3frac (problem 3.3.3)

Percentage Accurate: 70.2% → 99.8%

Time: 10.9s

Alternatives: 10

Speedup: 1.4×

Specification

\[\left|x\right| > 1\]

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: 70.2% 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: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

   1. +-commutative 64.3%

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

   2. associate-+r- 64.3%

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

   3. sub-neg 64.3%

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

   4. remove-double-neg 64.3%

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

   5. neg-sub0 64.3%

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

   6. associate-+l- 64.3%

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

   7. neg-sub0 64.3%

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

   8. distribute-neg-frac2 64.3%

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

   9. distribute-frac-neg2 64.3%

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

   10. associate-+r+ 64.3%

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

   11. +-commutative 64.3%

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

   12. remove-double-neg 64.3%

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

   13. distribute-neg-frac2 64.3%

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

   14. sub0-neg 64.3%

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

   15. associate-+l- 64.3%

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

   16. neg-sub0 64.3%

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

3. Simplified 64.3%

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

   1. frac-sub 20.9%

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

   2. frac-add 21.8%

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

   3. *-un-lft-identity 21.8%

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

   4. fma-define 19.8%

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

   5. *-rgt-identity 19.8%

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

   6. fmm-def 19.8%

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

6. Applied egg-rr 19.8%

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

   1. fmm-undef 19.8%

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

   2. associate-*r* 19.8%

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

   3. *-commutative 19.8%

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

8. Simplified 19.8%

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

9. Taylor expanded in x around 0 99.6%

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

    1. associate-/r* 99.8%

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

    2. div-inv 99.7%

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

11. Applied egg-rr 99.7%

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

    1. associate-*l/ 99.8%

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

    2. associate-/r* 99.8%

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

    3. un-div-inv 99.8%

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

13. Applied egg-rr 99.8%

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

14. Final simplification 99.8%

    \[\leadsto \frac{\frac{\frac{-2}{-1 - x}}{x}}{-1 + x} \]
15. Add Preprocessing

Alternative 2: 99.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

   1. +-commutative 64.3%

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

   2. associate-+r- 64.3%

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

   3. sub-neg 64.3%

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

   4. remove-double-neg 64.3%

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

   5. neg-sub0 64.3%

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

   6. associate-+l- 64.3%

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

   7. neg-sub0 64.3%

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

   8. distribute-neg-frac2 64.3%

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

   9. distribute-frac-neg2 64.3%

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

   10. associate-+r+ 64.3%

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

   11. +-commutative 64.3%

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

   12. remove-double-neg 64.3%

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

   13. distribute-neg-frac2 64.3%

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

   14. sub0-neg 64.3%

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

   15. associate-+l- 64.3%

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

   16. neg-sub0 64.3%

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

3. Simplified 64.3%

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

   1. frac-sub 20.9%

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

   2. frac-add 21.8%

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

   3. *-un-lft-identity 21.8%

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

   4. fma-define 19.8%

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

   5. *-rgt-identity 19.8%

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

   6. fmm-def 19.8%

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

6. Applied egg-rr 19.8%

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

   1. fmm-undef 19.8%

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

   2. associate-*r* 19.8%

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

   3. *-commutative 19.8%

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

8. Simplified 19.8%

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

9. Taylor expanded in x around 0 99.6%

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

10. Final simplification 99.6%

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

Alternative 3: 99.8% accurate, 1.4× speedup

Math

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

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(-2.0 / Float64(x * Float64(-1.0 + x))) / Float64(-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[(-2.0 / N[(x * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.3%

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

   1. +-commutative 64.3%

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

   2. associate-+r- 64.3%

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

   3. sub-neg 64.3%

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

   4. remove-double-neg 64.3%

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

   5. neg-sub0 64.3%

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

   6. associate-+l- 64.3%

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

   7. neg-sub0 64.3%

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

   8. distribute-neg-frac2 64.3%

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

   9. distribute-frac-neg2 64.3%

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

   10. associate-+r+ 64.3%

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

   11. +-commutative 64.3%

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

   12. remove-double-neg 64.3%

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

   13. distribute-neg-frac2 64.3%

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

   14. sub0-neg 64.3%

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

   15. associate-+l- 64.3%

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

   16. neg-sub0 64.3%

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

3. Simplified 64.3%

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

   1. frac-sub 20.9%

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

   2. frac-add 21.8%

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

   3. *-un-lft-identity 21.8%

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

   4. fma-define 19.8%

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

   5. *-rgt-identity 19.8%

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

   6. fmm-def 19.8%

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

6. Applied egg-rr 19.8%

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

   1. fmm-undef 19.8%

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

   2. associate-*r* 19.8%

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

   3. *-commutative 19.8%

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

8. Simplified 19.8%

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

9. Taylor expanded in x around 0 99.6%

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

    1. div-inv 99.6%

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

11. Applied egg-rr 99.6%

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

    1. associate-*r/ 99.6%

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

    2. metadata-eval 99.6%

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

    3. associate-/r* 99.8%

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

    4. associate-/r* 99.8%

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

13. Simplified 99.8%

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

15. Applied egg-rr 99.8%

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

    1. distribute-neg-frac 99.8%

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

    2. metadata-eval 99.8%

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

    3. associate-*r/ 99.8%

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

    4. metadata-eval 99.8%

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

17. Simplified 99.8%

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

18. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

   1. +-commutative 64.3%

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

   2. associate-+r- 64.3%

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

   3. sub-neg 64.3%

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

   4. remove-double-neg 64.3%

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

   5. neg-sub0 64.3%

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

   6. associate-+l- 64.3%

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

   7. neg-sub0 64.3%

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

   8. distribute-neg-frac2 64.3%

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

   9. distribute-frac-neg2 64.3%

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

   10. associate-+r+ 64.3%

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

   11. +-commutative 64.3%

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

   12. remove-double-neg 64.3%

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

   13. distribute-neg-frac2 64.3%

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

   14. sub0-neg 64.3%

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

   15. associate-+l- 64.3%

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

   16. neg-sub0 64.3%

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

3. Simplified 64.3%

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

   1. frac-sub 20.9%

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

   2. frac-add 21.8%

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

   3. *-un-lft-identity 21.8%

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

   4. fma-define 19.8%

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

   5. *-rgt-identity 19.8%

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

   6. fmm-def 19.8%

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

6. Applied egg-rr 19.8%

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

   1. fmm-undef 19.8%

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

   2. associate-*r* 19.8%

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

   3. *-commutative 19.8%

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

8. Simplified 19.8%

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

9. Taylor expanded in x around 0 99.6%

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

    1. associate-/r* 99.8%

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

    2. div-inv 99.7%

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

11. Applied egg-rr 99.7%

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

    1. associate-*l/ 99.8%

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

    2. un-div-inv 99.8%

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

13. Applied egg-rr 99.8%

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

14. Final simplification 99.8%

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

Alternative 5: 99.8% 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 / x) / Float64(-1.0 + x)) / Float64(-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 / x), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.3%

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

   1. +-commutative 64.3%

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

   2. associate-+r- 64.3%

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

   3. sub-neg 64.3%

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

   4. remove-double-neg 64.3%

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

   5. neg-sub0 64.3%

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

   6. associate-+l- 64.3%

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

   7. neg-sub0 64.3%

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

   8. distribute-neg-frac2 64.3%

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

   9. distribute-frac-neg2 64.3%

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

   10. associate-+r+ 64.3%

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

   11. +-commutative 64.3%

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

   12. remove-double-neg 64.3%

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

   13. distribute-neg-frac2 64.3%

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

   14. sub0-neg 64.3%

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

   15. associate-+l- 64.3%

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

   16. neg-sub0 64.3%

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

3. Simplified 64.3%

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

   1. frac-sub 20.9%

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

   2. frac-add 21.8%

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

   3. *-un-lft-identity 21.8%

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

   4. fma-define 19.8%

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

   5. *-rgt-identity 19.8%

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

   6. fmm-def 19.8%

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

6. Applied egg-rr 19.8%

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

   1. fmm-undef 19.8%

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

   2. associate-*r* 19.8%

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

   3. *-commutative 19.8%

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

8. Simplified 19.8%

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

9. Taylor expanded in x around 0 99.6%

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

    1. div-inv 99.6%

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

11. Applied egg-rr 99.6%

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

    1. associate-*r/ 99.6%

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

    2. metadata-eval 99.6%

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

    3. associate-/r* 99.8%

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

    4. associate-/r* 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

    \[\leadsto \frac{\frac{\frac{-2}{x}}{-1 + x}}{-1 - x} \]
15. Add Preprocessing

Alternative 6: 68.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

   1. +-commutative 64.3%

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

   2. associate-+r- 64.3%

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

   3. sub-neg 64.3%

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

   4. remove-double-neg 64.3%

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

   5. neg-sub0 64.3%

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

   6. associate-+l- 64.3%

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

   7. neg-sub0 64.3%

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

   8. distribute-neg-frac2 64.3%

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

   9. distribute-frac-neg2 64.3%

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

   10. associate-+r+ 64.3%

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

   11. +-commutative 64.3%

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

   12. remove-double-neg 64.3%

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

   13. distribute-neg-frac2 64.3%

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

   14. sub0-neg 64.3%

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

   15. associate-+l- 64.3%

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

   16. neg-sub0 64.3%

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

3. Simplified 64.3%

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

5. Taylor expanded in x around inf 61.4%

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

6. Final simplification 61.4%

   \[\leadsto \frac{1}{-1 + x} + \frac{-1}{x} \]
7. Add Preprocessing

Alternative 7: 52.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

   1. +-commutative 64.3%

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

   2. associate-+r- 64.3%

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

   3. sub-neg 64.3%

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

   4. remove-double-neg 64.3%

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

   5. neg-sub0 64.3%

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

   6. associate-+l- 64.3%

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

   7. neg-sub0 64.3%

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

   8. distribute-neg-frac2 64.3%

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

   9. distribute-frac-neg2 64.3%

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

   10. associate-+r+ 64.3%

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

   11. +-commutative 64.3%

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

   12. remove-double-neg 64.3%

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

   13. distribute-neg-frac2 64.3%

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

   14. sub0-neg 64.3%

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

   15. associate-+l- 64.3%

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

   16. neg-sub0 64.3%

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

3. Simplified 64.3%

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

   1. frac-sub 20.9%

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

   2. frac-add 21.8%

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

   3. *-un-lft-identity 21.8%

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

   4. fma-define 19.8%

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

   5. *-rgt-identity 19.8%

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

   6. fmm-def 19.8%

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

6. Applied egg-rr 19.8%

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

   1. fmm-undef 19.8%

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

   2. associate-*r* 19.8%

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

   3. *-commutative 19.8%

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

8. Simplified 19.8%

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

9. Taylor expanded in x around 0 99.6%

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

    1. div-inv 99.6%

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

11. Applied egg-rr 99.6%

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

    1. associate-*r/ 99.6%

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

    2. metadata-eval 99.6%

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

    3. associate-/r* 99.8%

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

    4. associate-/r* 99.8%

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

13. Simplified 99.8%

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

14. Taylor expanded in x around 0 47.1%

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

15. Final simplification 47.1%

    \[\leadsto \frac{\frac{2}{x}}{-1 - x} \]
16. Add Preprocessing

Alternative 8: 5.1% 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 64.3%

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

   1. +-commutative 64.3%

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

   2. associate-+r- 64.3%

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

   3. sub-neg 64.3%

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

   4. remove-double-neg 64.3%

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

   5. neg-sub0 64.3%

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

   6. associate-+l- 64.3%

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

   7. neg-sub0 64.3%

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

   8. distribute-neg-frac2 64.3%

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

   9. distribute-frac-neg2 64.3%

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

   10. associate-+r+ 64.3%

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

   11. +-commutative 64.3%

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

   12. remove-double-neg 64.3%

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

   13. distribute-neg-frac2 64.3%

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

   14. sub0-neg 64.3%

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

   15. associate-+l- 64.3%

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

   16. neg-sub0 64.3%

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

3. Simplified 64.3%

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

5. Taylor expanded in x around 0 4.6%

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

6. Final simplification 4.6%

   \[\leadsto \frac{-2}{x} \]
7. Add Preprocessing

Alternative 9: 5.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

   1. +-commutative 64.3%

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

   2. associate-+r- 64.3%

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

   3. sub-neg 64.3%

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

   4. remove-double-neg 64.3%

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

   5. neg-sub0 64.3%

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

   6. associate-+l- 64.3%

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

   7. neg-sub0 64.3%

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

   8. distribute-neg-frac2 64.3%

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

   9. distribute-frac-neg2 64.3%

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

   10. associate-+r+ 64.3%

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

   11. +-commutative 64.3%

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

   12. remove-double-neg 64.3%

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

   13. distribute-neg-frac2 64.3%

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

   14. sub0-neg 64.3%

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

   15. associate-+l- 64.3%

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

   16. neg-sub0 64.3%

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

3. Simplified 64.3%

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

5. Taylor expanded in x around inf 61.4%

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

6. Taylor expanded in x around 0 4.7%

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

7. Final simplification 4.7%

   \[\leadsto \frac{-1}{x} \]
8. Add Preprocessing

Alternative 10: 6.3% 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 64.3%

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

   1. +-commutative 64.3%

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

   2. associate-+r- 64.3%

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

   3. sub-neg 64.3%

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

   4. remove-double-neg 64.3%

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

   5. neg-sub0 64.3%

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

   6. associate-+l- 64.3%

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

   7. neg-sub0 64.3%

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

   8. distribute-neg-frac2 64.3%

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

   9. distribute-frac-neg2 64.3%

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

   10. associate-+r+ 64.3%

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

   11. +-commutative 64.3%

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

   12. remove-double-neg 64.3%

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

   13. distribute-neg-frac2 64.3%

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

   14. sub0-neg 64.3%

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

   15. associate-+l- 64.3%

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

   16. neg-sub0 64.3%

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

3. Simplified 64.3%

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

   1. +-commutative 64.3%

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

   2. associate-+l- 64.3%

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

6. Applied egg-rr 64.3%

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

   1. sub-neg 64.3%

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

   2. frac-2neg 64.3%

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

   3. metadata-eval 64.3%

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

   4. add-sqr-sqrt 20.5%

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

   5. sqrt-unprod 14.1%

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

   6. sqr-neg 14.1%

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

   7. sqrt-prod 3.1%

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

   8. add-sqr-sqrt 6.2%

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

8. Applied egg-rr 6.2%

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

   1. unsub-neg 6.2%

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

   2. +-inverses 6.2%

      \[\leadsto \frac{2}{x} - \color{blue}{0} \]

   3. --rgt-identity 6.2%

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

10. Simplified 6.2%

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

11. Final simplification 6.2%

    \[\leadsto \frac{2}{x} \]
12. Add Preprocessing

Developer target: 99.1% 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 2024095 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (/ 2.0 (* x (- (* x x) 1.0)))

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