Asymptote A

Percentage Accurate: 77.2% → 99.9%

Time: 5.9s

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: 77.2% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. *-un-lft-identity 77.0%

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

   2. *-commutative 77.0%

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

   3. frac-sub 78.1%

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

   4. associate-/r* 78.1%

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

   5. associate-/r/ 78.1%

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

   6. clear-num 78.1%

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

   7. associate-/r/ 78.1%

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

3. Applied egg-rr 78.1%

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

   1. associate-*r/ 78.1%

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

   2. associate-*l/ 78.1%

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

   3. *-commutative 78.1%

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

   4. neg-mul-1 78.1%

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

   5. neg-sub0 78.1%

      \[\leadsto \frac{\frac{\color{blue}{0 - \left(x + \left(-2 - x\right)\right)}}{1 + x}}{1 - x} \]

   6. +-commutative 78.1%

      \[\leadsto \frac{\frac{0 - \color{blue}{\left(\left(-2 - x\right) + x\right)}}{1 + x}}{1 - x} \]

   7. associate--r+ 78.1%

      \[\leadsto \frac{\frac{\color{blue}{\left(0 - \left(-2 - x\right)\right) - x}}{1 + x}}{1 - x} \]

   8. neg-sub0 78.1%

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

   9. sub-neg 78.1%

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

   10. mul-1-neg 78.1%

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

   11. distribute-neg-in 78.1%

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

   12. metadata-eval 78.1%

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

   13. mul-1-neg 78.1%

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

   14. remove-double-neg 78.1%

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

   15. +-commutative 78.1%

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

5. Simplified 78.1%

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

6. Taylor expanded in x around 0 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55) (+ (- 1.0 x) (/ -1.0 (+ x -1.0))) (/ (/ 2.0 x) (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = (1.0 - x) + (-1.0 / (x + -1.0));
	} 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 <= 1.55d0) then
        tmp = (1.0d0 - x) + ((-1.0d0) / (x + (-1.0d0)))
    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 <= 1.55) {
		tmp = (1.0 - x) + (-1.0 / (x + -1.0));
	} else {
		tmp = (2.0 / x) / (1.0 - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = (1.0 - x) + (-1.0 / (x + -1.0))
	else:
		tmp = (2.0 / x) / (1.0 - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = Float64(Float64(1.0 - x) + Float64(-1.0 / Float64(x + -1.0)));
	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 <= 1.55)
		tmp = (1.0 - x) + (-1.0 / (x + -1.0));
	else
		tmp = (2.0 / x) / (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 85.6%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]

   2. Taylor expanded in x around 0 62.9%

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

      1. mul-1-neg 62.9%

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

      2. sub-neg 62.9%

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

   4. Simplified 62.9%

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

   if 1.55000000000000004 < x

   1. Initial program 50.5%

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

      1. *-un-lft-identity 50.5%

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

      2. *-commutative 50.5%

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

      3. frac-sub 50.7%

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

      4. associate-/r* 50.7%

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

      5. associate-/r/ 50.7%

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

      6. clear-num 50.7%

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

      7. associate-/r/ 50.7%

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

   3. Applied egg-rr 50.7%

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

      1. associate-*r/ 50.7%

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

      2. associate-*l/ 50.7%

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

      3. *-commutative 50.7%

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

      4. neg-mul-1 50.7%

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

      5. neg-sub0 50.7%

         \[\leadsto \frac{\frac{\color{blue}{0 - \left(x + \left(-2 - x\right)\right)}}{1 + x}}{1 - x} \]

      6. +-commutative 50.7%

         \[\leadsto \frac{\frac{0 - \color{blue}{\left(\left(-2 - x\right) + x\right)}}{1 + x}}{1 - x} \]

      7. associate--r+ 50.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(0 - \left(-2 - x\right)\right) - x}}{1 + x}}{1 - x} \]

      8. neg-sub0 50.7%

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

      9. sub-neg 50.7%

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

      10. mul-1-neg 50.7%

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

      11. distribute-neg-in 50.7%

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

      12. metadata-eval 50.7%

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

      13. mul-1-neg 50.7%

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

      14. remove-double-neg 50.7%

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

      15. +-commutative 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in x around inf 98.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{1 - x}\\ \end{array} \]

Alternative 3: 73.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.76:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x \cdot \left(x + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.76) 2.0 (/ -2.0 (* x (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.76) {
		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 <= 0.76d0) 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 <= 0.76) {
		tmp = 2.0;
	} else {
		tmp = -2.0 / (x * (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.76:
		tmp = 2.0
	else:
		tmp = -2.0 / (x * (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.76)
		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 <= 0.76)
		tmp = 2.0;
	else
		tmp = -2.0 / (x * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.76], 2.0, N[(-2.0 / N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.76000000000000001

   1. Initial program 85.6%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]

   2. Taylor expanded in x around 0 63.1%

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

   if 0.76000000000000001 < x

   1. Initial program 50.5%

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

      1. *-un-lft-identity 50.5%

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

      2. *-commutative 50.5%

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

      3. frac-sub 50.7%

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

      4. associate-/r* 50.7%

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

      5. associate-/r/ 50.7%

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

      6. clear-num 50.7%

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

      7. associate-/r/ 50.7%

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

   3. Applied egg-rr 50.7%

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

      1. associate-*r/ 50.7%

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

      2. associate-*l/ 50.7%

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

      3. *-commutative 50.7%

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

      4. neg-mul-1 50.7%

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

      5. neg-sub0 50.7%

         \[\leadsto \frac{\frac{\color{blue}{0 - \left(x + \left(-2 - x\right)\right)}}{1 + x}}{1 - x} \]

      6. +-commutative 50.7%

         \[\leadsto \frac{\frac{0 - \color{blue}{\left(\left(-2 - x\right) + x\right)}}{1 + x}}{1 - x} \]

      7. associate--r+ 50.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(0 - \left(-2 - x\right)\right) - x}}{1 + x}}{1 - x} \]

      8. neg-sub0 50.7%

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

      9. sub-neg 50.7%

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

      10. mul-1-neg 50.7%

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

      11. distribute-neg-in 50.7%

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

      12. metadata-eval 50.7%

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

      13. mul-1-neg 50.7%

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

      14. remove-double-neg 50.7%

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

      15. +-commutative 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in x around inf 98.7%

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

      1. expm1-log1p-u 98.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{x}}{1 - x}\right)\right)} \]

      2. expm1-udef 49.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{2}{x}}{1 - x}\right)} - 1} \]

      3. frac-2neg 49.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-\frac{2}{x}}{-\left(1 - x\right)}}\right)} - 1 \]

      4. distribute-neg-frac 49.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{-2}{x}}}{-\left(1 - x\right)}\right)} - 1 \]

      5. metadata-eval 49.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{-2}}{x}}{-\left(1 - x\right)}\right)} - 1 \]

      6. neg-sub0 49.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-2}{x}}{\color{blue}{0 - \left(1 - x\right)}}\right)} - 1 \]

      7. associate--r- 49.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-2}{x}}{\color{blue}{\left(0 - 1\right) + x}}\right)} - 1 \]

      8. metadata-eval 49.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-2}{x}}{\color{blue}{-1} + x}\right)} - 1 \]

      9. metadata-eval 49.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-2}{x}}{\color{blue}{\left(-1\right)} + x}\right)} - 1 \]

      10. +-commutative 49.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-2}{x}}{\color{blue}{x + \left(-1\right)}}\right)} - 1 \]

      11. metadata-eval 49.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{-2}{x}}{x + \color{blue}{-1}}\right)} - 1 \]

   8. Applied egg-rr 49.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-2}{x}}{x + -1}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 98.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-2}{x}}{x + -1}\right)\right)} \]

      2. expm1-log1p 98.7%

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

      3. associate-/l/ 97.5%

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

   10. Simplified 97.5%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.76:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x \cdot \left(x + -1\right)}\\ \end{array} \]

Alternative 4: 74.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 85.6%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]

   2. Taylor expanded in x around 0 63.1%

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

   if 1 < x

   1. Initial program 50.5%

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

      1. *-un-lft-identity 50.5%

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

      2. *-commutative 50.5%

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

      3. frac-sub 50.7%

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

      4. associate-/r* 50.7%

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

      5. associate-/r/ 50.7%

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

      6. clear-num 50.7%

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

      7. associate-/r/ 50.7%

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

   3. Applied egg-rr 50.7%

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

      1. associate-*r/ 50.7%

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

      2. associate-*l/ 50.7%

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

      3. *-commutative 50.7%

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

      4. neg-mul-1 50.7%

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

      5. neg-sub0 50.7%

         \[\leadsto \frac{\frac{\color{blue}{0 - \left(x + \left(-2 - x\right)\right)}}{1 + x}}{1 - x} \]

      6. +-commutative 50.7%

         \[\leadsto \frac{\frac{0 - \color{blue}{\left(\left(-2 - x\right) + x\right)}}{1 + x}}{1 - x} \]

      7. associate--r+ 50.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(0 - \left(-2 - x\right)\right) - x}}{1 + x}}{1 - x} \]

      8. neg-sub0 50.7%

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

      9. sub-neg 50.7%

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

      10. mul-1-neg 50.7%

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

      11. distribute-neg-in 50.7%

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

      12. metadata-eval 50.7%

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

      13. mul-1-neg 50.7%

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

      14. remove-double-neg 50.7%

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

      15. +-commutative 50.7%

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

   5. Simplified 50.7%

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

      1. expm1-log1p-u 50.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left(2 + x\right) - x}{x + 1}}{1 - x}\right)\right)} \]

      2. expm1-udef 50.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\left(2 + x\right) - x}{x + 1}}{1 - x}\right)} - 1} \]

      3. associate-/l/ 50.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(2 + x\right) - x}{\left(1 - x\right) \cdot \left(x + 1\right)}}\right)} - 1 \]

      4. associate--l+ 50.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{2 + \left(x - x\right)}}{\left(1 - x\right) \cdot \left(x + 1\right)}\right)} - 1 \]

      5. +-inverses 50.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2 + \color{blue}{0}}{\left(1 - x\right) \cdot \left(x + 1\right)}\right)} - 1 \]

      6. metadata-eval 50.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{2}}{\left(1 - x\right) \cdot \left(x + 1\right)}\right)} - 1 \]

      7. *-commutative 50.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(x + 1\right) \cdot \left(1 - x\right)}}\right)} - 1 \]

   7. Applied egg-rr 50.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(x + 1\right) \cdot \left(1 - x\right)}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 98.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(x + 1\right) \cdot \left(1 - x\right)}\right)\right)} \]

      2. expm1-log1p 98.5%

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

      3. *-commutative 98.5%

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

      4. associate-/r* 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around inf 98.7%

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

4. Final simplification 71.9%

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

Alternative 5: 74.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.76:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.76) 2.0 (/ (/ 2.0 x) (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.76) {
		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 <= 0.76d0) 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 <= 0.76) {
		tmp = 2.0;
	} else {
		tmp = (2.0 / x) / (1.0 - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.76:
		tmp = 2.0
	else:
		tmp = (2.0 / x) / (1.0 - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.76)
		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 <= 0.76)
		tmp = 2.0;
	else
		tmp = (2.0 / x) / (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.76], 2.0, N[(N[(2.0 / x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.76000000000000001

   1. Initial program 85.6%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]

   2. Taylor expanded in x around 0 63.1%

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

   if 0.76000000000000001 < x

   1. Initial program 50.5%

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

      1. *-un-lft-identity 50.5%

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

      2. *-commutative 50.5%

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

      3. frac-sub 50.7%

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

      4. associate-/r* 50.7%

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

      5. associate-/r/ 50.7%

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

      6. clear-num 50.7%

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

      7. associate-/r/ 50.7%

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

   3. Applied egg-rr 50.7%

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

      1. associate-*r/ 50.7%

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

      2. associate-*l/ 50.7%

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

      3. *-commutative 50.7%

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

      4. neg-mul-1 50.7%

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

      5. neg-sub0 50.7%

         \[\leadsto \frac{\frac{\color{blue}{0 - \left(x + \left(-2 - x\right)\right)}}{1 + x}}{1 - x} \]

      6. +-commutative 50.7%

         \[\leadsto \frac{\frac{0 - \color{blue}{\left(\left(-2 - x\right) + x\right)}}{1 + x}}{1 - x} \]

      7. associate--r+ 50.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(0 - \left(-2 - x\right)\right) - x}}{1 + x}}{1 - x} \]

      8. neg-sub0 50.7%

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

      9. sub-neg 50.7%

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

      10. mul-1-neg 50.7%

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

      11. distribute-neg-in 50.7%

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

      12. metadata-eval 50.7%

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

      13. mul-1-neg 50.7%

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

      14. remove-double-neg 50.7%

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

      15. +-commutative 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in x around inf 98.7%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.76:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{1 - x}\\ \end{array} \]

Alternative 6: 99.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. *-un-lft-identity 77.0%

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

   2. *-commutative 77.0%

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

   3. frac-sub 78.1%

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

   4. associate-/r* 78.1%

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

   5. associate-/r/ 78.1%

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

   6. clear-num 78.1%

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

   7. associate-/r/ 78.1%

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

3. Applied egg-rr 78.1%

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

   1. associate-*r/ 78.1%

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

   2. associate-*l/ 78.1%

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

   3. *-commutative 78.1%

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

   4. neg-mul-1 78.1%

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

   5. neg-sub0 78.1%

      \[\leadsto \frac{\frac{\color{blue}{0 - \left(x + \left(-2 - x\right)\right)}}{1 + x}}{1 - x} \]

   6. +-commutative 78.1%

      \[\leadsto \frac{\frac{0 - \color{blue}{\left(\left(-2 - x\right) + x\right)}}{1 + x}}{1 - x} \]

   7. associate--r+ 78.1%

      \[\leadsto \frac{\frac{\color{blue}{\left(0 - \left(-2 - x\right)\right) - x}}{1 + x}}{1 - x} \]

   8. neg-sub0 78.1%

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

   9. sub-neg 78.1%

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

   10. mul-1-neg 78.1%

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

   11. distribute-neg-in 78.1%

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

   12. metadata-eval 78.1%

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

   13. mul-1-neg 78.1%

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

   14. remove-double-neg 78.1%

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

   15. +-commutative 78.1%

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

5. Simplified 78.1%

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

6. Taylor expanded in x around 0 99.9%

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

   1. div-inv 99.7%

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

   2. div-inv 99.7%

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

   3. associate-*l* 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate-*r* 99.7%

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

   2. associate-*r/ 99.7%

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

   3. metadata-eval 99.7%

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

   4. associate-*l/ 99.9%

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

   5. associate-*r/ 99.9%

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

   6. metadata-eval 99.9%

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

   7. associate-/l/ 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 7: 51.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = 2.0;
	else
		tmp = Float64(-2.0 / 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;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 85.6%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]

   2. Taylor expanded in x around 0 63.1%

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

   if 1 < x

   1. Initial program 50.5%

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

      1. *-un-lft-identity 50.5%

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

      2. *-commutative 50.5%

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

      3. frac-sub 50.7%

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

      4. associate-/r* 50.7%

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

      5. associate-/r/ 50.7%

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

      6. clear-num 50.7%

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

      7. associate-/r/ 50.7%

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

   3. Applied egg-rr 50.7%

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

      1. associate-*r/ 50.7%

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

      2. associate-*l/ 50.7%

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

      3. *-commutative 50.7%

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

      4. neg-mul-1 50.7%

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

      5. neg-sub0 50.7%

         \[\leadsto \frac{\frac{\color{blue}{0 - \left(x + \left(-2 - x\right)\right)}}{1 + x}}{1 - x} \]

      6. +-commutative 50.7%

         \[\leadsto \frac{\frac{0 - \color{blue}{\left(\left(-2 - x\right) + x\right)}}{1 + x}}{1 - x} \]

      7. associate--r+ 50.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(0 - \left(-2 - x\right)\right) - x}}{1 + x}}{1 - x} \]

      8. neg-sub0 50.7%

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

      9. sub-neg 50.7%

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

      10. mul-1-neg 50.7%

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

      11. distribute-neg-in 50.7%

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

      12. metadata-eval 50.7%

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

      13. mul-1-neg 50.7%

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

      14. remove-double-neg 50.7%

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

      15. +-commutative 50.7%

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

   5. Simplified 50.7%

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

      1. expm1-log1p-u 50.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left(2 + x\right) - x}{x + 1}}{1 - x}\right)\right)} \]

      2. expm1-udef 50.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\left(2 + x\right) - x}{x + 1}}{1 - x}\right)} - 1} \]

      3. associate-/l/ 50.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(2 + x\right) - x}{\left(1 - x\right) \cdot \left(x + 1\right)}}\right)} - 1 \]

      4. associate--l+ 50.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{2 + \left(x - x\right)}}{\left(1 - x\right) \cdot \left(x + 1\right)}\right)} - 1 \]

      5. +-inverses 50.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2 + \color{blue}{0}}{\left(1 - x\right) \cdot \left(x + 1\right)}\right)} - 1 \]

      6. metadata-eval 50.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{2}}{\left(1 - x\right) \cdot \left(x + 1\right)}\right)} - 1 \]

      7. *-commutative 50.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(x + 1\right) \cdot \left(1 - x\right)}}\right)} - 1 \]

   7. Applied egg-rr 50.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(x + 1\right) \cdot \left(1 - x\right)}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 98.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(x + 1\right) \cdot \left(1 - x\right)}\right)\right)} \]

      2. expm1-log1p 98.5%

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

      3. *-commutative 98.5%

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

      4. associate-/r* 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around inf 98.7%

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

   11. Taylor expanded in x around 0 6.6%

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

4. Final simplification 49.2%

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

Alternative 8: 10.6% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 77.0%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]

2. Taylor expanded in x around 0 47.8%

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

3. Taylor expanded in x around inf 10.3%

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

4. Final simplification 10.3%

   \[\leadsto 1 \]

Alternative 9: 49.9% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

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

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 77.0%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]

2. Taylor expanded in x around 0 48.3%

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

3. Final simplification 48.3%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))