3frac (problem 3.3.3)

Percentage Accurate: 84.5% → 99.9%

Time: 7.9s

Alternatives: 9

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: 84.5% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Simplified 87.6%

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

   1. frac-sub 56.8%

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

   2. frac-sub 58.9%

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

   3. *-un-lft-identity 58.9%

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

   4. distribute-rgt-in 58.9%

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

   5. neg-mul-1 58.9%

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

   6. sub-neg 58.9%

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

   7. *-rgt-identity 58.9%

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

   8. distribute-rgt-in 58.9%

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

   9. metadata-eval 58.9%

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

   10. metadata-eval 58.9%

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

   11. fma-def 58.9%

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

   12. metadata-eval 58.9%

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

   13. distribute-rgt-in 58.9%

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

   14. neg-mul-1 58.9%

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

   15. sub-neg 58.9%

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

5. Applied egg-rr 58.9%

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

   1. +-commutative 58.9%

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

   2. remove-double-neg 58.9%

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

   3. metadata-eval 58.9%

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

   4. distribute-neg-in 58.9%

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

   5. neg-mul-1 58.9%

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

   6. *-commutative 58.9%

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

   7. fma-udef 58.9%

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

   8. distribute-lft-neg-in 58.9%

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

   9. distribute-lft-neg-in 58.9%

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

   10. fma-udef 58.9%

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

   11. *-commutative 58.9%

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

   12. neg-mul-1 58.9%

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

   13. distribute-neg-in 58.9%

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

   14. remove-double-neg 58.9%

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

   15. metadata-eval 58.9%

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

   16. +-commutative 58.9%

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

7. Simplified 58.9%

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

8. Taylor expanded in x around 0 99.3%

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

   1. associate-/r* 99.9%

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

   2. div-inv 99.9%

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

   3. *-un-lft-identity 99.9%

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

   4. distribute-rgt-out-- 99.9%

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

   5. sub-neg 99.9%

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

   6. metadata-eval 99.9%

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

10. Applied egg-rr 99.9%

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

    1. div-inv 99.9%

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

    2. *-commutative 99.9%

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

    3. associate-/r* 99.9%

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

12. Applied egg-rr 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 83.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.65) (not (<= x 1.0)))
   (+ (/ 1.0 (+ x 1.0)) (/ -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 / (x + 1.0)) + (-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 / (x + 1.0d0)) + ((-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 / (x + 1.0)) + (-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 / (x + 1.0)) + (-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(x + 1.0)) + 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 / (x + 1.0)) + (-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[(x + 1.0), $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 76.7%

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

   3. Simplified 76.7%

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

      1. frac-2neg 76.7%

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

      2. frac-2neg 76.7%

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

      3. metadata-eval 76.7%

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

      4. frac-sub 18.7%

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

      5. metadata-eval 18.7%

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

      6. +-commutative 18.7%

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

      7. distribute-neg-in 18.7%

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

      8. metadata-eval 18.7%

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

      9. sub-neg 18.7%

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

      10. +-commutative 18.7%

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

      11. distribute-neg-in 18.7%

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

      12. metadata-eval 18.7%

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

      13. sub-neg 18.7%

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

   5. Applied egg-rr 18.7%

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

      1. cancel-sign-sub 18.7%

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

      2. *-commutative 18.7%

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

      3. neg-mul-1 18.7%

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

      4. unsub-neg 18.7%

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

      5. sub-neg 18.7%

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

      6. distribute-lft-in 18.7%

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

      7. *-rgt-identity 18.7%

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

      8. sqr-neg 18.7%

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

      9. unpow2 18.7%

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

      10. +-commutative 18.7%

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

      11. sub-neg 18.7%

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

      12. unpow2 18.7%

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

   7. Simplified 18.7%

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

   8. Taylor expanded in x around inf 76.0%

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

   if -0.650000000000000022 < x < 1

   1. Initial program 100.0%

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

   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 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 87.2%

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

Alternative 3: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.86 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{2}{x + 1}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.86) (not (<= x 1.0)))
   (/ (/ 2.0 (+ x 1.0)) (* x x))
   (- (* x -2.0) (/ 2.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.86) || !(x <= 1.0)) {
		tmp = (2.0 / (x + 1.0)) / (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 <= (-0.86d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (2.0d0 / (x + 1.0d0)) / (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 <= -0.86) || !(x <= 1.0)) {
		tmp = (2.0 / (x + 1.0)) / (x * x);
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.86) or not (x <= 1.0):
		tmp = (2.0 / (x + 1.0)) / (x * x)
	else:
		tmp = (x * -2.0) - (2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.86) || !(x <= 1.0))
		tmp = Float64(Float64(2.0 / Float64(x + 1.0)) / 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 <= -0.86) || ~((x <= 1.0)))
		tmp = (2.0 / (x + 1.0)) / (x * x);
	else
		tmp = (x * -2.0) - (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.86], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(2.0 / N[(x + 1.0), $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 < -0.859999999999999987 or 1 < x

   1. Initial program 76.7%

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

   3. Simplified 76.7%

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

      1. frac-sub 18.7%

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

      2. frac-sub 22.8%

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

      3. *-un-lft-identity 22.8%

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

      4. distribute-rgt-in 22.8%

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

      5. neg-mul-1 22.8%

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

      6. sub-neg 22.8%

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

      7. *-rgt-identity 22.8%

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

      8. distribute-rgt-in 22.8%

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

      9. metadata-eval 22.8%

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

      10. metadata-eval 22.8%

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

      11. fma-def 22.8%

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

      12. metadata-eval 22.8%

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

      13. distribute-rgt-in 22.8%

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

      14. neg-mul-1 22.8%

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

      15. sub-neg 22.8%

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

   5. Applied egg-rr 22.8%

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

      1. +-commutative 22.8%

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

      2. remove-double-neg 22.8%

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

      3. metadata-eval 22.8%

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

      4. distribute-neg-in 22.8%

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

      5. neg-mul-1 22.8%

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

      6. *-commutative 22.8%

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

      7. fma-udef 22.8%

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

      8. distribute-lft-neg-in 22.8%

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

      9. distribute-lft-neg-in 22.8%

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

      10. fma-udef 22.8%

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

      11. *-commutative 22.8%

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

      12. neg-mul-1 22.8%

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

      13. distribute-neg-in 22.8%

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

      14. remove-double-neg 22.8%

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

      15. metadata-eval 22.8%

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

      16. +-commutative 22.8%

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

   7. Simplified 22.8%

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

   8. Taylor expanded in x around 0 98.6%

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

      1. expm1-log1p-u 98.6%

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

      2. expm1-udef 76.5%

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

      3. *-un-lft-identity 76.5%

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

      4. distribute-rgt-out-- 76.5%

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

      5. sub-neg 76.5%

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

      6. metadata-eval 76.5%

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

   10. Applied egg-rr 76.5%

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

       1. expm1-def 98.6%

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

       2. expm1-log1p 98.6%

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

       3. associate-/r* 99.9%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in x around inf 98.8%

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

       1. unpow2 98.8%

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

   15. Simplified 98.8%

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

   if -0.859999999999999987 < x < 1

   1. Initial program 100.0%

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

   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 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.86 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{2}{x + 1}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \]

Alternative 4: 76.2% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ -1.0 (* x x))
   (if (<= x 1.0) (- (* x -2.0) (/ 2.0 x)) (/ (/ 1.0 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 = (1.0 / 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 = (1.0d0 / 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 = (1.0 / 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 = (1.0 / 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(Float64(1.0 / 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 = (1.0 / 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[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in x around inf 80.5%

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

   5. Taylor expanded in x around inf 59.7%

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

      1. unpow2 59.7%

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

   7. Simplified 59.7%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   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 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

   if 1 < x

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in x around inf 71.4%

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

   5. Taylor expanded in x around inf 54.8%

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

      1. unpow2 54.8%

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

   7. Simplified 54.8%

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

      1. associate-/r* 54.8%

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

      2. div-inv 54.8%

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

   9. Applied egg-rr 54.8%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 56.0%

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

       3. frac-times 56.0%

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

       4. metadata-eval 56.0%

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

       5. metadata-eval 56.0%

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

       6. frac-times 56.0%

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

       7. sqrt-unprod 56.0%

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

       8. add-sqr-sqrt 56.0%

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

       9. un-div-inv 56.0%

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

   11. Applied egg-rr 56.0%

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

4. Final simplification 77.6%

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

Alternative 5: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Simplified 87.6%

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

   1. frac-sub 56.8%

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

   2. frac-sub 58.9%

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

   3. *-un-lft-identity 58.9%

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

   4. distribute-rgt-in 58.9%

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

   5. neg-mul-1 58.9%

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

   6. sub-neg 58.9%

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

   7. *-rgt-identity 58.9%

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

   8. distribute-rgt-in 58.9%

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

   9. metadata-eval 58.9%

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

   10. metadata-eval 58.9%

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

   11. fma-def 58.9%

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

   12. metadata-eval 58.9%

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

   13. distribute-rgt-in 58.9%

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

   14. neg-mul-1 58.9%

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

   15. sub-neg 58.9%

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

5. Applied egg-rr 58.9%

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

   1. +-commutative 58.9%

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

   2. remove-double-neg 58.9%

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

   3. metadata-eval 58.9%

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

   4. distribute-neg-in 58.9%

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

   5. neg-mul-1 58.9%

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

   6. *-commutative 58.9%

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

   7. fma-udef 58.9%

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

   8. distribute-lft-neg-in 58.9%

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

   9. distribute-lft-neg-in 58.9%

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

   10. fma-udef 58.9%

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

   11. *-commutative 58.9%

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

   12. neg-mul-1 58.9%

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

   13. distribute-neg-in 58.9%

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

   14. remove-double-neg 58.9%

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

   15. metadata-eval 58.9%

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

   16. +-commutative 58.9%

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

7. Simplified 58.9%

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

8. Taylor expanded in x around 0 99.3%

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

9. Final simplification 99.3%

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

Alternative 6: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Simplified 87.6%

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

   1. frac-sub 56.8%

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

   2. frac-sub 58.9%

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

   3. *-un-lft-identity 58.9%

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

   4. distribute-rgt-in 58.9%

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

   5. neg-mul-1 58.9%

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

   6. sub-neg 58.9%

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

   7. *-rgt-identity 58.9%

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

   8. distribute-rgt-in 58.9%

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

   9. metadata-eval 58.9%

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

   10. metadata-eval 58.9%

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

   11. fma-def 58.9%

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

   12. metadata-eval 58.9%

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

   13. distribute-rgt-in 58.9%

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

   14. neg-mul-1 58.9%

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

   15. sub-neg 58.9%

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

5. Applied egg-rr 58.9%

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

   1. +-commutative 58.9%

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

   2. remove-double-neg 58.9%

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

   3. metadata-eval 58.9%

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

   4. distribute-neg-in 58.9%

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

   5. neg-mul-1 58.9%

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

   6. *-commutative 58.9%

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

   7. fma-udef 58.9%

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

   8. distribute-lft-neg-in 58.9%

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

   9. distribute-lft-neg-in 58.9%

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

   10. fma-udef 58.9%

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

   11. *-commutative 58.9%

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

   12. neg-mul-1 58.9%

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

   13. distribute-neg-in 58.9%

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

   14. remove-double-neg 58.9%

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

   15. metadata-eval 58.9%

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

   16. +-commutative 58.9%

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

7. Simplified 58.9%

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

8. Taylor expanded in x around 0 99.3%

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

   1. expm1-log1p-u 77.0%

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

   2. expm1-udef 65.3%

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

   3. *-un-lft-identity 65.3%

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

   4. distribute-rgt-out-- 65.3%

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

   5. sub-neg 65.3%

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

   6. metadata-eval 65.3%

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

10. Applied egg-rr 65.3%

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

    1. expm1-def 77.0%

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

    2. expm1-log1p 99.3%

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

    3. associate-/r* 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

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

Alternative 7: 75.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.37\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.37))) (/ -1.0 (* x x)) (/ -2.0 x)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.37)) {
		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 <= 0.37d0))) 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 <= 0.37)) {
		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 <= 0.37):
		tmp = -1.0 / (x * x)
	else:
		tmp = -2.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.37))
		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 <= 0.37)))
		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, 0.37]], $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 0.37 < x

   1. Initial program 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in x around inf 76.0%

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

   5. Taylor expanded in x around inf 57.3%

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

      1. unpow2 57.3%

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

   7. Simplified 57.3%

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

   if -1 < x < 0.37

   1. Initial program 100.0%

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

   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 98.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.37\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 8: 76.0% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) (/ -1.0 (* x x)) (if (<= x 1.0) (/ -2.0 x) (/ (/ 1.0 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 = (1.0 / 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 = (1.0d0 / 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 = (1.0 / 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 = (1.0 / 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(Float64(1.0 / 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 = (1.0 / 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[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in x around inf 80.5%

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

   5. Taylor expanded in x around inf 59.7%

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

      1. unpow2 59.7%

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

   7. Simplified 59.7%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   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 98.8%

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

   if 1 < x

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in x around inf 71.4%

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

   5. Taylor expanded in x around inf 54.8%

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

      1. unpow2 54.8%

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

   7. Simplified 54.8%

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

      1. associate-/r* 54.8%

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

      2. div-inv 54.8%

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

   9. Applied egg-rr 54.8%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 56.0%

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

       3. frac-times 56.0%

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

       4. metadata-eval 56.0%

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

       5. metadata-eval 56.0%

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

       6. frac-times 56.0%

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

       7. sqrt-unprod 56.0%

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

       8. add-sqr-sqrt 56.0%

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

       9. un-div-inv 56.0%

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

   11. Applied egg-rr 56.0%

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

4. Final simplification 77.0%

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

Alternative 9: 51.4% 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 87.6%

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

3. Simplified 87.6%

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

4. Taylor expanded in x around 0 49.1%

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

5. Final simplification 49.1%

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

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 2023278 
(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))))