3frac (problem 3.3.3)

Percentage Accurate: 68.5% → 99.2%

Time: 9.5s

Alternatives: 7

Speedup: 1.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.2%

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

   1. sub-neg 69.2%

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

   2. distribute-neg-frac 69.2%

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

   3. metadata-eval 69.2%

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

   4. metadata-eval 69.2%

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

   5. metadata-eval 69.2%

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

   6. associate-/r* 69.2%

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

   7. metadata-eval 69.2%

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

   8. neg-mul-1 69.2%

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

   9. +-commutative 69.2%

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

   10. associate-+l+ 69.2%

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

   11. +-commutative 69.2%

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

   12. neg-mul-1 69.2%

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

   13. metadata-eval 69.2%

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

   14. associate-/r* 69.2%

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

   15. metadata-eval 69.2%

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

   16. metadata-eval 69.2%

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

   17. +-commutative 69.2%

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

   18. +-commutative 69.2%

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

   19. sub-neg 69.2%

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

   20. metadata-eval 69.2%

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

3. Simplified 69.2%

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

   1. +-commutative 69.2%

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

   2. frac-add 21.0%

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

   3. frac-add 20.6%

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

   4. *-un-lft-identity 20.6%

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

   5. *-rgt-identity 20.6%

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

   6. +-commutative 20.6%

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

   7. +-commutative 20.6%

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

   8. +-commutative 20.6%

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

   9. +-commutative 20.6%

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

6. Applied egg-rr 20.6%

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

7. Taylor expanded in x around 0 99.6%

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

   1. expm1-log1p-u 99.6%

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

   2. expm1-udef 68.8%

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

   3. *-commutative 68.8%

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

   4. metadata-eval 68.8%

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

   5. sub-neg 68.8%

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

   6. difference-of-sqr-1 68.8%

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

   7. distribute-rgt-out-- 68.8%

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

   8. unpow3 68.8%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{{x}^{3}} - 1 \cdot x}\right)} - 1 \]

   9. *-un-lft-identity 68.8%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{{x}^{3} - \color{blue}{x}}\right)} - 1 \]

9. Applied egg-rr 68.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{x}^{3} - x}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 99.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{3} - x}\right)\right)} \]

    2. expm1-log1p 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

    \[\leadsto \frac{2}{{x}^{3} - x} \]
13. Add Preprocessing

Alternative 2: 99.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.2%

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

   1. sub-neg 69.2%

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

   2. distribute-neg-frac 69.2%

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

   3. metadata-eval 69.2%

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

   4. metadata-eval 69.2%

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

   5. metadata-eval 69.2%

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

   6. associate-/r* 69.2%

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

   7. metadata-eval 69.2%

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

   8. neg-mul-1 69.2%

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

   9. +-commutative 69.2%

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

   10. associate-+l+ 69.2%

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

   11. +-commutative 69.2%

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

   12. neg-mul-1 69.2%

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

   13. metadata-eval 69.2%

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

   14. associate-/r* 69.2%

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

   15. metadata-eval 69.2%

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

   16. metadata-eval 69.2%

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

   17. +-commutative 69.2%

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

   18. +-commutative 69.2%

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

   19. sub-neg 69.2%

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

   20. metadata-eval 69.2%

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

3. Simplified 69.2%

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

   1. +-commutative 69.2%

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

   2. frac-add 21.0%

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

   3. frac-add 20.6%

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

   4. *-un-lft-identity 20.6%

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

   5. *-rgt-identity 20.6%

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

   6. +-commutative 20.6%

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

   7. +-commutative 20.6%

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

   8. +-commutative 20.6%

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

   9. +-commutative 20.6%

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

6. Applied egg-rr 20.6%

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

7. Taylor expanded in x around 0 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.2%

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

   1. sub-neg 69.2%

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

   2. distribute-neg-frac 69.2%

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

   3. metadata-eval 69.2%

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

   4. metadata-eval 69.2%

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

   5. metadata-eval 69.2%

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

   6. associate-/r* 69.2%

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

   7. metadata-eval 69.2%

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

   8. neg-mul-1 69.2%

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

   9. +-commutative 69.2%

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

   10. associate-+l+ 69.2%

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

   11. +-commutative 69.2%

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

   12. neg-mul-1 69.2%

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

   13. metadata-eval 69.2%

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

   14. associate-/r* 69.2%

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

   15. metadata-eval 69.2%

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

   16. metadata-eval 69.2%

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

   17. +-commutative 69.2%

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

   18. +-commutative 69.2%

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

   19. sub-neg 69.2%

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

   20. metadata-eval 69.2%

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

3. Simplified 69.2%

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

   1. +-commutative 69.2%

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

   2. frac-add 21.0%

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

   3. frac-add 20.6%

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

   4. *-un-lft-identity 20.6%

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

   5. *-rgt-identity 20.6%

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

   6. +-commutative 20.6%

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

   7. +-commutative 20.6%

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

   8. +-commutative 20.6%

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

   9. +-commutative 20.6%

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

6. Applied egg-rr 20.6%

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

7. Taylor expanded in x around 0 99.6%

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

   1. expm1-log1p-u 48.9%

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

   2. expm1-udef 48.9%

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

   3. associate-*l* 48.9%

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

   4. +-commutative 48.9%

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

9. Applied egg-rr 48.9%

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

    1. expm1-def 48.9%

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

    2. expm1-log1p 99.6%

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

    3. +-commutative 99.6%

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

    4. *-commutative 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 4: 67.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.2%

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

   1. sub-neg 69.2%

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

   2. distribute-neg-frac 69.2%

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

   3. metadata-eval 69.2%

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

   4. metadata-eval 69.2%

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

   5. metadata-eval 69.2%

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

   6. associate-/r* 69.2%

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

   7. metadata-eval 69.2%

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

   8. neg-mul-1 69.2%

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

   9. +-commutative 69.2%

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

   10. associate-+l+ 69.2%

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

   11. +-commutative 69.2%

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

   12. neg-mul-1 69.2%

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

   13. metadata-eval 69.2%

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

   14. associate-/r* 69.2%

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

   15. metadata-eval 69.2%

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

   16. metadata-eval 69.2%

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

   17. +-commutative 69.2%

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

   18. +-commutative 69.2%

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

   19. sub-neg 69.2%

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

   20. metadata-eval 69.2%

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

3. Simplified 69.2%

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

5. Taylor expanded in x around inf 67.2%

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

6. Final simplification 67.2%

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

Alternative 5: 5.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.2%

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

   1. sub-neg 69.2%

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

   2. distribute-neg-frac 69.2%

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

   3. metadata-eval 69.2%

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

   4. metadata-eval 69.2%

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

   5. metadata-eval 69.2%

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

   6. associate-/r* 69.2%

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

   7. metadata-eval 69.2%

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

   8. neg-mul-1 69.2%

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

   9. +-commutative 69.2%

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

   10. associate-+l+ 69.2%

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

   11. +-commutative 69.2%

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

   12. neg-mul-1 69.2%

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

   13. metadata-eval 69.2%

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

   14. associate-/r* 69.2%

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

   15. metadata-eval 69.2%

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

   16. metadata-eval 69.2%

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

   17. +-commutative 69.2%

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

   18. +-commutative 69.2%

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

   19. sub-neg 69.2%

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

   20. metadata-eval 69.2%

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

3. Simplified 69.2%

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

5. Taylor expanded in x around 0 4.9%

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

6. Final simplification 4.9%

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

Alternative 6: 3.3% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -9 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -9.0)

C

double code(double x) {
	return -9.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -9.0d0
end function

Java

public static double code(double x) {
	return -9.0;
}

Python

def code(x):
	return -9.0

Julia

function code(x)
	return -9.0
end

MATLAB

function tmp = code(x)
	tmp = -9.0;
end

Wolfram

code[x_] := -9.0

Derivation

1. Initial program 69.2%

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

   1. sub-neg 69.2%

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

   2. distribute-neg-frac 69.2%

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

   3. metadata-eval 69.2%

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

   4. metadata-eval 69.2%

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

   5. metadata-eval 69.2%

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

   6. associate-/r* 69.2%

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

   7. metadata-eval 69.2%

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

   8. neg-mul-1 69.2%

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

   9. +-commutative 69.2%

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

   10. associate-+l+ 69.2%

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

   11. +-commutative 69.2%

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

   12. neg-mul-1 69.2%

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

   13. metadata-eval 69.2%

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

   14. associate-/r* 69.2%

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

   15. metadata-eval 69.2%

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

   16. metadata-eval 69.2%

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

   17. +-commutative 69.2%

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

   18. +-commutative 69.2%

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

   19. sub-neg 69.2%

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

   20. metadata-eval 69.2%

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

3. Simplified 69.2%

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

   1. frac-add 21.0%

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

   2. div-inv 16.5%

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

   3. *-un-lft-identity 16.5%

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

   4. *-rgt-identity 16.5%

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

   5. +-commutative 16.5%

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

   6. +-commutative 16.5%

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

   7. metadata-eval 16.5%

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

   8. frac-times 19.0%

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

   9. clear-num 19.0%

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

   10. frac-times 16.5%

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

   11. metadata-eval 16.5%

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

   12. +-commutative 16.5%

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

   13. /-rgt-identity 16.5%

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

6. Applied egg-rr 16.5%

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

   1. associate-*r/ 21.0%

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

   2. *-rgt-identity 21.0%

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

   3. associate-+l+ 21.0%

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

   4. +-commutative 21.0%

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

   5. associate-+r+ 21.0%

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

   6. metadata-eval 21.0%

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

   7. *-commutative 21.0%

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

8. Simplified 21.0%

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

10. Applied egg-rr 0.0%

    \[\leadsto \color{blue}{\frac{\frac{-8}{{x}^{3}} + \frac{\frac{0}{0}}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{3}}}{\frac{4}{{x}^{2}} + \frac{0}{0 \cdot \mathsf{fma}\left(x, x, -1\right)} \cdot \left(\frac{0}{0 \cdot \mathsf{fma}\left(x, x, -1\right)} - \frac{-2}{x}\right)}} \]

12. Simplified 3.4%

    \[\leadsto \color{blue}{-9} \]

13. Final simplification 3.4%

    \[\leadsto -9 \]
14. Add Preprocessing

Alternative 7: 3.4% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 69.2%

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

   1. associate-+l- 69.2%

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

   2. sub-neg 69.2%

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

   3. +-commutative 69.2%

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

   4. neg-sub0 69.2%

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

   5. associate-+l- 69.2%

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

   6. neg-sub0 69.2%

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

   7. distribute-neg-frac 69.2%

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

   8. metadata-eval 69.2%

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

   9. sub-neg 69.2%

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

   10. metadata-eval 69.2%

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

3. Simplified 69.2%

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

   1. clear-num 69.2%

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

   2. frac-2neg 69.2%

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

   3. metadata-eval 69.2%

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

   4. frac-add 21.1%

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

   5. *-un-lft-identity 21.1%

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

   6. +-commutative 21.1%

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

   7. distribute-neg-in 21.1%

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

   8. metadata-eval 21.1%

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

   9. sub-neg 21.1%

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

   10. div-inv 21.1%

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

   11. metadata-eval 21.1%

       \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot \color{blue}{-0.5}\right) \cdot -1}{\frac{x}{-2} \cdot \left(-\left(x + -1\right)\right)} \]

   12. div-inv 21.1%

       \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\color{blue}{\left(x \cdot \frac{1}{-2}\right)} \cdot \left(-\left(x + -1\right)\right)} \]

   13. metadata-eval 21.1%

       \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\left(x \cdot \color{blue}{-0.5}\right) \cdot \left(-\left(x + -1\right)\right)} \]

   14. +-commutative 21.1%

       \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\left(x \cdot -0.5\right) \cdot \left(-\color{blue}{\left(-1 + x\right)}\right)} \]

   15. distribute-neg-in 21.1%

       \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\left(x \cdot -0.5\right) \cdot \color{blue}{\left(\left(--1\right) + \left(-x\right)\right)}} \]

   16. metadata-eval 21.1%

       \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\left(x \cdot -0.5\right) \cdot \left(\color{blue}{1} + \left(-x\right)\right)} \]

   17. sub-neg 21.1%

       \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\left(x \cdot -0.5\right) \cdot \color{blue}{\left(1 - x\right)}} \]

6. Applied egg-rr 21.1%

   \[\leadsto \frac{1}{1 + x} + \color{blue}{\frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)}} \]
7. Step-by-step derivation

   1. associate-+l- 21.1%

      \[\leadsto \frac{1}{1 + x} + \frac{\color{blue}{1 - \left(x - \left(x \cdot -0.5\right) \cdot -1\right)}}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)} \]

   2. *-commutative 21.1%

      \[\leadsto \frac{1}{1 + x} + \frac{1 - \left(x - \color{blue}{-1 \cdot \left(x \cdot -0.5\right)}\right)}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)} \]

   3. associate-*r* 21.1%

      \[\leadsto \frac{1}{1 + x} + \frac{1 - \left(x - \color{blue}{\left(-1 \cdot x\right) \cdot -0.5}\right)}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)} \]

   4. neg-mul-1 21.1%

      \[\leadsto \frac{1}{1 + x} + \frac{1 - \left(x - \color{blue}{\left(-x\right)} \cdot -0.5\right)}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)} \]

   5. cancel-sign-sub 21.1%

      \[\leadsto \frac{1}{1 + x} + \frac{1 - \color{blue}{\left(x + x \cdot -0.5\right)}}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)} \]

   6. +-commutative 21.1%

      \[\leadsto \frac{1}{1 + x} + \frac{1 - \color{blue}{\left(x \cdot -0.5 + x\right)}}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)} \]

   7. *-commutative 21.1%

      \[\leadsto \frac{1}{1 + x} + \frac{1 - \left(\color{blue}{-0.5 \cdot x} + x\right)}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)} \]

   8. distribute-lft1-in 21.1%

      \[\leadsto \frac{1}{1 + x} + \frac{1 - \color{blue}{\left(-0.5 + 1\right) \cdot x}}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)} \]

   9. metadata-eval 21.1%

      \[\leadsto \frac{1}{1 + x} + \frac{1 - \color{blue}{0.5} \cdot x}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)} \]

   10. metadata-eval 21.1%

       \[\leadsto \frac{1}{1 + x} + \frac{1 - \color{blue}{\frac{1}{2}} \cdot x}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)} \]

   11. *-commutative 21.1%

       \[\leadsto \frac{1}{1 + x} + \frac{1 - \color{blue}{x \cdot \frac{1}{2}}}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)} \]

   12. metadata-eval 21.1%

       \[\leadsto \frac{1}{1 + x} + \frac{1 - x \cdot \color{blue}{0.5}}{\left(x \cdot -0.5\right) \cdot \left(1 - x\right)} \]

   13. associate-*l* 21.1%

       \[\leadsto \frac{1}{1 + x} + \frac{1 - x \cdot 0.5}{\color{blue}{x \cdot \left(-0.5 \cdot \left(1 - x\right)\right)}} \]

8. Simplified 21.1%

   \[\leadsto \frac{1}{1 + x} + \color{blue}{\frac{1 - x \cdot 0.5}{x \cdot \left(-0.5 \cdot \left(1 - x\right)\right)}} \]

9. Taylor expanded in x around 0 3.3%

   \[\leadsto \frac{1}{1 + x} + \frac{1 - x \cdot 0.5}{\color{blue}{-0.5 \cdot x}} \]

10. Taylor expanded in x around inf 3.3%

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

11. Final simplification 3.3%

    \[\leadsto 1 \]
12. Add Preprocessing

Developer target: 99.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024020 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

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

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