3frac (problem 3.3.3)

Percentage Accurate: 85.0% → 99.7%

Time: 9.1s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{2}{x}}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x + -1\right) - x \cdot \left(x \cdot \left(1 - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e+69)
   (/ (/ 2.0 x) (fma x x x))
   (/ 2.0 (- (* x (+ x -1.0)) (* x (* x (- 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -4e+69) {
		tmp = (2.0 / x) / fma(x, x, x);
	} else {
		tmp = 2.0 / ((x * (x + -1.0)) - (x * (x * (1.0 - x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e+69)
		tmp = Float64(Float64(2.0 / x) / fma(x, x, x));
	else
		tmp = Float64(2.0 / Float64(Float64(x * Float64(x + -1.0)) - Float64(x * Float64(x * Float64(1.0 - x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4e+69], N[(N[(2.0 / x), $MachinePrecision] / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - N[(x * N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.0000000000000003e69

   1. Initial program 82.6%

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

      1. remove-double-neg 82.6%

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

      2. sub-neg 82.6%

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

      3. sub-neg 82.6%

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

      4. distribute-neg-frac 82.6%

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

      5. metadata-eval 82.6%

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

      6. metadata-eval 82.6%

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

      7. metadata-eval 82.6%

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

      8. associate-/r* 82.6%

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

      9. metadata-eval 82.6%

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

      10. neg-mul-1 82.6%

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

      11. associate--l+ 82.6%

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

      12. +-commutative 82.6%

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

      13. distribute-neg-frac 82.6%

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

      14. metadata-eval 82.6%

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

      15. metadata-eval 82.6%

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

      16. metadata-eval 82.6%

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

      17. associate-/r* 82.6%

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

      18. metadata-eval 82.6%

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

      19. neg-mul-1 82.6%

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

      20. sub0-neg 82.6%

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

      21. associate-+l- 82.6%

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

      22. neg-sub0 82.6%

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

   3. Simplified 82.6%

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

      1. frac-2neg 82.6%

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

      2. metadata-eval 82.6%

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

      3. frac-sub 20.8%

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

      4. frac-add 22.6%

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

      5. fma-def 22.6%

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

      6. +-commutative 22.6%

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

      7. distribute-neg-in 22.6%

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

      8. neg-mul-1 22.6%

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

      9. metadata-eval 22.6%

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

      10. fma-def 22.6%

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

      11. *-rgt-identity 22.6%

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

      12. +-commutative 22.6%

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

      13. distribute-neg-in 22.6%

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

      14. neg-mul-1 22.6%

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

      15. metadata-eval 22.6%

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

      16. fma-def 22.6%

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

   5. Applied egg-rr 22.6%

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

   6. Taylor expanded in x around 0 98.5%

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

      1. *-un-lft-identity 98.5%

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

      2. *-commutative 98.5%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around inf 99.9%

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

   if -4.0000000000000003e69 < x

   1. Initial program 83.1%

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

      1. remove-double-neg 83.1%

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

      2. sub-neg 83.1%

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

      3. sub-neg 83.1%

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

      4. distribute-neg-frac 83.1%

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

      5. metadata-eval 83.1%

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

      6. metadata-eval 83.1%

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

      7. metadata-eval 83.1%

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

      8. associate-/r* 83.1%

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

      9. metadata-eval 83.1%

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

      10. neg-mul-1 83.1%

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

      11. associate--l+ 83.1%

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

      12. +-commutative 83.1%

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

      13. distribute-neg-frac 83.1%

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

      14. metadata-eval 83.1%

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

      15. metadata-eval 83.1%

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

      16. metadata-eval 83.1%

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

      17. associate-/r* 83.1%

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

      18. metadata-eval 83.1%

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

      19. neg-mul-1 83.1%

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

      20. sub0-neg 83.1%

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

      21. associate-+l- 83.1%

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

      22. neg-sub0 83.1%

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

   3. Simplified 83.1%

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

      1. frac-2neg 83.1%

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

      2. metadata-eval 83.1%

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

      3. frac-sub 68.4%

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

      4. frac-add 68.4%

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

      5. fma-def 68.4%

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

      6. +-commutative 68.4%

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

      7. distribute-neg-in 68.4%

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

      8. neg-mul-1 68.4%

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

      9. metadata-eval 68.4%

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

      10. fma-def 68.4%

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

      11. *-rgt-identity 68.4%

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

      12. +-commutative 68.4%

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

      13. distribute-neg-in 68.4%

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

      14. neg-mul-1 68.4%

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

      15. metadata-eval 68.4%

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

      16. fma-def 68.4%

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

   5. Applied egg-rr 68.4%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-udef 99.9%

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

      3. neg-mul-1 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. *-commutative 99.9%

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

      6. neg-mul-1 99.9%

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

      7. *-commutative 99.9%

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

      8. distribute-rgt-neg-in 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{2}{x}}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x + -1\right) - x \cdot \left(x \cdot \left(1 - x\right)\right)}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.0%

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

   1. remove-double-neg 83.0%

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

   2. sub-neg 83.0%

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

   3. sub-neg 83.0%

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

   4. distribute-neg-frac 83.0%

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

   5. metadata-eval 83.0%

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

   6. metadata-eval 83.0%

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

   7. metadata-eval 83.0%

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

   8. associate-/r* 83.0%

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

   9. metadata-eval 83.0%

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

   10. neg-mul-1 83.0%

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

   11. associate--l+ 83.0%

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

   12. +-commutative 83.0%

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

   13. distribute-neg-frac 83.0%

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

   14. metadata-eval 83.0%

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

   15. metadata-eval 83.0%

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

   16. metadata-eval 83.0%

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

   17. associate-/r* 83.0%

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

   18. metadata-eval 83.0%

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

   19. neg-mul-1 83.0%

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

   20. sub0-neg 83.0%

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

   21. associate-+l- 83.0%

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

   22. neg-sub0 83.0%

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

3. Simplified 83.0%

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

   1. frac-2neg 83.0%

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

   2. metadata-eval 83.0%

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

   3. frac-sub 61.0%

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

   4. frac-add 61.2%

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

   5. fma-def 61.2%

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

   6. +-commutative 61.2%

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

   7. distribute-neg-in 61.2%

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

   8. neg-mul-1 61.2%

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

   9. metadata-eval 61.2%

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

   10. fma-def 61.2%

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

   11. *-rgt-identity 61.2%

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

   12. +-commutative 61.2%

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

   13. distribute-neg-in 61.2%

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

   14. neg-mul-1 61.2%

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

   15. metadata-eval 61.2%

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

   16. fma-def 61.2%

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

5. Applied egg-rr 61.2%

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

6. Taylor expanded in x around 0 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. *-commutative 99.7%

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

8. Applied egg-rr 99.9%

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

10. Applied egg-rr 99.9%

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

    1. associate-*l/ 99.9%

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

    2. associate-*r/ 99.9%

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

    3. metadata-eval 99.9%

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

    4. +-commutative 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.0%

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

   1. remove-double-neg 83.0%

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

   2. sub-neg 83.0%

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

   3. sub-neg 83.0%

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

   4. distribute-neg-frac 83.0%

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

   5. metadata-eval 83.0%

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

   6. metadata-eval 83.0%

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

   7. metadata-eval 83.0%

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

   8. associate-/r* 83.0%

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

   9. metadata-eval 83.0%

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

   10. neg-mul-1 83.0%

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

   11. associate--l+ 83.0%

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

   12. +-commutative 83.0%

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

   13. distribute-neg-frac 83.0%

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

   14. metadata-eval 83.0%

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

   15. metadata-eval 83.0%

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

   16. metadata-eval 83.0%

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

   17. associate-/r* 83.0%

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

   18. metadata-eval 83.0%

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

   19. neg-mul-1 83.0%

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

   20. sub0-neg 83.0%

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

   21. associate-+l- 83.0%

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

   22. neg-sub0 83.0%

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

3. Simplified 83.0%

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

   1. frac-2neg 83.0%

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

   2. metadata-eval 83.0%

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

   3. frac-sub 61.0%

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

   4. frac-add 61.2%

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

   5. fma-def 61.2%

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

   6. +-commutative 61.2%

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

   7. distribute-neg-in 61.2%

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

   8. neg-mul-1 61.2%

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

   9. metadata-eval 61.2%

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

   10. fma-def 61.2%

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

   11. *-rgt-identity 61.2%

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

   12. +-commutative 61.2%

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

   13. distribute-neg-in 61.2%

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

   14. neg-mul-1 61.2%

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

   15. metadata-eval 61.2%

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

   16. fma-def 61.2%

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

5. Applied egg-rr 61.2%

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

6. Taylor expanded in x around 0 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. *-commutative 99.7%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 4: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x + -1\right) - x \cdot \left(x \cdot \left(1 - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e+17)
   (/ 2.0 (pow x 3.0))
   (/ 2.0 (- (* x (+ x -1.0)) (* x (* x (- 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1e+17) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = 2.0 / ((x * (x + -1.0)) - (x * (x * (1.0 - x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d+17)) then
        tmp = 2.0d0 / (x ** 3.0d0)
    else
        tmp = 2.0d0 / ((x * (x + (-1.0d0))) - (x * (x * (1.0d0 - x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1e+17) {
		tmp = 2.0 / Math.pow(x, 3.0);
	} else {
		tmp = 2.0 / ((x * (x + -1.0)) - (x * (x * (1.0 - x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1e+17:
		tmp = 2.0 / math.pow(x, 3.0)
	else:
		tmp = 2.0 / ((x * (x + -1.0)) - (x * (x * (1.0 - x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e+17)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(x * Float64(x + -1.0)) - Float64(x * Float64(x * Float64(1.0 - x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1e+17)
		tmp = 2.0 / (x ^ 3.0);
	else
		tmp = 2.0 / ((x * (x + -1.0)) - (x * (x * (1.0 - x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1e+17], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - N[(x * N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e17

   1. Initial program 60.3%

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

      1. remove-double-neg 60.3%

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

      2. sub-neg 60.3%

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

      3. sub-neg 60.3%

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

      4. distribute-neg-frac 60.3%

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

      5. metadata-eval 60.3%

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

      6. metadata-eval 60.3%

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

      7. metadata-eval 60.3%

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

      8. associate-/r* 60.3%

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

      9. metadata-eval 60.3%

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

      10. neg-mul-1 60.3%

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

      11. associate--l+ 60.3%

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

      12. +-commutative 60.3%

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

      13. distribute-neg-frac 60.3%

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

      14. metadata-eval 60.3%

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

      15. metadata-eval 60.3%

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

      16. metadata-eval 60.3%

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

      17. associate-/r* 60.3%

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

      18. metadata-eval 60.3%

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

      19. neg-mul-1 60.3%

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

      20. sub0-neg 60.3%

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

      21. associate-+l- 60.3%

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

      22. neg-sub0 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in x around inf 98.8%

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

   if -1e17 < x

   1. Initial program 89.4%

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

      1. remove-double-neg 89.4%

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

      2. sub-neg 89.4%

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

      3. sub-neg 89.4%

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

      4. distribute-neg-frac 89.4%

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

      5. metadata-eval 89.4%

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

      6. metadata-eval 89.4%

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

      7. metadata-eval 89.4%

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

      8. associate-/r* 89.4%

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

      9. metadata-eval 89.4%

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

      10. neg-mul-1 89.4%

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

      11. associate--l+ 89.4%

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

      12. +-commutative 89.4%

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

      13. distribute-neg-frac 89.4%

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

      14. metadata-eval 89.4%

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

      15. metadata-eval 89.4%

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

      16. metadata-eval 89.4%

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

      17. associate-/r* 89.4%

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

      18. metadata-eval 89.4%

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

      19. neg-mul-1 89.4%

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

      20. sub0-neg 89.4%

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

      21. associate-+l- 89.4%

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

      22. neg-sub0 89.4%

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

   3. Simplified 89.4%

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

      1. frac-2neg 89.4%

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

      2. metadata-eval 89.4%

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

      3. frac-sub 73.5%

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

      4. frac-add 73.5%

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

      5. fma-def 73.5%

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

      6. +-commutative 73.5%

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

      7. distribute-neg-in 73.5%

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

      8. neg-mul-1 73.5%

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

      9. metadata-eval 73.5%

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

      10. fma-def 73.5%

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

      11. *-rgt-identity 73.5%

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

      12. +-commutative 73.5%

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

      13. distribute-neg-in 73.5%

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

      14. neg-mul-1 73.5%

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

      15. metadata-eval 73.5%

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

      16. fma-def 73.5%

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

   5. Applied egg-rr 73.5%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-udef 99.9%

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

      3. neg-mul-1 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. *-commutative 99.9%

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

      6. neg-mul-1 99.9%

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

      7. *-commutative 99.9%

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

      8. distribute-rgt-neg-in 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x + -1\right) - x \cdot \left(x \cdot \left(1 - x\right)\right)}\\ \end{array} \]

Alternative 5: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+104}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x + -1\right) - x \cdot \left(x \cdot \left(1 - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e+104) 0.0 (/ 2.0 (- (* x (+ x -1.0)) (* x (* x (- 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -2e+104) {
		tmp = 0.0;
	} else {
		tmp = 2.0 / ((x * (x + -1.0)) - (x * (x * (1.0 - x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d+104)) then
        tmp = 0.0d0
    else
        tmp = 2.0d0 / ((x * (x + (-1.0d0))) - (x * (x * (1.0d0 - x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2e+104) {
		tmp = 0.0;
	} else {
		tmp = 2.0 / ((x * (x + -1.0)) - (x * (x * (1.0 - x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2e+104:
		tmp = 0.0
	else:
		tmp = 2.0 / ((x * (x + -1.0)) - (x * (x * (1.0 - x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e+104)
		tmp = 0.0;
	else
		tmp = Float64(2.0 / Float64(Float64(x * Float64(x + -1.0)) - Float64(x * Float64(x * Float64(1.0 - x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2e+104)
		tmp = 0.0;
	else
		tmp = 2.0 / ((x * (x + -1.0)) - (x * (x * (1.0 - x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e+104], 0.0, N[(2.0 / N[(N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - N[(x * N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e104

   1. Initial program 98.4%

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

      1. remove-double-neg 98.4%

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

      2. sub-neg 98.4%

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

      3. sub-neg 98.4%

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

      4. distribute-neg-frac 98.4%

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

      5. metadata-eval 98.4%

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

      6. metadata-eval 98.4%

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

      7. metadata-eval 98.4%

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

      8. associate-/r* 98.4%

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

      9. metadata-eval 98.4%

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

      10. neg-mul-1 98.4%

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

      11. associate--l+ 98.4%

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

      12. +-commutative 98.4%

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

      13. distribute-neg-frac 98.4%

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

      14. metadata-eval 98.4%

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

      15. metadata-eval 98.4%

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

      16. metadata-eval 98.4%

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

      17. associate-/r* 98.4%

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

      18. metadata-eval 98.4%

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

      19. neg-mul-1 98.4%

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

      20. sub0-neg 98.4%

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

      21. associate-+l- 98.4%

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

      22. neg-sub0 98.4%

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

   3. Simplified 98.4%

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

      1. frac-sub 23.6%

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

      2. div-inv 22.0%

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

      3. *-rgt-identity 22.0%

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

   5. Applied egg-rr 22.0%

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

   6. Taylor expanded in x around inf 22.0%

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

   8. Applied egg-rr 25.7%

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

      1. *-commutative 25.7%

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

      2. neg-mul-1 25.7%

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

      3. sub-neg 25.7%

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

      4. div-sub 25.7%

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

      5. +-inverses 98.4%

         \[\leadsto \color{blue}{0} \]

   10. Simplified 98.4%

       \[\leadsto \color{blue}{0} \]

   if -2e104 < x

   1. Initial program 80.7%

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

      1. remove-double-neg 80.7%

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

      2. sub-neg 80.7%

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

      3. sub-neg 80.7%

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

      4. distribute-neg-frac 80.7%

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

      5. metadata-eval 80.7%

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

      6. metadata-eval 80.7%

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

      7. metadata-eval 80.7%

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

      8. associate-/r* 80.7%

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

      9. metadata-eval 80.7%

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

      10. neg-mul-1 80.7%

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

      11. associate--l+ 80.7%

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

      12. +-commutative 80.7%

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

      13. distribute-neg-frac 80.7%

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

      14. metadata-eval 80.7%

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

      15. metadata-eval 80.7%

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

      16. metadata-eval 80.7%

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

      17. associate-/r* 80.7%

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

      18. metadata-eval 80.7%

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

      19. neg-mul-1 80.7%

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

      20. sub0-neg 80.7%

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

      21. associate-+l- 80.7%

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

      22. neg-sub0 80.7%

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

   3. Simplified 80.7%

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

      1. frac-2neg 80.7%

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

      2. metadata-eval 80.7%

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

      3. frac-sub 66.5%

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

      4. frac-add 66.5%

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

      5. fma-def 66.5%

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

      6. +-commutative 66.5%

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

      7. distribute-neg-in 66.5%

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

      8. neg-mul-1 66.5%

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

      9. metadata-eval 66.5%

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

      10. fma-def 66.5%

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

      11. *-rgt-identity 66.5%

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

      12. +-commutative 66.5%

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

      13. distribute-neg-in 66.5%

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

      14. neg-mul-1 66.5%

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

      15. metadata-eval 66.5%

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

      16. fma-def 66.5%

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

   5. Applied egg-rr 66.5%

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

   6. Taylor expanded in x around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. fma-udef 99.8%

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

      3. neg-mul-1 99.8%

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

      4. distribute-lft-in 99.9%

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

      5. *-commutative 99.9%

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

      6. neg-mul-1 99.9%

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

      7. *-commutative 99.9%

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

      8. distribute-rgt-neg-in 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+104}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x + -1\right) - x \cdot \left(x \cdot \left(1 - x\right)\right)}\\ \end{array} \]

Alternative 6: 85.0% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 83.0%

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

2. Final simplification 83.0%

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

Alternative 7: 83.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 1.0) (/ -2.0 x) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = -2.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 0.0d0
    else if (x <= 1.0d0) then
        tmp = (-2.0d0) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = -2.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0
	elif x <= 1.0:
		tmp = -2.0 / x
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 1.0], N[(-2.0 / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 63.4%

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

      1. remove-double-neg 63.4%

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

      2. sub-neg 63.4%

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

      3. sub-neg 63.4%

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

      4. distribute-neg-frac 63.4%

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

      5. metadata-eval 63.4%

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

      6. metadata-eval 63.4%

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

      7. metadata-eval 63.4%

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

      8. associate-/r* 63.4%

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

      9. metadata-eval 63.4%

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

      10. neg-mul-1 63.4%

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

      11. associate--l+ 63.4%

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

      12. +-commutative 63.4%

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

      13. distribute-neg-frac 63.4%

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

      14. metadata-eval 63.4%

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

      15. metadata-eval 63.4%

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

      16. metadata-eval 63.4%

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

      17. associate-/r* 63.4%

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

      18. metadata-eval 63.4%

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

      19. neg-mul-1 63.4%

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

      20. sub0-neg 63.4%

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

      21. associate-+l- 63.4%

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

      22. neg-sub0 63.4%

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

   3. Simplified 63.4%

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

      1. frac-sub 16.1%

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

      2. div-inv 14.8%

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

      3. *-rgt-identity 14.8%

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

   5. Applied egg-rr 14.8%

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

   6. Taylor expanded in x around inf 13.8%

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

   8. Applied egg-rr 15.1%

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

      1. *-commutative 15.1%

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

      2. neg-mul-1 15.1%

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

      3. sub-neg 15.1%

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

      4. div-sub 15.1%

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

      5. +-inverses 62.1%

         \[\leadsto \color{blue}{0} \]

   10. Simplified 62.1%

       \[\leadsto \color{blue}{0} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate--l+ 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. associate-/r* 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. sub0-neg 100.0%

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

      21. associate-+l- 100.0%

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

      22. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.1%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 83.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.0%

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

   1. remove-double-neg 83.0%

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

   2. sub-neg 83.0%

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

   3. sub-neg 83.0%

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

   4. distribute-neg-frac 83.0%

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

   5. metadata-eval 83.0%

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

   6. metadata-eval 83.0%

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

   7. metadata-eval 83.0%

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

   8. associate-/r* 83.0%

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

   9. metadata-eval 83.0%

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

   10. neg-mul-1 83.0%

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

   11. associate--l+ 83.0%

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

   12. +-commutative 83.0%

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

   13. distribute-neg-frac 83.0%

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

   14. metadata-eval 83.0%

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

   15. metadata-eval 83.0%

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

   16. metadata-eval 83.0%

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

   17. associate-/r* 83.0%

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

   18. metadata-eval 83.0%

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

   19. neg-mul-1 83.0%

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

   20. sub0-neg 83.0%

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

   21. associate-+l- 83.0%

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

   22. neg-sub0 83.0%

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

3. Simplified 83.0%

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

4. Taylor expanded in x around 0 54.5%

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

5. Taylor expanded in x around 0 81.9%

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

   1. +-commutative 81.9%

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

   2. distribute-neg-in 81.9%

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

   3. associate-*r/ 81.9%

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

   4. metadata-eval 81.9%

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

   5. distribute-neg-frac 81.9%

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

   6. metadata-eval 81.9%

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

   7. metadata-eval 81.9%

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

7. Simplified 81.9%

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

8. Final simplification 81.9%

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

Alternative 9: 35.8% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 83.0%

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

   1. remove-double-neg 83.0%

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

   2. sub-neg 83.0%

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

   3. sub-neg 83.0%

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

   4. distribute-neg-frac 83.0%

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

   5. metadata-eval 83.0%

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

   6. metadata-eval 83.0%

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

   7. metadata-eval 83.0%

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

   8. associate-/r* 83.0%

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

   9. metadata-eval 83.0%

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

   10. neg-mul-1 83.0%

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

   11. associate--l+ 83.0%

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

   12. +-commutative 83.0%

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

   13. distribute-neg-frac 83.0%

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

   14. metadata-eval 83.0%

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

   15. metadata-eval 83.0%

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

   16. metadata-eval 83.0%

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

   17. associate-/r* 83.0%

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

   18. metadata-eval 83.0%

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

   19. neg-mul-1 83.0%

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

   20. sub0-neg 83.0%

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

   21. associate-+l- 83.0%

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

   22. neg-sub0 83.0%

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

3. Simplified 83.0%

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

   1. frac-sub 61.0%

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

   2. div-inv 60.4%

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

   3. *-rgt-identity 60.4%

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

5. Applied egg-rr 60.4%

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

6. Taylor expanded in x around inf 8.1%

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

8. Applied egg-rr 8.2%

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

   1. *-commutative 8.2%

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

   2. neg-mul-1 8.2%

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

   3. sub-neg 8.2%

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

   4. div-sub 8.2%

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

   5. +-inverses 30.1%

      \[\leadsto \color{blue}{0} \]

10. Simplified 30.1%

    \[\leadsto \color{blue}{0} \]

11. Final simplification 30.1%

    \[\leadsto 0 \]

Developer target: 99.5% 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 2023310 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

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

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