Asymptote A

Percentage Accurate: 77.5% → 99.4%

Time: 8.3s

Alternatives: 5

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: 77.5% 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: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

   1. sub-neg 74.6%

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

   2. +-commutative 74.6%

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

   3. distribute-neg-frac2 74.6%

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

   4. neg-sub0 74.6%

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

   5. associate-+l- 74.6%

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

   6. neg-sub0 74.6%

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

   7. remove-double-neg 74.6%

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

   8. distribute-neg-in 74.6%

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

   9. sub-neg 74.6%

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

   10. distribute-neg-frac2 74.6%

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

   11. sub-neg 74.6%

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

   12. +-commutative 74.6%

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

   13. unsub-neg 74.6%

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

   14. sub-neg 74.6%

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

   15. +-commutative 74.6%

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

   16. unsub-neg 74.6%

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

   17. metadata-eval 74.6%

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

3. Simplified 74.6%

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

   1. sub-neg 74.6%

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

   2. distribute-neg-frac 74.6%

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

   3. metadata-eval 74.6%

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

6. Applied egg-rr 74.6%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 74.7% 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 82.9%

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

      1. sub-neg 82.9%

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

      2. +-commutative 82.9%

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

      3. distribute-neg-frac2 82.9%

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

      4. neg-sub0 82.9%

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

      5. associate-+l- 82.9%

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

      6. neg-sub0 82.9%

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

      7. remove-double-neg 82.9%

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

      8. distribute-neg-in 82.9%

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

      9. sub-neg 82.9%

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

      10. distribute-neg-frac2 82.9%

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

      11. sub-neg 82.9%

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

      12. +-commutative 82.9%

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

      13. unsub-neg 82.9%

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

      14. sub-neg 82.9%

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

      15. +-commutative 82.9%

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

      16. unsub-neg 82.9%

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

      17. metadata-eval 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in x around 0 64.9%

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

   if 0.75 < x

   1. Initial program 49.3%

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

      1. sub-neg 49.3%

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

      2. +-commutative 49.3%

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

      3. distribute-neg-frac2 49.3%

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

      4. neg-sub0 49.3%

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

      5. associate-+l- 49.3%

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

      6. neg-sub0 49.3%

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

      7. remove-double-neg 49.3%

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

      8. distribute-neg-in 49.3%

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

      9. sub-neg 49.3%

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

      10. distribute-neg-frac2 49.3%

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

      11. sub-neg 49.3%

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

      12. +-commutative 49.3%

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

      13. unsub-neg 49.3%

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

      14. sub-neg 49.3%

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

      15. +-commutative 49.3%

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

      16. unsub-neg 49.3%

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

      17. metadata-eval 49.3%

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

   3. Simplified 49.3%

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

      1. sub-neg 49.3%

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

      2. distribute-neg-frac 49.3%

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

      3. metadata-eval 49.3%

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

   6. Applied egg-rr 49.3%

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

   8. Simplified 99.7%

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

      1. flip3-- 45.0%

         \[\leadsto \frac{-2}{\left(1 - x\right) \cdot \color{blue}{\frac{{-1}^{3} - {x}^{3}}{-1 \cdot -1 + \left(x \cdot x + -1 \cdot x\right)}}} \]

      2. associate-*r/ 34.4%

         \[\leadsto \frac{-2}{\color{blue}{\frac{\left(1 - x\right) \cdot \left({-1}^{3} - {x}^{3}\right)}{-1 \cdot -1 + \left(x \cdot x + -1 \cdot x\right)}}} \]

      3. metadata-eval 34.4%

         \[\leadsto \frac{-2}{\frac{\left(1 - x\right) \cdot \left(\color{blue}{-1} - {x}^{3}\right)}{-1 \cdot -1 + \left(x \cdot x + -1 \cdot x\right)}} \]

      4. metadata-eval 34.4%

         \[\leadsto \frac{-2}{\frac{\left(1 - x\right) \cdot \left(-1 - {x}^{3}\right)}{\color{blue}{1} + \left(x \cdot x + -1 \cdot x\right)}} \]

      5. distribute-rgt-out 34.4%

         \[\leadsto \frac{-2}{\frac{\left(1 - x\right) \cdot \left(-1 - {x}^{3}\right)}{1 + \color{blue}{x \cdot \left(x + -1\right)}}} \]

   10. Applied egg-rr 34.4%

       \[\leadsto \frac{-2}{\color{blue}{\frac{\left(1 - x\right) \cdot \left(-1 - {x}^{3}\right)}{1 + x \cdot \left(x + -1\right)}}} \]
   11. Step-by-step derivation

       1. associate-/l* 45.0%

          \[\leadsto \frac{-2}{\color{blue}{\left(1 - x\right) \cdot \frac{-1 - {x}^{3}}{1 + x \cdot \left(x + -1\right)}}} \]

   12. Simplified 45.0%

       \[\leadsto \frac{-2}{\color{blue}{\left(1 - x\right) \cdot \frac{-1 - {x}^{3}}{1 + x \cdot \left(x + -1\right)}}} \]

   13. Taylor expanded in x around inf 98.3%

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

       1. neg-mul-1 98.3%

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

   15. Simplified 98.3%

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

   16. Taylor expanded in x around 0 98.3%

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

4. Final simplification 73.1%

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

Alternative 3: 75.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], 2.0, 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 82.9%

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

      1. sub-neg 82.9%

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

      2. +-commutative 82.9%

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

      3. distribute-neg-frac2 82.9%

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

      4. neg-sub0 82.9%

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

      5. associate-+l- 82.9%

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

      6. neg-sub0 82.9%

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

      7. remove-double-neg 82.9%

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

      8. distribute-neg-in 82.9%

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

      9. sub-neg 82.9%

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

      10. distribute-neg-frac2 82.9%

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

      11. sub-neg 82.9%

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

      12. +-commutative 82.9%

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

      13. unsub-neg 82.9%

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

      14. sub-neg 82.9%

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

      15. +-commutative 82.9%

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

      16. unsub-neg 82.9%

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

      17. metadata-eval 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in x around 0 64.9%

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

   if 1 < x

   1. Initial program 49.3%

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

      1. sub-neg 49.3%

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

      2. +-commutative 49.3%

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

      3. distribute-neg-frac2 49.3%

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

      4. neg-sub0 49.3%

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

      5. associate-+l- 49.3%

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

      6. neg-sub0 49.3%

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

      7. remove-double-neg 49.3%

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

      8. distribute-neg-in 49.3%

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

      9. sub-neg 49.3%

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

      10. distribute-neg-frac2 49.3%

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

      11. sub-neg 49.3%

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

      12. +-commutative 49.3%

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

      13. unsub-neg 49.3%

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

      14. sub-neg 49.3%

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

      15. +-commutative 49.3%

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

      16. unsub-neg 49.3%

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

      17. metadata-eval 49.3%

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

   3. Simplified 49.3%

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

      1. frac-sub 50.4%

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

      2. *-rgt-identity 50.4%

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

      3. metadata-eval 50.4%

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

      4. div-inv 50.4%

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

      5. associate-/r* 50.4%

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

      6. metadata-eval 50.4%

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

      7. div-inv 50.4%

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

      8. *-un-lft-identity 50.4%

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

      9. associate--l- 57.5%

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

      10. div-inv 57.5%

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

      11. metadata-eval 57.5%

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

      12. *-rgt-identity 57.5%

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

      13. div-inv 57.5%

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

      14. metadata-eval 57.5%

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

      15. *-rgt-identity 57.5%

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

   6. Applied egg-rr 57.5%

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

   7. Taylor expanded in x around inf 98.3%

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

4. Final simplification 73.1%

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

Alternative 4: 62.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) 2.0 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = 0.0;
	}
	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 = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 2.0
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], 2.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 82.9%

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

      1. sub-neg 82.9%

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

      2. +-commutative 82.9%

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

      3. distribute-neg-frac2 82.9%

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

      4. neg-sub0 82.9%

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

      5. associate-+l- 82.9%

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

      6. neg-sub0 82.9%

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

      7. remove-double-neg 82.9%

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

      8. distribute-neg-in 82.9%

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

      9. sub-neg 82.9%

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

      10. distribute-neg-frac2 82.9%

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

      11. sub-neg 82.9%

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

      12. +-commutative 82.9%

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

      13. unsub-neg 82.9%

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

      14. sub-neg 82.9%

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

      15. +-commutative 82.9%

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

      16. unsub-neg 82.9%

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

      17. metadata-eval 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in x around 0 64.9%

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

   if 1 < x

   1. Initial program 49.3%

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

      1. sub-neg 49.3%

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

      2. +-commutative 49.3%

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

      3. distribute-neg-frac2 49.3%

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

      4. neg-sub0 49.3%

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

      5. associate-+l- 49.3%

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

      6. neg-sub0 49.3%

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

      7. remove-double-neg 49.3%

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

      8. distribute-neg-in 49.3%

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

      9. sub-neg 49.3%

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

      10. distribute-neg-frac2 49.3%

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

      11. sub-neg 49.3%

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

      12. +-commutative 49.3%

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

      13. unsub-neg 49.3%

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

      14. sub-neg 49.3%

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

      15. +-commutative 49.3%

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

      16. unsub-neg 49.3%

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

      17. metadata-eval 49.3%

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

   3. Simplified 49.3%

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

      1. flip-- 8.1%

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

      2. metadata-eval 8.1%

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

      3. metadata-eval 8.1%

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

      4. associate-/r/ 8.1%

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

      5. metadata-eval 8.1%

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

      6. pow2 8.1%

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

   6. Applied egg-rr 8.1%

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

      1. associate-*l/ 8.2%

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

      2. *-lft-identity 8.2%

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

      3. +-commutative 8.2%

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

   8. Simplified 8.2%

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

      1. clear-num 8.1%

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

      2. metadata-eval 8.1%

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

      3. unpow2 8.1%

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

      4. +-commutative 8.1%

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

      5. flip-- 49.3%

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

      6. frac-sub 50.4%

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

      7. *-commutative 50.4%

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

      8. distribute-lft-out-- 50.4%

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

      9. associate--r+ 57.5%

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

      10. flip-- 54.3%

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

      11. *-un-lft-identity 54.3%

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

      12. associate-/l/ 54.3%

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

   10. Applied egg-rr 0.0%

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

       1. div-sub 0.0%

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

       2. unpow2 0.0%

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

       3. +-inverses 0.0%

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

       4. metadata-eval 0.0%

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

       5. +-inverses 0.0%

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

       6. metadata-eval 0.0%

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

       7. metadata-eval 0.0%

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

       8. +-inverses 47.0%

          \[\leadsto \color{blue}{0} \]

   12. Simplified 47.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 27.4% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 74.6%

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

   1. sub-neg 74.6%

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

   2. +-commutative 74.6%

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

   3. distribute-neg-frac2 74.6%

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

   4. neg-sub0 74.6%

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

   5. associate-+l- 74.6%

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

   6. neg-sub0 74.6%

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

   7. remove-double-neg 74.6%

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

   8. distribute-neg-in 74.6%

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

   9. sub-neg 74.6%

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

   10. distribute-neg-frac2 74.6%

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

   11. sub-neg 74.6%

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

   12. +-commutative 74.6%

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

   13. unsub-neg 74.6%

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

   14. sub-neg 74.6%

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

   15. +-commutative 74.6%

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

   16. unsub-neg 74.6%

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

   17. metadata-eval 74.6%

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

3. Simplified 74.6%

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

   1. flip-- 53.5%

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

   2. metadata-eval 53.5%

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

   3. metadata-eval 53.5%

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

   4. associate-/r/ 53.4%

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

   5. metadata-eval 53.4%

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

   6. pow2 53.4%

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

6. Applied egg-rr 53.4%

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

   1. associate-*l/ 53.6%

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

   2. *-lft-identity 53.6%

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

   3. +-commutative 53.6%

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

8. Simplified 53.6%

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

   1. clear-num 53.5%

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

   2. metadata-eval 53.5%

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

   3. unpow2 53.5%

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

   4. +-commutative 53.5%

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

   5. flip-- 74.6%

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

   6. frac-sub 75.0%

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

   7. *-commutative 75.0%

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

   8. distribute-lft-out-- 75.0%

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

   9. associate--r+ 78.4%

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

   10. flip-- 28.4%

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

   11. *-un-lft-identity 28.4%

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

   12. associate-/l/ 28.4%

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

10. Applied egg-rr 0.0%

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

    1. div-sub 0.0%

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

    2. unpow2 0.0%

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

    3. +-inverses 0.0%

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

    4. metadata-eval 0.0%

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

    5. +-inverses 0.0%

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

    6. metadata-eval 0.0%

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

    7. metadata-eval 0.0%

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

    8. +-inverses 25.7%

       \[\leadsto \color{blue}{0} \]

12. Simplified 25.7%

    \[\leadsto \color{blue}{0} \]

13. Final simplification 25.7%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))