3frac (problem 3.3.3)

Percentage Accurate: 70.4% → 99.5%

Time: 11.9s

Alternatives: 17

Speedup: 1.0×

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: 70.4% 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: 99.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot {x}^{-3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fma 2.0 (pow x -2.0) 2.0) (pow x -3.0)))

C

double code(double x) {
	return fma(2.0, pow(x, -2.0), 2.0) * pow(x, -3.0);
}

Julia

function code(x)
	return Float64(fma(2.0, (x ^ -2.0), 2.0) * (x ^ -3.0))
end

Wolfram

code[x_] := N[(N[(2.0 * N[Power[x, -2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in x around inf 99.2%

   \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{3}}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.2%

      \[\leadsto \frac{2 + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}}}{{x}^{3}} \]

   2. metadata-eval 99.2%

      \[\leadsto \frac{2 + \frac{\color{blue}{2}}{{x}^{2}}}{{x}^{3}} \]

7. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{2 + \frac{2}{{x}^{2}}}{{x}^{3}}} \]
8. Step-by-step derivation

   1. div-inv 99.2%

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

   2. +-commutative 99.2%

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

   3. div-inv 99.2%

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

   4. fma-define 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{2}}, 2\right)} \cdot \frac{1}{{x}^{3}} \]

   5. pow-flip 99.2%

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

   6. metadata-eval 99.2%

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

   7. pow-flip 99.7%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot \color{blue}{{x}^{\left(-3\right)}} \]

   8. metadata-eval 99.7%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot {x}^{\color{blue}{-3}} \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot {x}^{-3}} \]
10. Add Preprocessing

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\frac{\frac{16 \cdot \frac{1}{x} - 8}{x} + 4}{x} - 2}{x}}{x \cdot \frac{x + 1}{2 + x}}}{1 - x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/
   (/ (- (/ (+ (/ (- (* 16.0 (/ 1.0 x)) 8.0) x) 4.0) x) 2.0) x)
   (* x (/ (+ x 1.0) (+ 2.0 x))))
  (- 1.0 x)))

C

double code(double x) {
	return ((((((((16.0 * (1.0 / x)) - 8.0) / x) + 4.0) / x) - 2.0) / x) / (x * ((x + 1.0) / (2.0 + x)))) / (1.0 - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((((((16.0d0 * (1.0d0 / x)) - 8.0d0) / x) + 4.0d0) / x) - 2.0d0) / x) / (x * ((x + 1.0d0) / (2.0d0 + x)))) / (1.0d0 - x)
end function

Java

public static double code(double x) {
	return ((((((((16.0 * (1.0 / x)) - 8.0) / x) + 4.0) / x) - 2.0) / x) / (x * ((x + 1.0) / (2.0 + x)))) / (1.0 - x);
}

Python

def code(x):
	return ((((((((16.0 * (1.0 / x)) - 8.0) / x) + 4.0) / x) - 2.0) / x) / (x * ((x + 1.0) / (2.0 + x)))) / (1.0 - x)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(16.0 * Float64(1.0 / x)) - 8.0) / x) + 4.0) / x) - 2.0) / x) / Float64(x * Float64(Float64(x + 1.0) / Float64(2.0 + x)))) / Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = ((((((((16.0 * (1.0 / x)) - 8.0) / x) + 4.0) / x) - 2.0) / x) / (x * ((x + 1.0) / (2.0 + x)))) / (1.0 - x);
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(16.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 8.0), $MachinePrecision] / x), $MachinePrecision] + 4.0), $MachinePrecision] / x), $MachinePrecision] - 2.0), $MachinePrecision] / x), $MachinePrecision] / N[(x * N[(N[(x + 1.0), $MachinePrecision] / N[(2.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

   1. frac-2neg 72.0%

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

   2. frac-2neg 72.0%

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

   3. metadata-eval 72.0%

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

   4. frac-sub 15.7%

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

   5. metadata-eval 15.7%

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

   6. sub-neg 15.7%

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

   7. distribute-neg-in 15.7%

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

   8. metadata-eval 15.7%

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

   9. neg-mul-1 15.7%

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

   10. *-commutative 15.7%

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

   11. sub-neg 15.7%

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

   12. *-commutative 15.7%

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

   13. neg-mul-1 15.7%

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

   14. sub-neg 15.7%

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

   15. distribute-neg-in 15.7%

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

   16. metadata-eval 15.7%

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

   17. neg-mul-1 15.7%

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

   18. *-commutative 15.7%

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

   19. sub-neg 15.7%

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

   20. *-commutative 15.7%

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

6. Applied egg-rr 15.7%

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

   1. *-commutative 15.7%

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

   2. sub-neg 15.7%

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

   3. remove-double-neg 15.7%

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

   4. distribute-lft-in 15.7%

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

   5. metadata-eval 15.7%

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

   6. neg-mul-1 15.7%

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

   7. remove-double-neg 15.7%

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

   8. sub-neg 15.7%

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

   9. remove-double-neg 15.7%

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

   10. distribute-lft-in 15.7%

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

   11. *-rgt-identity 15.7%

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

   12. neg-mul-1 15.7%

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

   13. distribute-rgt-in 15.7%

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

   14. sub-neg 15.7%

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

8. Simplified 15.7%

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

   1. frac-2neg 15.7%

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

   2. metadata-eval 15.7%

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

   3. clear-num 18.2%

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

   4. frac-add 14.1%

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

   5. +-commutative 14.1%

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

   6. fma-define 14.1%

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

   7. +-commutative 14.1%

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

   8. fma-define 14.1%

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

10. Applied egg-rr 14.1%

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

12. Simplified 72.0%

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

13. Taylor expanded in x around -inf 99.5%

    \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{16 \cdot \frac{1}{x} - 8}{x} - 4}{x} - 2}{x}}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

14. Final simplification 99.5%

    \[\leadsto \frac{\frac{\frac{\frac{\frac{16 \cdot \frac{1}{x} - 8}{x} + 4}{x} - 2}{x}}{x \cdot \frac{x + 1}{2 + x}}}{1 - x} \]
15. Add Preprocessing

Alternative 3: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{-2 + \frac{4 + \frac{-8}{x}}{x}}{x}}{x \cdot \frac{x + 1}{2 + x}}}{1 - x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/ (/ (+ -2.0 (/ (+ 4.0 (/ -8.0 x)) x)) x) (* x (/ (+ x 1.0) (+ 2.0 x))))
  (- 1.0 x)))

C

double code(double x) {
	return (((-2.0 + ((4.0 + (-8.0 / x)) / x)) / x) / (x * ((x + 1.0) / (2.0 + x)))) / (1.0 - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((-2.0d0) + ((4.0d0 + ((-8.0d0) / x)) / x)) / x) / (x * ((x + 1.0d0) / (2.0d0 + x)))) / (1.0d0 - x)
end function

Java

public static double code(double x) {
	return (((-2.0 + ((4.0 + (-8.0 / x)) / x)) / x) / (x * ((x + 1.0) / (2.0 + x)))) / (1.0 - x);
}

Python

def code(x):
	return (((-2.0 + ((4.0 + (-8.0 / x)) / x)) / x) / (x * ((x + 1.0) / (2.0 + x)))) / (1.0 - x)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(-2.0 + Float64(Float64(4.0 + Float64(-8.0 / x)) / x)) / x) / Float64(x * Float64(Float64(x + 1.0) / Float64(2.0 + x)))) / Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = (((-2.0 + ((4.0 + (-8.0 / x)) / x)) / x) / (x * ((x + 1.0) / (2.0 + x)))) / (1.0 - x);
end

Wolfram

code[x_] := N[(N[(N[(N[(-2.0 + N[(N[(4.0 + N[(-8.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(x * N[(N[(x + 1.0), $MachinePrecision] / N[(2.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

   1. frac-2neg 72.0%

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

   2. frac-2neg 72.0%

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

   3. metadata-eval 72.0%

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

   4. frac-sub 15.7%

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

   5. metadata-eval 15.7%

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

   6. sub-neg 15.7%

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

   7. distribute-neg-in 15.7%

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

   8. metadata-eval 15.7%

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

   9. neg-mul-1 15.7%

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

   10. *-commutative 15.7%

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

   11. sub-neg 15.7%

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

   12. *-commutative 15.7%

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

   13. neg-mul-1 15.7%

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

   14. sub-neg 15.7%

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

   15. distribute-neg-in 15.7%

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

   16. metadata-eval 15.7%

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

   17. neg-mul-1 15.7%

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

   18. *-commutative 15.7%

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

   19. sub-neg 15.7%

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

   20. *-commutative 15.7%

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

6. Applied egg-rr 15.7%

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

   1. *-commutative 15.7%

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

   2. sub-neg 15.7%

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

   3. remove-double-neg 15.7%

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

   4. distribute-lft-in 15.7%

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

   5. metadata-eval 15.7%

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

   6. neg-mul-1 15.7%

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

   7. remove-double-neg 15.7%

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

   8. sub-neg 15.7%

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

   9. remove-double-neg 15.7%

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

   10. distribute-lft-in 15.7%

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

   11. *-rgt-identity 15.7%

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

   12. neg-mul-1 15.7%

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

   13. distribute-rgt-in 15.7%

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

   14. sub-neg 15.7%

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

8. Simplified 15.7%

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

   1. frac-2neg 15.7%

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

   2. metadata-eval 15.7%

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

   3. clear-num 18.2%

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

   4. frac-add 14.1%

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

   5. +-commutative 14.1%

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

   6. fma-define 14.1%

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

   7. +-commutative 14.1%

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

   8. fma-define 14.1%

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

10. Applied egg-rr 14.1%

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

12. Simplified 72.0%

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

13. Taylor expanded in x around -inf 99.5%

    \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{8 \cdot \frac{1}{x} - 4}{x} - 2}{x}}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]
14. Step-by-step derivation

    1. mul-1-neg 99.5%

       \[\leadsto \frac{\frac{\color{blue}{-\frac{-1 \cdot \frac{8 \cdot \frac{1}{x} - 4}{x} - 2}{x}}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

    2. distribute-neg-frac2 99.5%

       \[\leadsto \frac{\frac{\color{blue}{\frac{-1 \cdot \frac{8 \cdot \frac{1}{x} - 4}{x} - 2}{-x}}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

    3. sub-neg 99.5%

       \[\leadsto \frac{\frac{\frac{\color{blue}{-1 \cdot \frac{8 \cdot \frac{1}{x} - 4}{x} + \left(-2\right)}}{-x}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

    4. associate-*r/ 99.5%

       \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{-1 \cdot \left(8 \cdot \frac{1}{x} - 4\right)}{x}} + \left(-2\right)}{-x}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

    5. sub-neg 99.5%

       \[\leadsto \frac{\frac{\frac{\frac{-1 \cdot \color{blue}{\left(8 \cdot \frac{1}{x} + \left(-4\right)\right)}}{x} + \left(-2\right)}{-x}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

    6. distribute-lft-in 99.5%

       \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{-1 \cdot \left(8 \cdot \frac{1}{x}\right) + -1 \cdot \left(-4\right)}}{x} + \left(-2\right)}{-x}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

    7. neg-mul-1 99.5%

       \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{\left(-8 \cdot \frac{1}{x}\right)} + -1 \cdot \left(-4\right)}{x} + \left(-2\right)}{-x}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

    8. associate-*r/ 99.5%

       \[\leadsto \frac{\frac{\frac{\frac{\left(-\color{blue}{\frac{8 \cdot 1}{x}}\right) + -1 \cdot \left(-4\right)}{x} + \left(-2\right)}{-x}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

    9. metadata-eval 99.5%

       \[\leadsto \frac{\frac{\frac{\frac{\left(-\frac{\color{blue}{8}}{x}\right) + -1 \cdot \left(-4\right)}{x} + \left(-2\right)}{-x}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

    10. distribute-neg-frac 99.5%

        \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{\frac{-8}{x}} + -1 \cdot \left(-4\right)}{x} + \left(-2\right)}{-x}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

    11. metadata-eval 99.5%

        \[\leadsto \frac{\frac{\frac{\frac{\frac{\color{blue}{-8}}{x} + -1 \cdot \left(-4\right)}{x} + \left(-2\right)}{-x}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

    12. metadata-eval 99.5%

        \[\leadsto \frac{\frac{\frac{\frac{\frac{-8}{x} + -1 \cdot \color{blue}{-4}}{x} + \left(-2\right)}{-x}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

    13. metadata-eval 99.5%

        \[\leadsto \frac{\frac{\frac{\frac{\frac{-8}{x} + \color{blue}{4}}{x} + \left(-2\right)}{-x}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

    14. metadata-eval 99.5%

        \[\leadsto \frac{\frac{\frac{\frac{\frac{-8}{x} + 4}{x} + \color{blue}{-2}}{-x}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

15. Simplified 99.5%

    \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{\frac{-8}{x} + 4}{x} + -2}{-x}}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

16. Final simplification 99.5%

    \[\leadsto \frac{\frac{\frac{-2 + \frac{4 + \frac{-8}{x}}{x}}{x}}{x \cdot \frac{x + 1}{2 + x}}}{1 - x} \]
17. Add Preprocessing

Alternative 4: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\frac{4}{x} - 2}{x}}{x \cdot \frac{x + 1}{2 + x}}}{1 - x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (/ (- (/ 4.0 x) 2.0) x) (* x (/ (+ x 1.0) (+ 2.0 x)))) (- 1.0 x)))

C

double code(double x) {
	return ((((4.0 / x) - 2.0) / x) / (x * ((x + 1.0) / (2.0 + x)))) / (1.0 - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((4.0d0 / x) - 2.0d0) / x) / (x * ((x + 1.0d0) / (2.0d0 + x)))) / (1.0d0 - x)
end function

Java

public static double code(double x) {
	return ((((4.0 / x) - 2.0) / x) / (x * ((x + 1.0) / (2.0 + x)))) / (1.0 - x);
}

Python

def code(x):
	return ((((4.0 / x) - 2.0) / x) / (x * ((x + 1.0) / (2.0 + x)))) / (1.0 - x)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(4.0 / x) - 2.0) / x) / Float64(x * Float64(Float64(x + 1.0) / Float64(2.0 + x)))) / Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = ((((4.0 / x) - 2.0) / x) / (x * ((x + 1.0) / (2.0 + x)))) / (1.0 - x);
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(4.0 / x), $MachinePrecision] - 2.0), $MachinePrecision] / x), $MachinePrecision] / N[(x * N[(N[(x + 1.0), $MachinePrecision] / N[(2.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

   1. frac-2neg 72.0%

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

   2. frac-2neg 72.0%

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

   3. metadata-eval 72.0%

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

   4. frac-sub 15.7%

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

   5. metadata-eval 15.7%

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

   6. sub-neg 15.7%

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

   7. distribute-neg-in 15.7%

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

   8. metadata-eval 15.7%

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

   9. neg-mul-1 15.7%

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

   10. *-commutative 15.7%

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

   11. sub-neg 15.7%

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

   12. *-commutative 15.7%

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

   13. neg-mul-1 15.7%

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

   14. sub-neg 15.7%

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

   15. distribute-neg-in 15.7%

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

   16. metadata-eval 15.7%

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

   17. neg-mul-1 15.7%

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

   18. *-commutative 15.7%

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

   19. sub-neg 15.7%

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

   20. *-commutative 15.7%

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

6. Applied egg-rr 15.7%

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

   1. *-commutative 15.7%

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

   2. sub-neg 15.7%

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

   3. remove-double-neg 15.7%

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

   4. distribute-lft-in 15.7%

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

   5. metadata-eval 15.7%

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

   6. neg-mul-1 15.7%

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

   7. remove-double-neg 15.7%

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

   8. sub-neg 15.7%

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

   9. remove-double-neg 15.7%

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

   10. distribute-lft-in 15.7%

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

   11. *-rgt-identity 15.7%

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

   12. neg-mul-1 15.7%

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

   13. distribute-rgt-in 15.7%

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

   14. sub-neg 15.7%

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

8. Simplified 15.7%

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

   1. frac-2neg 15.7%

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

   2. metadata-eval 15.7%

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

   3. clear-num 18.2%

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

   4. frac-add 14.1%

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

   5. +-commutative 14.1%

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

   6. fma-define 14.1%

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

   7. +-commutative 14.1%

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

   8. fma-define 14.1%

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

10. Applied egg-rr 14.1%

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

12. Simplified 72.0%

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

13. Taylor expanded in x around inf 99.0%

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

    1. associate-*r/ 99.0%

       \[\leadsto \frac{\frac{\frac{2 - \color{blue}{\frac{4 \cdot 1}{x}}}{x}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

    2. metadata-eval 99.0%

       \[\leadsto \frac{\frac{\frac{2 - \frac{\color{blue}{4}}{x}}{x}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

15. Simplified 99.0%

    \[\leadsto \frac{\frac{\color{blue}{\frac{2 - \frac{4}{x}}{x}}}{x \cdot \frac{-1 - x}{2 + x}}}{1 - x} \]

16. Final simplification 99.0%

    \[\leadsto \frac{\frac{\frac{\frac{4}{x} - 2}{x}}{x \cdot \frac{x + 1}{2 + x}}}{1 - x} \]
17. Add Preprocessing

Alternative 5: 97.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

   1. frac-2neg 72.0%

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

   2. frac-2neg 72.0%

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

   3. metadata-eval 72.0%

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

   4. frac-sub 15.7%

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

   5. metadata-eval 15.7%

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

   6. sub-neg 15.7%

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

   7. distribute-neg-in 15.7%

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

   8. metadata-eval 15.7%

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

   9. neg-mul-1 15.7%

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

   10. *-commutative 15.7%

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

   11. sub-neg 15.7%

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

   12. *-commutative 15.7%

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

   13. neg-mul-1 15.7%

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

   14. sub-neg 15.7%

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

   15. distribute-neg-in 15.7%

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

   16. metadata-eval 15.7%

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

   17. neg-mul-1 15.7%

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

   18. *-commutative 15.7%

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

   19. sub-neg 15.7%

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

   20. *-commutative 15.7%

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

6. Applied egg-rr 15.7%

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

   1. *-commutative 15.7%

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

   2. sub-neg 15.7%

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

   3. remove-double-neg 15.7%

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

   4. distribute-lft-in 15.7%

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

   5. metadata-eval 15.7%

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

   6. neg-mul-1 15.7%

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

   7. remove-double-neg 15.7%

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

   8. sub-neg 15.7%

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

   9. remove-double-neg 15.7%

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

   10. distribute-lft-in 15.7%

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

   11. *-rgt-identity 15.7%

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

   12. neg-mul-1 15.7%

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

   13. distribute-rgt-in 15.7%

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

   14. sub-neg 15.7%

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

8. Simplified 15.7%

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

   1. frac-2neg 15.7%

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

   2. metadata-eval 15.7%

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

   3. clear-num 18.2%

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

   4. frac-add 14.1%

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

   5. +-commutative 14.1%

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

   6. fma-define 14.1%

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

   7. +-commutative 14.1%

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

   8. fma-define 14.1%

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

10. Applied egg-rr 14.1%

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

12. Simplified 72.0%

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

13. Taylor expanded in x around inf 97.9%

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

14. Final simplification 97.9%

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

Alternative 6: 70.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

   1. +-commutative 72.0%

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

   2. frac-sub 15.7%

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

   3. div-inv 13.0%

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

   4. fma-define 7.2%

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

   5. *-rgt-identity 7.2%

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

   6. fmm-def 7.2%

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

6. Applied egg-rr 7.2%

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

   1. fma-undefine 13.0%

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

   2. +-commutative 13.0%

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

   3. associate-*r/ 15.7%

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

   4. *-rgt-identity 15.7%

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

   5. associate-/r* 72.0%

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

   6. fmm-undef 72.0%

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

   7. div-sub 72.0%

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

   8. *-inverses 72.0%

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

8. Simplified 72.0%

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

9. Taylor expanded in x around inf 72.0%

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

    1. associate-*r/ 72.0%

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

    2. metadata-eval 72.0%

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

11. Simplified 72.0%

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

12. Final simplification 72.0%

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

Alternative 7: 70.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

   1. +-commutative 72.0%

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

   2. frac-sub 15.7%

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

   3. div-inv 13.0%

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

   4. fma-define 7.2%

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

   5. *-rgt-identity 7.2%

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

   6. fmm-def 7.2%

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

6. Applied egg-rr 7.2%

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

   1. fma-undefine 13.0%

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

   2. +-commutative 13.0%

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

   3. associate-*r/ 15.7%

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

   4. *-rgt-identity 15.7%

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

   5. associate-/r* 72.0%

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

   6. fmm-undef 72.0%

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

   7. div-sub 72.0%

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

   8. *-inverses 72.0%

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

8. Simplified 72.0%

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

9. Taylor expanded in x around 0 72.0%

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

10. Final simplification 72.0%

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

Alternative 8: 70.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

3. Final simplification 72.0%

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

Alternative 9: 69.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in x around inf 71.1%

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

6. Taylor expanded in x around inf 70.9%

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

   1. div-inv 70.9%

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

8. Applied egg-rr 70.9%

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

9. Final simplification 70.9%

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

Alternative 10: 69.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in x around inf 71.1%

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

6. Taylor expanded in x around inf 70.9%

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

   1. div-inv 70.9%

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

8. Applied egg-rr 70.9%

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

   1. +-commutative 70.9%

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

   2. div-inv 70.9%

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

   3. fma-define 70.9%

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

   4. un-div-inv 70.9%

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

10. Applied egg-rr 70.9%

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

    1. fma-undefine 70.9%

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

    2. *-rgt-identity 70.9%

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

    3. associate-/l* 70.9%

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

    4. distribute-rgt-out 70.9%

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

12. Simplified 70.9%

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

Alternative 11: 69.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in x around inf 71.1%

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

6. Taylor expanded in x around inf 70.9%

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

   1. +-commutative 70.9%

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

   2. frac-2neg 70.9%

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

   3. metadata-eval 70.9%

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

   4. frac-add 70.9%

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

   5. *-un-lft-identity 70.9%

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

8. Applied egg-rr 70.9%

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

   1. add-sqr-sqrt 31.2%

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

   2. fma-define 27.8%

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

   3. distribute-lft-neg-out 27.8%

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

   4. add-sqr-sqrt 27.9%

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

   5. sqrt-unprod 1.2%

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

   6. sqr-neg 1.2%

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

   7. sqrt-unprod 0.0%

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

   8. add-sqr-sqrt 27.3%

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

   9. fmm-def 27.3%

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

   10. add-sqr-sqrt 58.7%

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

   11. add-sqr-sqrt 31.5%

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

   12. sqrt-unprod 7.2%

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

   13. sqr-neg 7.2%

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

   14. sqrt-unprod 31.2%

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

   15. add-sqr-sqrt 70.9%

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

10. Applied egg-rr 70.9%

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

    1. distribute-lft-in 70.9%

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

    2. *-rgt-identity 70.9%

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

    3. rgt-mult-inverse 70.9%

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

12. Simplified 70.9%

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

13. Final simplification 70.9%

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

Alternative 12: 69.0% 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 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in x around inf 70.9%

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

6. Final simplification 70.9%

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

Alternative 13: 68.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in x around inf 71.1%

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

6. Taylor expanded in x around inf 70.9%

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

   1. div-inv 70.9%

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

8. Applied egg-rr 70.9%

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

9. Taylor expanded in x around inf 70.7%

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

Alternative 14: 68.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in x around 0 3.4%

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

6. Taylor expanded in x around 0 70.7%

   \[\leadsto \color{blue}{-1} + \frac{x - 2}{x} \]
7. Add Preprocessing

Alternative 15: 54.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in x around inf 71.1%

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

6. Taylor expanded in x around inf 70.9%

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

   1. +-commutative 70.9%

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

   2. frac-2neg 70.9%

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

   3. metadata-eval 70.9%

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

   4. frac-add 70.9%

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

   5. *-un-lft-identity 70.9%

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

8. Applied egg-rr 70.9%

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

9. Taylor expanded in x around 0 60.2%

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

10. Final simplification 60.2%

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

Alternative 16: 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 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in x around 0 5.4%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]
6. Add Preprocessing

Alternative 17: 3.4% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

   2. associate-+r- 71.9%

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

   3. sub-neg 71.9%

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

   4. remove-double-neg 71.9%

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

   5. neg-sub0 71.9%

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

   6. associate-+l- 71.9%

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

   7. neg-sub0 71.9%

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

   8. distribute-neg-frac2 71.9%

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

   9. distribute-frac-neg2 71.9%

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

   10. associate-+r+ 72.0%

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

   11. +-commutative 72.0%

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

   12. remove-double-neg 72.0%

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

   13. distribute-neg-frac2 72.0%

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

   14. sub0-neg 72.0%

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

   15. associate-+l- 72.0%

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

   16. neg-sub0 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in x around 0 3.4%

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

6. Taylor expanded in x around inf 3.4%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Developer Target 1: 99.2% 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 2024170 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (! :herbie-platform default (/ 2 (* x (- (* x x) 1))))

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