3frac (problem 3.3.3)

Percentage Accurate: 84.7% → 99.9%

Time: 6.3s

Alternatives: 11

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. associate-+l- 86.0%

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

   2. sub-neg 86.0%

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

   3. neg-mul-1 86.0%

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

   4. metadata-eval 86.0%

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

   5. cancel-sign-sub-inv 86.0%

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

   6. +-commutative 86.0%

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

   7. *-lft-identity 86.0%

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

   8. sub-neg 86.0%

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

   9. metadata-eval 86.0%

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

3. Simplified 86.0%

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

   1. frac-sub 57.1%

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

   2. frac-sub 59.6%

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

   3. *-un-lft-identity 59.6%

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

   4. distribute-rgt-in 59.7%

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

   5. neg-mul-1 59.7%

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

   6. sub-neg 59.7%

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

   7. *-rgt-identity 59.7%

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

   8. distribute-rgt-in 59.7%

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

   9. metadata-eval 59.7%

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

   10. metadata-eval 59.7%

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

   11. fma-def 59.7%

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

   12. metadata-eval 59.7%

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

   13. distribute-rgt-in 59.7%

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

   14. neg-mul-1 59.7%

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

   15. sub-neg 59.7%

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

5. Applied egg-rr 59.7%

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

6. Taylor expanded in x around 0 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 83.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.65 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.65) (not (<= x 1.0)))
   (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))
   (- (* x -2.0) (/ 2.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.65) || !(x <= 1.0)) {
		tmp = (1.0 / (1.0 + x)) + (-1.0 / x);
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.65d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    else
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.65) || !(x <= 1.0)) {
		tmp = (1.0 / (1.0 + x)) + (-1.0 / x);
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.65) or not (x <= 1.0):
		tmp = (1.0 / (1.0 + x)) + (-1.0 / x)
	else:
		tmp = (x * -2.0) - (2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.65) || !(x <= 1.0))
		tmp = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x));
	else
		tmp = Float64(Float64(x * -2.0) - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.65) || ~((x <= 1.0)))
		tmp = (1.0 / (1.0 + x)) + (-1.0 / x);
	else
		tmp = (x * -2.0) - (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.65], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.650000000000000022 or 1 < x

   1. Initial program 72.9%

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

      1. associate-+l- 72.9%

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

      2. sub-neg 72.9%

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

      3. neg-mul-1 72.9%

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

      4. metadata-eval 72.9%

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

      5. cancel-sign-sub-inv 72.9%

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

      6. +-commutative 72.9%

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

      7. *-lft-identity 72.9%

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

      8. sub-neg 72.9%

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

      9. metadata-eval 72.9%

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

   3. Simplified 72.9%

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

   4. Taylor expanded in x around inf 71.8%

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

   if -0.650000000000000022 < x < 1

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. metadata-eval 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. *-lft-identity 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.1%

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

      1. associate-*r/ 99.1%

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

      2. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.65 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \]

Alternative 3: 98.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.85) (not (<= x 1.0)))
   (/ 2.0 (* (+ 1.0 x) (* x x)))
   (- (* x -2.0) (/ 2.0 x))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.85d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = 2.0d0 / ((1.0d0 + x) * (x * x))
    else
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -0.85) or not (x <= 1.0):
		tmp = 2.0 / ((1.0 + x) * (x * x))
	else:
		tmp = (x * -2.0) - (2.0 / x)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -0.85], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(2.0 / N[(N[(1.0 + x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.849999999999999978 or 1 < x

   1. Initial program 72.9%

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

      1. associate-+l- 72.9%

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

      2. sub-neg 72.9%

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

      3. neg-mul-1 72.9%

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

      4. metadata-eval 72.9%

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

      5. cancel-sign-sub-inv 72.9%

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

      6. +-commutative 72.9%

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

      7. *-lft-identity 72.9%

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

      8. sub-neg 72.9%

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

      9. metadata-eval 72.9%

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

   3. Simplified 72.9%

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

      1. frac-sub 16.8%

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

      2. frac-sub 21.8%

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

      3. *-un-lft-identity 21.8%

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

      4. distribute-rgt-in 21.8%

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

      5. neg-mul-1 21.8%

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

      6. sub-neg 21.8%

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

      7. *-rgt-identity 21.8%

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

      8. distribute-rgt-in 21.8%

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

      9. metadata-eval 21.8%

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

      10. metadata-eval 21.8%

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

      11. fma-def 21.8%

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

      12. metadata-eval 21.8%

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

      13. distribute-rgt-in 21.8%

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

      14. neg-mul-1 21.8%

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

      15. sub-neg 21.8%

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

   5. Applied egg-rr 21.8%

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

   6. Taylor expanded in x around 0 99.5%

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

   7. Taylor expanded in x around inf 97.9%

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

      1. unpow2 97.9%

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

   9. Simplified 97.9%

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

   if -0.849999999999999978 < x < 1

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. metadata-eval 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. *-lft-identity 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.1%

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

      1. associate-*r/ 99.1%

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

      2. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

4. Final simplification 98.5%

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

Alternative 4: 98.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.85) (not (<= x 1.0)))
   (/ (/ 2.0 (+ 1.0 x)) (* x x))
   (- (* x -2.0) (/ 2.0 x))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.85d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (2.0d0 / (1.0d0 + x)) / (x * x)
    else
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -0.85) or not (x <= 1.0):
		tmp = (2.0 / (1.0 + x)) / (x * x)
	else:
		tmp = (x * -2.0) - (2.0 / x)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -0.85], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.849999999999999978 or 1 < x

   1. Initial program 72.9%

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

      1. associate-+l- 72.9%

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

      2. sub-neg 72.9%

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

      3. neg-mul-1 72.9%

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

      4. metadata-eval 72.9%

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

      5. cancel-sign-sub-inv 72.9%

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

      6. +-commutative 72.9%

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

      7. *-lft-identity 72.9%

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

      8. sub-neg 72.9%

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

      9. metadata-eval 72.9%

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

   3. Simplified 72.9%

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

      1. frac-sub 16.8%

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

      2. frac-sub 21.8%

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

      3. *-un-lft-identity 21.8%

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

      4. distribute-rgt-in 21.8%

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

      5. neg-mul-1 21.8%

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

      6. sub-neg 21.8%

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

      7. *-rgt-identity 21.8%

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

      8. distribute-rgt-in 21.8%

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

      9. metadata-eval 21.8%

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

      10. metadata-eval 21.8%

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

      11. fma-def 21.8%

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

      12. metadata-eval 21.8%

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

      13. distribute-rgt-in 21.8%

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

      14. neg-mul-1 21.8%

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

      15. sub-neg 21.8%

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

   5. Applied egg-rr 21.8%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/r* 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in x around inf 98.2%

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

       1. unpow2 97.9%

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

   11. Simplified 98.2%

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

   if -0.849999999999999978 < x < 1

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. metadata-eval 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. *-lft-identity 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.1%

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

      1. associate-*r/ 99.1%

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

      2. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

4. Final simplification 98.7%

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

Alternative 5: 98.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -0.849999999999999978

   1. Initial program 69.9%

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

      1. associate-+l- 69.9%

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

      2. sub-neg 69.9%

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

      3. neg-mul-1 69.9%

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

      4. metadata-eval 69.9%

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

      5. cancel-sign-sub-inv 69.9%

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

      6. +-commutative 69.9%

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

      7. *-lft-identity 69.9%

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

      8. sub-neg 69.9%

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

      9. metadata-eval 69.9%

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

   3. Simplified 69.9%

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

      1. frac-sub 15.6%

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

      2. frac-sub 21.6%

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

      3. *-un-lft-identity 21.6%

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

      4. distribute-rgt-in 21.7%

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

      5. neg-mul-1 21.7%

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

      6. sub-neg 21.7%

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

      7. *-rgt-identity 21.7%

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

      8. distribute-rgt-in 21.7%

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

      9. metadata-eval 21.7%

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

      10. metadata-eval 21.7%

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

      11. fma-def 21.7%

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

      12. metadata-eval 21.7%

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

      13. distribute-rgt-in 21.7%

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

      14. neg-mul-1 21.7%

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

      15. sub-neg 21.7%

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

   5. Applied egg-rr 21.7%

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

   6. Taylor expanded in x around 0 99.3%

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

   7. Taylor expanded in x around inf 98.7%

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

      1. unpow2 98.7%

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

   9. Simplified 98.7%

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

       1. associate-/r* 99.2%

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

       2. div-inv 99.2%

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

       3. times-frac 99.2%

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

   11. Applied egg-rr 99.2%

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

   if -0.849999999999999978 < x < 1

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. metadata-eval 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. *-lft-identity 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.1%

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

      1. associate-*r/ 99.1%

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

      2. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

   if 1 < x

   1. Initial program 75.8%

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

      1. associate-+l- 75.8%

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

      2. sub-neg 75.8%

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

      3. neg-mul-1 75.8%

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

      4. metadata-eval 75.8%

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

      5. cancel-sign-sub-inv 75.8%

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

      6. +-commutative 75.8%

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

      7. *-lft-identity 75.8%

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

      8. sub-neg 75.8%

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

      9. metadata-eval 75.8%

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

   3. Simplified 75.8%

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

      1. frac-sub 17.9%

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

      2. frac-sub 21.9%

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

      3. *-un-lft-identity 21.9%

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

      4. distribute-rgt-in 21.9%

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

      5. neg-mul-1 21.9%

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

      6. sub-neg 21.9%

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

      7. *-rgt-identity 21.9%

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

      8. distribute-rgt-in 21.9%

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

      9. metadata-eval 21.9%

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

      10. metadata-eval 21.9%

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

      11. fma-def 21.9%

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

      12. metadata-eval 21.9%

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

      13. distribute-rgt-in 21.9%

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

      14. neg-mul-1 21.9%

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

      15. sub-neg 21.9%

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

   5. Applied egg-rr 21.9%

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

   6. Taylor expanded in x around 0 99.7%

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

   7. Taylor expanded in x around inf 97.3%

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

      1. unpow2 97.3%

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

   9. Simplified 97.3%

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

4. Final simplification 98.6%

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

Alternative 6: 77.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 69.9%

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

      1. associate-+l- 69.9%

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

      2. sub-neg 69.9%

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

      3. neg-mul-1 69.9%

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

      4. metadata-eval 69.9%

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

      5. cancel-sign-sub-inv 69.9%

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

      6. +-commutative 69.9%

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

      7. *-lft-identity 69.9%

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

      8. sub-neg 69.9%

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

      9. metadata-eval 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in x around inf 70.1%

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

   5. Taylor expanded in x around inf 52.4%

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

      1. unpow2 52.4%

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

   7. Simplified 52.4%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. metadata-eval 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. *-lft-identity 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.1%

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

      1. associate-*r/ 99.1%

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

      2. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

   if 1 < x

   1. Initial program 75.8%

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

      1. associate-+l- 75.8%

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

      2. sub-neg 75.8%

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

      3. neg-mul-1 75.8%

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

      4. metadata-eval 75.8%

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

      5. cancel-sign-sub-inv 75.8%

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

      6. +-commutative 75.8%

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

      7. *-lft-identity 75.8%

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

      8. sub-neg 75.8%

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

      9. metadata-eval 75.8%

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

   3. Simplified 75.8%

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

      1. frac-sub 17.9%

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

      2. frac-sub 21.9%

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

      3. *-un-lft-identity 21.9%

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

      4. distribute-rgt-in 21.9%

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

      5. neg-mul-1 21.9%

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

      6. sub-neg 21.9%

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

      7. *-rgt-identity 21.9%

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

      8. distribute-rgt-in 21.9%

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

      9. metadata-eval 21.9%

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

      10. metadata-eval 21.9%

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

      11. fma-def 21.9%

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

      12. metadata-eval 21.9%

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

      13. distribute-rgt-in 21.9%

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

      14. neg-mul-1 21.9%

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

      15. sub-neg 21.9%

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

   5. Applied egg-rr 21.9%

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

   6. Taylor expanded in x around 0 99.7%

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

   7. Taylor expanded in x around inf 97.3%

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

      1. unpow2 97.3%

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

   9. Simplified 97.3%

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

   10. Taylor expanded in x around 0 58.1%

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

       1. unpow2 58.1%

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

   12. Simplified 58.1%

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

4. Final simplification 76.5%

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

Alternative 7: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. associate-+l- 86.0%

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

   2. sub-neg 86.0%

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

   3. neg-mul-1 86.0%

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

   4. metadata-eval 86.0%

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

   5. cancel-sign-sub-inv 86.0%

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

   6. +-commutative 86.0%

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

   7. *-lft-identity 86.0%

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

   8. sub-neg 86.0%

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

   9. metadata-eval 86.0%

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

3. Simplified 86.0%

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

   1. frac-sub 57.1%

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

   2. frac-sub 59.6%

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

   3. *-un-lft-identity 59.6%

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

   4. distribute-rgt-in 59.7%

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

   5. neg-mul-1 59.7%

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

   6. sub-neg 59.7%

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

   7. *-rgt-identity 59.7%

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

   8. distribute-rgt-in 59.7%

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

   9. metadata-eval 59.7%

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

   10. metadata-eval 59.7%

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

   11. fma-def 59.7%

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

   12. metadata-eval 59.7%

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

   13. distribute-rgt-in 59.7%

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

   14. neg-mul-1 59.7%

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

   15. sub-neg 59.7%

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

5. Applied egg-rr 59.7%

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

6. Taylor expanded in x around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 8: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. associate-+l- 86.0%

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

   2. sub-neg 86.0%

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

   3. neg-mul-1 86.0%

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

   4. metadata-eval 86.0%

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

   5. cancel-sign-sub-inv 86.0%

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

   6. +-commutative 86.0%

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

   7. *-lft-identity 86.0%

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

   8. sub-neg 86.0%

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

   9. metadata-eval 86.0%

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

3. Simplified 86.0%

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

   1. frac-sub 57.1%

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

   2. frac-sub 59.6%

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

   3. *-un-lft-identity 59.6%

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

   4. distribute-rgt-in 59.7%

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

   5. neg-mul-1 59.7%

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

   6. sub-neg 59.7%

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

   7. *-rgt-identity 59.7%

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

   8. distribute-rgt-in 59.7%

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

   9. metadata-eval 59.7%

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

   10. metadata-eval 59.7%

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

   11. fma-def 59.7%

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

   12. metadata-eval 59.7%

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

   13. distribute-rgt-in 59.7%

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

   14. neg-mul-1 59.7%

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

   15. sub-neg 59.7%

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

5. Applied egg-rr 59.7%

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

6. Taylor expanded in x around 0 99.7%

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

   1. expm1-log1p-u 72.6%

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

   2. expm1-udef 58.8%

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

   3. associate-/r* 58.8%

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

8. Applied egg-rr 58.8%

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

   1. expm1-def 72.7%

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

   2. expm1-log1p 99.9%

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

   3. associate-/l/ 99.7%

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

   4. associate-/r* 99.9%

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

   5. +-commutative 99.9%

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

10. Simplified 99.9%

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

11. Final simplification 99.9%

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

Alternative 9: 76.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.38\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.38))) (/ -1.0 (* x x)) (/ -2.0 x)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.38)) {
		tmp = -1.0 / (x * x);
	} 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)) .or. (.not. (x <= 0.38d0))) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = (-2.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.38)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = -2.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.38):
		tmp = -1.0 / (x * x)
	else:
		tmp = -2.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.38))
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(-2.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.38)))
		tmp = -1.0 / (x * x);
	else
		tmp = -2.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.38]], $MachinePrecision]], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(-2.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.38 < x

   1. Initial program 73.1%

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

      1. associate-+l- 73.1%

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

      2. sub-neg 73.1%

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

      3. neg-mul-1 73.1%

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

      4. metadata-eval 73.1%

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

      5. cancel-sign-sub-inv 73.1%

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

      6. +-commutative 73.1%

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

      7. *-lft-identity 73.1%

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

      8. sub-neg 73.1%

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

      9. metadata-eval 73.1%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in x around inf 71.4%

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

   5. Taylor expanded in x around inf 54.4%

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

      1. unpow2 54.4%

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

   7. Simplified 54.4%

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

   if -1 < x < 0.38

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. metadata-eval 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. *-lft-identity 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.5%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.38\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 10: 76.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 69.9%

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

      1. associate-+l- 69.9%

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

      2. sub-neg 69.9%

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

      3. neg-mul-1 69.9%

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

      4. metadata-eval 69.9%

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

      5. cancel-sign-sub-inv 69.9%

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

      6. +-commutative 69.9%

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

      7. *-lft-identity 69.9%

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

      8. sub-neg 69.9%

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

      9. metadata-eval 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in x around inf 70.1%

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

   5. Taylor expanded in x around inf 52.4%

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

      1. unpow2 52.4%

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

   7. Simplified 52.4%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. metadata-eval 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. *-lft-identity 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.9%

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

   if 1 < x

   1. Initial program 75.8%

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

      1. associate-+l- 75.8%

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

      2. sub-neg 75.8%

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

      3. neg-mul-1 75.8%

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

      4. metadata-eval 75.8%

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

      5. cancel-sign-sub-inv 75.8%

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

      6. +-commutative 75.8%

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

      7. *-lft-identity 75.8%

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

      8. sub-neg 75.8%

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

      9. metadata-eval 75.8%

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

   3. Simplified 75.8%

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

      1. frac-sub 17.9%

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

      2. frac-sub 21.9%

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

      3. *-un-lft-identity 21.9%

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

      4. distribute-rgt-in 21.9%

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

      5. neg-mul-1 21.9%

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

      6. sub-neg 21.9%

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

      7. *-rgt-identity 21.9%

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

      8. distribute-rgt-in 21.9%

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

      9. metadata-eval 21.9%

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

      10. metadata-eval 21.9%

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

      11. fma-def 21.9%

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

      12. metadata-eval 21.9%

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

      13. distribute-rgt-in 21.9%

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

      14. neg-mul-1 21.9%

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

      15. sub-neg 21.9%

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

   5. Applied egg-rr 21.9%

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

   6. Taylor expanded in x around 0 99.7%

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

   7. Taylor expanded in x around inf 97.3%

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

      1. unpow2 97.3%

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

   9. Simplified 97.3%

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

   10. Taylor expanded in x around 0 58.1%

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

       1. unpow2 58.1%

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

   12. Simplified 58.1%

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

4. Final simplification 76.4%

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

Alternative 11: 52.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. associate-+l- 86.0%

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

   2. sub-neg 86.0%

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

   3. neg-mul-1 86.0%

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

   4. metadata-eval 86.0%

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

   5. cancel-sign-sub-inv 86.0%

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

   6. +-commutative 86.0%

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

   7. *-lft-identity 86.0%

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

   8. sub-neg 86.0%

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

   9. metadata-eval 86.0%

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

3. Simplified 86.0%

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

4. Taylor expanded in x around 0 50.5%

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

5. Final simplification 50.5%

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

Developer target: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023178 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))