 80.1%

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

   16. unsub-neg 80.1%

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

   17. metadata-eval 80.1%

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

3. Simplified 80.1%

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

   1. sub-neg 80.1%

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

   2. distribute-neg-frac 80.1%

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

   3. metadata-eval 80.1%

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

6. Applied egg-rr 80.1%

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

8. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{-2}{\left(x + 1\right) \cdot \left(x + -1\right)}} \]
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-+ 99.4%

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

   6. sub-neg 99.4%

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

   7. metadata-eval 99.4%

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

   8. neg-mul-1 99.4%

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

   9. distribute-lft-in 99.4%

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

   10. +-commutative 99.4%

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

   11. mul-1-neg 99.4%

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

   12. distribute-neg-frac2 99.4%

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

   13. metadata-eval 99.4%

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

   14. metadata-eval 99.4%

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

   15. +-commutative 99.4%

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

   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. clear-num 99.8%

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

    2. un-div-inv 99.9%

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

    3. flip-- 99.5%

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

    4. metadata-eval 99.5%

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

    5. metadata-eval 99.5%

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

    6. +-commutative 99.5%

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

    7. associate-/l/ 99.5%

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

    8. neg-mul-1 99.5%

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

    9. +-commutative 99.5%

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

    10. distribute-neg-in 99.5%

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

    11. metadata-eval 99.5%

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

    12. sub-neg 99.5%

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

    13. flip-+ 99.9%

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

    14. +-commutative 99.9%

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

12. Applied egg-rr 99.9%

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

Alternative 2: 74.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.75:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.75) 2.0 (/ (/ -2.0 x) (+ x -1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.75) {
		tmp = 2.0;
	} 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 <= 0.75d0) then
        tmp = 2.0d0
    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 <= 0.75) {
		tmp = 2.0;
	} else {
		tmp = (-2.0 / x) / (x + -1.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.75:
		tmp = 2.0
	else:
		tmp = (-2.0 / x) / (x + -1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.75)
		tmp = 2.0;
	else
		tmp = Float64(Float64(-2.0 / x) / Float64(x + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.75)
		tmp = 2.0;
	else
		tmp = (-2.0 / x) / (x + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.75], 2.0, N[(N[(-2.0 / x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.75

   1. Initial program 89.4%

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

      1. sub-neg 89.4%

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

      2. +-commutative 89.4%

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

      3. distribute-neg-frac2 89.4%

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

      4. neg-sub0 89.4%

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

      5. associate-+l- 89.4%

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

      6. neg-sub0 89.4%

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

      7. remove-double-neg 89.4%

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

      8. distribute-neg-in 89.4%

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

      9. sub-neg 89.4%

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

      10. distribute-neg-frac2 89.4%

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

      11. sub-neg 89.4%

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

      12. +-commutative 89.4%

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

      13. unsub-neg 89.4%

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

      14. sub-neg 89.4%

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

      15. +-commutative 89.4%

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

      16. unsub-neg 89.4%

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

      17. metadata-eval 89.4%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in x around 0 69.4%

      \[\leadsto \color{blue}{2} \]

   if 0.75 < x

   1. Initial program 51.8%

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

      1. sub-neg 51.8%

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

      2. +-commutative 51.8%

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

      3. distribute-neg-frac2 51.8%

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

      4. neg-sub0 51.8%

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

      5. associate-+l- 51.8%

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

      6. neg-sub0 51.8%

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

      7. remove-double-neg 51.8%

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

      8. distribute-neg-in 51.8%

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

      9. sub-neg 51.8%

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

      10. distribute-neg-frac2 51.8%

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

      11. sub-neg 51.8%

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

      12. +-commutative 51.8%

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

      13. unsub-neg 51.8%

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

      14. sub-neg 51.8%

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

      15. +-commutative 51.8%

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

      16. unsub-neg 51.8%

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

      17. metadata-eval 51.8%

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

   3. Simplified 51.8%

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

      1. sub-neg 51.8%

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

      2. distribute-neg-frac 51.8%

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

      3. metadata-eval 51.8%

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

   6. Applied egg-rr 51.8%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in x around inf 96.4%

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

       1. div-inv 96.4%

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

       2. associate-/r* 97.1%

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

   11. Applied egg-rr 97.1%

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

       1. associate-/r* 96.4%

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

       2. associate-*r/ 96.4%

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

       3. metadata-eval 96.4%

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

       4. associate-/r* 97.1%

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

   13. Simplified 97.1%

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

Alternative 3: 73.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.75:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x \cdot \left(x + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.75) 2.0 (/ -2.0 (* x (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.75) {
		tmp = 2.0;
	} 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 <= 0.75d0) then
        tmp = 2.0d0
    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 <= 0.75) {
		tmp = 2.0;
	} else {
		tmp = -2.0 / (x * (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.75:
		tmp = 2.0
	else:
		tmp = -2.0 / (x * (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.75)
		tmp = 2.0;
	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 <= 0.75)
		tmp = 2.0;
	else
		tmp = -2.0 / (x * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.75], 2.0, N[(-2.0 / N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.75

   1. Initial program 89.4%

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

      1. sub-neg 89.4%

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

      2. +-commutative 89.4%

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

      3. distribute-neg-frac2 89.4%

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

      4. neg-sub0 89.4%

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

      5. associate-+l- 89.4%

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

      6. neg-sub0 89.4%

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

      7. remove-double-neg 89.4%

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

      8. distribute-neg-in 89.4%

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

      9. sub-neg 89.4%

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

      10. distribute-neg-frac2 89.4%

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

      11. sub-neg 89.4%

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

      12. +-commutative 89.4%

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

      13. unsub-neg 89.4%

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

      14. sub-neg 89.4%

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

      15. +-commutative 89.4%

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

      16. unsub-neg 89.4%

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

      17. metadata-eval 89.4%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in x around 0 69.4%

      \[\leadsto \color{blue}{2} \]

   if 0.75 < x

   1. Initial program 51.8%

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

      1. sub-neg 51.8%

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

      2. +-commutative 51.8%

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

      3. distribute-neg-frac2 51.8%

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

      4. neg-sub0 51.8%

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

      5. associate-+l- 51.8%

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

      6. neg-sub0 51.8%

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

      7. remove-double-neg 51.8%

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

      8. distribute-neg-in 51.8%

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

      9. sub-neg 51.8%

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

      10. distribute-neg-frac2 51.8%

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

      11. sub-neg 51.8%

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

      12. +-commutative 51.8%

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

      13. unsub-neg 51.8%

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

      14. sub-neg 51.8%

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

      15. +-commutative 51.8%

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

      16. unsub-neg 51.8%

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

      17. metadata-eval 51.8%

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

   3. Simplified 51.8%

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

      1. sub-neg 51.8%

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

      2. distribute-neg-frac 51.8%

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

      3. metadata-eval 51.8%

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

   6. Applied egg-rr 51.8%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in x around inf 96.4%

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

Alternative 4: 99.9% 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 80.1%

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

   1. sub-neg 80.1%

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

   2. +-commutative 80.1%

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

   3. distribute-neg-frac2 80.1%

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

   4. neg-sub0 80.1%

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

   5. associate-+l- 80.1%

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

   6. neg-sub0 80.1%

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

   7. remove-double-neg 80.1%

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

   8. distribute-neg-in 80.1%

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

   9. sub-neg 80.1%

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

   10. distribute-neg-frac2 80.1%

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

   11. sub-neg 80.1%

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

   12. +-commutative 80.1%

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

   13. unsub-neg 80.1%

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

   14. sub-neg 80.1%

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

   15. +-commutative 80.1%

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

   16. unsub-neg 80.1%

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

   17. metadata-eval 80.1%

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

3. Simplified 80.1%

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

   1. sub-neg 80.1%

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

   2. distribute-neg-frac 80.1%

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

   3. metadata-eval 80.1%

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

6. Applied egg-rr 80.1%

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

8. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{-2}{\left(x + 1\right) \cdot \left(x + -1\right)}} \]
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-+ 99.4%

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

   6. sub-neg 99.4%

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

   7. metadata-eval 99.4%

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

   8. neg-mul-1 99.4%

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

   9. distribute-lft-in 99.4%

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

   10. +-commutative 99.4%

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

   11. mul-1-neg 99.4%

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

   12. distribute-neg-frac2 99.4%

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

   13. metadata-eval 99.4%

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

   14. metadata-eval 99.4%

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

   15. +-commutative 99.4%

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

   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. clear-num 99.9%

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

    3. un-div-inv 99.9%

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

    4. flip-- 99.5%

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

    5. metadata-eval 99.5%

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

    6. metadata-eval 99.5%

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

    7. +-commutative 99.5%

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

    8. associate-/l/ 99.5%

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

    9. neg-mul-1 99.5%

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

    10. +-commutative 99.5%

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

    11. distribute-neg-in 99.5%

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

    12. metadata-eval 99.5%

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

    13. sub-neg 99.5%

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

    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 5: 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 80.1%

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

   1. sub-neg 80.1%

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

   2. +-commutative 80.1%

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

   3. distribute-neg-frac2 80.1%

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

   4. neg-sub0 80.1%

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

   5. associate-+l- 80.1%

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

   6. neg-sub0 80.1%

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

   7. remove-double-neg 80.1%

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

   8. distribute-neg-in 80.1%

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

   9. sub-neg 80.1%

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

   10. distribute-neg-frac2 80.1%

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

   11. sub-neg 80.1%

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

   12. +-commutative 80.1%

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

   13. unsub-neg 80.1%

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

   14. sub-neg 80.1%

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

   15. +-commutative 80.1%

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

   16. unsub-neg 80.1%

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

   17. metadata-eval 80.1%

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

3. Simplified 80.1%

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

   1. sub-neg 80.1%

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

   2. distribute-neg-frac 80.1%

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

   3. metadata-eval 80.1%

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

6. Applied egg-rr 80.1%

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

8. Simplified 99.5%

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

Alternative 6: 50.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 89.4%

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

      1. sub-neg 89.4%

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

      2. +-commutative 89.4%

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

      3. distribute-neg-frac2 89.4%

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

      4. neg-sub0 89.4%

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

      5. associate-+l- 89.4%

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

      6. neg-sub0 89.4%

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

      7. remove-double-neg 89.4%

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

      8. distribute-neg-in 89.4%

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

      9. sub-neg 89.4%

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

      10. distribute-neg-frac2 89.4%

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

      11. sub-neg 89.4%

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

      12. +-commutative 89.4%

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

      13. unsub-neg 89.4%

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

      14. sub-neg 89.4%

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

      15. +-commutative 89.4%

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

      16. unsub-neg 89.4%

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

      17. metadata-eval 89.4%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in x around 0 69.4%

      \[\leadsto \color{blue}{2} \]

   if 1 < x

   1. Initial program 51.8%

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

      1. sub-neg 51.8%

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

      2. +-commutative 51.8%

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

      3. distribute-neg-frac2 51.8%

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

      4. neg-sub0 51.8%

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

      5. associate-+l- 51.8%

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

      6. neg-sub0 51.8%

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

      7. remove-double-neg 51.8%

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

      8. distribute-neg-in 51.8%

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

      9. sub-neg 51.8%

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

      10. distribute-neg-frac2 51.8%

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

      11. sub-neg 51.8%

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

      12. +-commutative 51.8%

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

      13. unsub-neg 51.8%

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

      14. sub-neg 51.8%

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

      15. +-commutative 51.8%

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

      16. unsub-neg 51.8%

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

      17. metadata-eval 51.8%

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

   3. Simplified 51.8%

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

   5. Taylor expanded in x around 0 2.7%

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

   6. Taylor expanded in x around inf 2.7%

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

   7. Taylor expanded in x around 0 6.7%

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

Alternative 7: 50.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

   1. sub-neg 80.1%

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

   2. +-commutative 80.1%

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

   3. distribute-neg-frac2 80.1%

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

   4. neg-sub0 80.1%

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

   5. associate-+l- 80.1%

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

   6. neg-sub0 80.1%

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

   7. remove-double-neg 80.1%

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

   8. distribute-neg-in 80.1%

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

   9. sub-neg 80.1%

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

   10. distribute-neg-frac2 80.1%

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

   11. sub-neg 80.1%

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

   12. +-commutative 80.1%

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

   13. unsub-neg 80.1%

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

   14. sub-neg 80.1%

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

   15. +-commutative 80.1%

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

   16. unsub-neg 80.1%

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

   17. metadata-eval 80.1%

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

3. Simplified 80.1%

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

   1. sub-neg 80.1%

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

   2. distribute-neg-frac 80.1%

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

   3. metadata-eval 80.1%

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

6. Applied egg-rr 80.1%

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

8. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{-2}{\left(x + 1\right) \cdot \left(x + -1\right)}} \]
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-+ 99.4%

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

   6. sub-neg 99.4%

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

   7. metadata-eval 99.4%

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

   8. neg-mul-1 99.4%

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

   9. distribute-lft-in 99.4%

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

   10. +-commutative 99.4%

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

   11. mul-1-neg 99.4%

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

   12. distribute-neg-frac2 99.4%

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

   13. metadata-eval 99.4%

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

   14. metadata-eval 99.4%

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

   15. +-commutative 99.4%

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

   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. clear-num 99.8%

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

    2. un-div-inv 99.9%

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

    3. flip-- 99.5%

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

    4. metadata-eval 99.5%

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

    5. metadata-eval 99.5%

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

    6. +-commutative 99.5%

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

    7. associate-/l/ 99.5%

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

    8. neg-mul-1 99.5%

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

    9. +-commutative 99.5%

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

    10. distribute-neg-in 99.5%

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

    11. metadata-eval 99.5%

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

    12. sub-neg 99.5%

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

    13. flip-+ 99.9%

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

    14. +-commutative 99.9%

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

12. Applied egg-rr 99.9%

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

13. Taylor expanded in x around 0 53.9%

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

Alternative 8: 49.6% 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 80.1%

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

   1. sub-neg 80.1%

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

   2. +-commutative 80.1%

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

   3. distribute-neg-frac2 80.1%

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

   4. neg-sub0 80.1%

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

   5. associate-+l- 80.1%

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

   6. neg-sub0 80.1%

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

   7. remove-double-neg 80.1%

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

   8. distribute-neg-in 80.1%

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

   9. sub-neg 80.1%

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

   10. distribute-neg-frac2 80.1%

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

   11. sub-neg 80.1%

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

   12. +-commutative 80.1%

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

   13. unsub-neg 80.1%

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

   14. sub-neg 80.1%

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

   15. +-commutative 80.1%

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

   16. unsub-neg 80.1%

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

   17. metadata-eval 80.1%

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

3. Simplified 80.1%

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

5. Taylor expanded in x around 0 53.0%

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

Alternative 9: 10.5% 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 80.1%

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

   1. sub-neg 80.1%

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

   2. +-commutative 80.1%

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

   3. distribute-neg-frac2 80.1%

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

   4. neg-sub0 80.1%

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

   5. associate-+l- 80.1%

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

   6. neg-sub0 80.1%

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

   7. remove-double-neg 80.1%

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

   8. distribute-neg-in 80.1%

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

   9. sub-neg 80.1%

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

   10. distribute-neg-frac2 80.1%

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

   11. sub-neg 80.1%

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

   12. +-commutative 80.1%

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

   13. unsub-neg 80.1%

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

   14. sub-neg 80.1%

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

   15. +-commutative 80.1%

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

   16. unsub-neg 80.1%

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

   17. metadata-eval 80.1%

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

3. Simplified 80.1%

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

5. Taylor expanded in x around 0 52.3%

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

6. Taylor expanded in x around inf 3.0%

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

7. Taylor expanded in x around inf 11.0%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))