3frac (problem 3.3.3)

Percentage Accurate: 70.4% → 99.8%

Time: 11.3s

Alternatives: 11

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: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 67.0%

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

   3. sub-neg 67.0%

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

   4. remove-double-neg 67.0%

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

   5. neg-sub0 67.0%

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

   6. associate-+l- 67.0%

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

   7. neg-sub0 67.0%

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

   8. distribute-neg-frac2 67.0%

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

   9. distribute-frac-neg2 67.0%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

   1. frac-2neg 67.0%

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

   2. metadata-eval 67.0%

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

   3. frac-sub 16.0%

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

   4. frac-add 16.8%

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

   5. fma-define 16.8%

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

   6. distribute-neg-in 16.8%

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

   7. neg-mul-1 16.8%

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

   8. *-commutative 16.8%

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

   9. metadata-eval 16.8%

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

   10. fma-define 16.8%

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

   11. *-rgt-identity 16.8%

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

   12. fma-neg 16.8%

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

   13. distribute-neg-in 16.8%

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

   14. neg-mul-1 16.8%

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

   15. *-commutative 16.8%

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

   16. metadata-eval 16.8%

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

   17. fma-define 16.8%

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

6. Applied egg-rr 16.8%

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

   1. fma-undefine 16.8%

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

   2. neg-mul-1 16.8%

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

   3. distribute-rgt-neg-in 16.8%

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

   4. fma-define 15.2%

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

   5. sub-neg 15.2%

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

   6. distribute-neg-in 15.2%

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

   7. metadata-eval 15.2%

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

   8. remove-double-neg 15.2%

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

   9. +-commutative 15.2%

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

   10. *-commutative 15.2%

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

   11. fma-neg 15.2%

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

   12. fma-undefine 15.2%

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

   13. *-commutative 15.2%

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

   14. neg-mul-1 15.2%

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

   15. +-commutative 15.2%

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

   16. unsub-neg 15.2%

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

   17. *-commutative 15.2%

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

   18. associate-*l* 15.2%

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

   19. *-commutative 15.2%

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

8. Simplified 15.2%

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

9. Taylor expanded in x around 0 99.7%

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

    1. associate-/r* 99.8%

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

    2. div-inv 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

    \[\leadsto \frac{2}{x} \cdot \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \]
13. Add Preprocessing

Alternative 2: 99.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 67.0%

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

   3. sub-neg 67.0%

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

   4. remove-double-neg 67.0%

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

   5. neg-sub0 67.0%

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

   6. associate-+l- 67.0%

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

   7. neg-sub0 67.0%

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

   8. distribute-neg-frac2 67.0%

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

   9. distribute-frac-neg2 67.0%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

   1. frac-2neg 67.0%

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

   2. metadata-eval 67.0%

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

   3. frac-sub 16.0%

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

   4. frac-add 16.8%

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

   5. fma-define 16.8%

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

   6. distribute-neg-in 16.8%

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

   7. neg-mul-1 16.8%

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

   8. *-commutative 16.8%

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

   9. metadata-eval 16.8%

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

   10. fma-define 16.8%

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

   11. *-rgt-identity 16.8%

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

   12. fma-neg 16.8%

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

   13. distribute-neg-in 16.8%

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

   14. neg-mul-1 16.8%

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

   15. *-commutative 16.8%

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

   16. metadata-eval 16.8%

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

   17. fma-define 16.8%

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

6. Applied egg-rr 16.8%

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

   1. fma-undefine 16.8%

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

   2. neg-mul-1 16.8%

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

   3. distribute-rgt-neg-in 16.8%

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

   4. fma-define 15.2%

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

   5. sub-neg 15.2%

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

   6. distribute-neg-in 15.2%

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

   7. metadata-eval 15.2%

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

   8. remove-double-neg 15.2%

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

   9. +-commutative 15.2%

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

   10. *-commutative 15.2%

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

   11. fma-neg 15.2%

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

   12. fma-undefine 15.2%

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

   13. *-commutative 15.2%

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

   14. neg-mul-1 15.2%

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

   15. +-commutative 15.2%

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

   16. unsub-neg 15.2%

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

   17. *-commutative 15.2%

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

   18. associate-*l* 15.2%

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

   19. *-commutative 15.2%

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

8. Simplified 15.2%

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

9. Taylor expanded in x around 0 99.7%

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

10. Final simplification 99.7%

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

Alternative 3: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot {x}^{-3} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * (x ** (-3.0d0))
end function

Java

public static double code(double x) {
	return 2.0 * Math.pow(x, -3.0);
}

Python

def code(x):
	return 2.0 * math.pow(x, -3.0)

Julia

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

MATLAB

function tmp = code(x)
	tmp = 2.0 * (x ^ -3.0);
end

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 67.0%

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

   3. sub-neg 67.0%

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

   4. remove-double-neg 67.0%

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

   5. neg-sub0 67.0%

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

   6. associate-+l- 67.0%

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

   7. neg-sub0 67.0%

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

   8. distribute-neg-frac2 67.0%

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

   9. distribute-frac-neg2 67.0%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around inf 98.9%

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

   1. div-inv 98.9%

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

   2. pow-flip 99.2%

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

   3. metadata-eval 99.2%

      \[\leadsto 2 \cdot {x}^{\color{blue}{-3}} \]

7. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{2 \cdot {x}^{-3}} \]

8. Final simplification 99.2%

   \[\leadsto 2 \cdot {x}^{-3} \]
9. Add Preprocessing

Alternative 4: 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]

Derivation

1. Initial program 67.0%

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

3. Final simplification 67.0%

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

Alternative 5: 69.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 67.0%

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

   3. sub-neg 67.0%

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

   4. remove-double-neg 67.0%

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

   5. neg-sub0 67.0%

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

   6. associate-+l- 67.0%

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

   7. neg-sub0 67.0%

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

   8. distribute-neg-frac2 67.0%

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

   9. distribute-frac-neg2 67.0%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around inf 66.1%

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

   1. associate-*r/ 66.1%

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

   2. neg-mul-1 66.1%

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

   3. distribute-neg-in 66.1%

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

   4. metadata-eval 66.1%

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

   5. distribute-neg-frac 66.1%

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

   6. metadata-eval 66.1%

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

7. Simplified 66.1%

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

8. Final simplification 66.1%

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

Alternative 6: 68.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 67.0%

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

   3. sub-neg 67.0%

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

   4. remove-double-neg 67.0%

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

   5. neg-sub0 67.0%

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

   6. associate-+l- 67.0%

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

   7. neg-sub0 67.0%

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

   8. distribute-neg-frac2 67.0%

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

   9. distribute-frac-neg2 67.0%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around inf 65.8%

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

   1. frac-add 65.8%

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

   2. *-un-lft-identity 65.8%

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

7. Applied egg-rr 65.8%

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

   1. *-commutative 65.8%

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

   2. distribute-lft-in 65.8%

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

   3. neg-mul-1 65.8%

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

   4. metadata-eval 65.8%

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

   5. +-commutative 65.8%

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

   6. sub-neg 65.8%

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

   7. *-commutative 65.8%

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

9. Simplified 65.8%

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

10. Final simplification 65.8%

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

Alternative 7: 68.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 67.0%

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

   3. sub-neg 67.0%

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

   4. remove-double-neg 67.0%

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

   5. neg-sub0 67.0%

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

   6. associate-+l- 67.0%

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

   7. neg-sub0 67.0%

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

   8. distribute-neg-frac2 67.0%

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

   9. distribute-frac-neg2 67.0%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around inf 65.8%

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

6. Final simplification 65.8%

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

Alternative 8: 52.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 67.0%

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

   3. sub-neg 67.0%

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

   4. remove-double-neg 67.0%

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

   5. neg-sub0 67.0%

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

   6. associate-+l- 67.0%

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

   7. neg-sub0 67.0%

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

   8. distribute-neg-frac2 67.0%

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

   9. distribute-frac-neg2 67.0%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around inf 65.8%

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

   1. frac-add 65.8%

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

   2. *-un-lft-identity 65.8%

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

7. Applied egg-rr 65.8%

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

   1. *-commutative 65.8%

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

   2. distribute-lft-in 65.8%

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

   3. neg-mul-1 65.8%

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

   4. metadata-eval 65.8%

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

   5. +-commutative 65.8%

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

   6. sub-neg 65.8%

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

   7. *-commutative 65.8%

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

9. Simplified 65.8%

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

    1. *-un-lft-identity 65.8%

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

11. Applied egg-rr 65.8%

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

    1. *-lft-identity 65.8%

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

    2. +-commutative 65.8%

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

    3. associate--r- 54.0%

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

    4. +-inverses 54.0%

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

    5. metadata-eval 54.0%

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

    6. associate-/r* 52.7%

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

13. Simplified 52.7%

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

14. Final simplification 52.7%

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

Alternative 9: 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 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 67.0%

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

   3. sub-neg 67.0%

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

   4. remove-double-neg 67.0%

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

   5. neg-sub0 67.0%

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

   6. associate-+l- 67.0%

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

   7. neg-sub0 67.0%

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

   8. distribute-neg-frac2 67.0%

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

   9. distribute-frac-neg2 67.0%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around 0 4.9%

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

6. Final simplification 4.9%

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

Alternative 10: 5.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 67.0%

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

   3. sub-neg 67.0%

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

   4. remove-double-neg 67.0%

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

   5. neg-sub0 67.0%

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

   6. associate-+l- 67.0%

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

   7. neg-sub0 67.0%

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

   8. distribute-neg-frac2 67.0%

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

   9. distribute-frac-neg2 67.0%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around inf 65.8%

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

6. Taylor expanded in x around 0 4.9%

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

7. Final simplification 4.9%

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

Alternative 11: 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 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 67.0%

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

   3. sub-neg 67.0%

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

   4. remove-double-neg 67.0%

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

   5. neg-sub0 67.0%

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

   6. associate-+l- 67.0%

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

   7. neg-sub0 67.0%

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

   8. distribute-neg-frac2 67.0%

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

   9. distribute-frac-neg2 67.0%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around 0 3.3%

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

6. Taylor expanded in x around inf 3.3%

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

7. Final simplification 3.3%

   \[\leadsto 1 \]
8. Add Preprocessing

Developer target: 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 2024069 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

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

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