3frac (problem 3.3.3)

Percentage Accurate: 69.3% → 99.5%

Time: 12.6s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. +-commutative 75.1%

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

   2. associate-+r- 75.0%

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

   3. sub-neg 75.0%

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

   4. remove-double-neg 75.0%

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

   5. neg-sub0 75.0%

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

   6. associate-+l- 75.0%

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

   7. neg-sub0 75.0%

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

   8. distribute-neg-frac2 75.0%

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

   9. distribute-frac-neg2 75.0%

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

   10. associate-+r+ 75.1%

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

   11. +-commutative 75.1%

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

   12. remove-double-neg 75.1%

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

   13. distribute-neg-frac2 75.1%

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

   14. sub0-neg 75.1%

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

   15. associate-+l- 75.1%

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

   16. neg-sub0 75.1%

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

3. Simplified 75.1%

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

5. Taylor expanded in x around inf 98.7%

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

   1. associate-*r/ 98.7%

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

   2. metadata-eval 98.7%

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

7. Simplified 98.7%

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

   1. div-inv 98.7%

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

   2. +-commutative 98.7%

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

   3. div-inv 98.7%

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

   4. fma-define 98.7%

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

   5. pow-flip 98.7%

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

   6. metadata-eval 98.7%

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

   7. pow-flip 99.2%

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

   8. metadata-eval 99.2%

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

9. Applied egg-rr 99.2%

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

Alternative 2: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{-3} \cdot \left(-4 + \frac{-4 + \frac{-4}{x}}{x}\right)}{-2 - \frac{2}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* (pow x -3.0) (+ -4.0 (/ (+ -4.0 (/ -4.0 x)) x))) (- -2.0 (/ 2.0 x))))

C

double code(double x) {
	return (pow(x, -3.0) * (-4.0 + ((-4.0 + (-4.0 / x)) / x))) / (-2.0 - (2.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x ** (-3.0d0)) * ((-4.0d0) + (((-4.0d0) + ((-4.0d0) / x)) / x))) / ((-2.0d0) - (2.0d0 / x))
end function

Java

public static double code(double x) {
	return (Math.pow(x, -3.0) * (-4.0 + ((-4.0 + (-4.0 / x)) / x))) / (-2.0 - (2.0 / x));
}

Python

def code(x):
	return (math.pow(x, -3.0) * (-4.0 + ((-4.0 + (-4.0 / x)) / x))) / (-2.0 - (2.0 / x))

Julia

function code(x)
	return Float64(Float64((x ^ -3.0) * Float64(-4.0 + Float64(Float64(-4.0 + Float64(-4.0 / x)) / x))) / Float64(-2.0 - Float64(2.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = ((x ^ -3.0) * (-4.0 + ((-4.0 + (-4.0 / x)) / x))) / (-2.0 - (2.0 / x));
end

Wolfram

code[x_] := N[(N[(N[Power[x, -3.0], $MachinePrecision] * N[(-4.0 + N[(N[(-4.0 + N[(-4.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-2.0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.1%

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

   1. +-commutative 75.1%

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

   2. associate-+r- 75.0%

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

   3. sub-neg 75.0%

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

   4. remove-double-neg 75.0%

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

   5. neg-sub0 75.0%

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

   6. associate-+l- 75.0%

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

   7. neg-sub0 75.0%

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

   8. distribute-neg-frac2 75.0%

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

   9. distribute-frac-neg2 75.0%

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

   10. associate-+r+ 75.1%

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

   11. +-commutative 75.1%

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

   12. remove-double-neg 75.1%

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

   13. distribute-neg-frac2 75.1%

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

   14. sub0-neg 75.1%

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

   15. associate-+l- 75.1%

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

   16. neg-sub0 75.1%

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

3. Simplified 75.1%

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

   1. +-commutative 75.1%

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

   2. flip-+ 75.1%

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

   3. pow2 75.1%

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

   4. sub-neg 75.1%

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

   5. distribute-neg-frac 75.1%

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

   6. metadata-eval 75.1%

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

   7. inv-pow 75.1%

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

   8. inv-pow 75.1%

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

   9. pow-prod-up 65.7%

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

   10. metadata-eval 65.7%

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

   11. sub-neg 65.7%

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

   12. distribute-neg-frac 65.7%

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

   13. metadata-eval 65.7%

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

6. Applied egg-rr 65.7%

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

7. Taylor expanded in x around inf 91.3%

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

   1. sub-neg 91.3%

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

   2. metadata-eval 91.3%

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

   3. +-commutative 91.3%

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

   4. associate-*r/ 91.3%

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

   5. neg-mul-1 91.3%

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

   6. distribute-neg-in 91.3%

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

   7. metadata-eval 91.3%

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

   8. unsub-neg 91.3%

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

   9. associate-*r/ 91.3%

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

   10. metadata-eval 91.3%

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

9. Simplified 91.3%

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

10. Taylor expanded in x around inf 91.3%

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

    1. associate-*r/ 91.3%

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

    2. neg-mul-1 91.3%

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

    3. associate-*r/ 91.3%

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

    4. metadata-eval 91.3%

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

    5. distribute-neg-in 91.3%

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

    6. metadata-eval 91.3%

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

    7. unsub-neg 91.3%

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

12. Simplified 91.3%

    \[\leadsto \frac{\frac{-4 + \frac{-4 - \frac{4}{x}}{x}}{{x}^{4}}}{\color{blue}{\frac{-2 - \frac{2}{x}}{x}}} \]
13. Step-by-step derivation

    1. *-un-lft-identity 91.3%

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

    2. associate-/r/ 91.3%

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

    3. div-inv 91.3%

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

    4. pow-flip 92.8%

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

    5. metadata-eval 92.8%

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

14. Applied egg-rr 92.8%

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

    1. *-lft-identity 92.8%

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

    2. associate-*l/ 92.8%

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

    3. associate-*l* 92.8%

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

    4. sub-neg 92.8%

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

    5. distribute-neg-frac 92.8%

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

    6. metadata-eval 92.8%

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

    7. pow-plus 99.1%

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

    8. metadata-eval 99.1%

       \[\leadsto \frac{\left(-4 + \frac{-4 + \frac{-4}{x}}{x}\right) \cdot {x}^{\color{blue}{-3}}}{-2 - \frac{2}{x}} \]

16. Simplified 99.1%

    \[\leadsto \color{blue}{\frac{\left(-4 + \frac{-4 + \frac{-4}{x}}{x}\right) \cdot {x}^{-3}}{-2 - \frac{2}{x}}} \]

17. Final simplification 99.1%

    \[\leadsto \frac{{x}^{-3} \cdot \left(-4 + \frac{-4 + \frac{-4}{x}}{x}\right)}{-2 - \frac{2}{x}} \]
18. Add Preprocessing

Alternative 3: 99.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. +-commutative 75.1%

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

   2. associate-+r- 75.0%

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

   3. sub-neg 75.0%

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

   4. remove-double-neg 75.0%

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

   5. neg-sub0 75.0%

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

   6. associate-+l- 75.0%

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

   7. neg-sub0 75.0%

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

   8. distribute-neg-frac2 75.0%

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

   9. distribute-frac-neg2 75.0%

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

   10. associate-+r+ 75.1%

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

   11. +-commutative 75.1%

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

   12. remove-double-neg 75.1%

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

   13. distribute-neg-frac2 75.1%

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

   14. sub0-neg 75.1%

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

   15. associate-+l- 75.1%

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

   16. neg-sub0 75.1%

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

3. Simplified 75.1%

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

5. Taylor expanded in x around inf 98.4%

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

   1. div-inv 98.4%

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

   2. pow-flip 98.9%

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

   3. metadata-eval 98.9%

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

7. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{2 \cdot {x}^{-3}} \]
8. Add Preprocessing

Alternative 4: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

3. Final simplification 75.1%

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

Alternative 5: 68.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. +-commutative 75.1%

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

   2. associate-+r- 75.0%

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

   3. sub-neg 75.0%

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

   4. remove-double-neg 75.0%

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

   5. neg-sub0 75.0%

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

   6. associate-+l- 75.0%

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

   7. neg-sub0 75.0%

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

   8. distribute-neg-frac2 75.0%

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

   9. distribute-frac-neg2 75.0%

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

   10. associate-+r+ 75.1%

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

   11. +-commutative 75.1%

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

   12. remove-double-neg 75.1%

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

   13. distribute-neg-frac2 75.1%

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

   14. sub0-neg 75.1%

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

   15. associate-+l- 75.1%

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

   16. neg-sub0 75.1%

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

3. Simplified 75.1%

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

5. Taylor expanded in x around inf 74.0%

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

   1. frac-add 74.0%

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

   2. *-un-lft-identity 74.0%

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

7. Applied egg-rr 74.0%

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

   1. *-commutative 74.0%

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

   2. +-commutative 74.0%

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

   3. distribute-lft-in 74.0%

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

   4. metadata-eval 74.0%

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

   5. mul-1-neg 74.0%

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

   6. unsub-neg 74.0%

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

   7. *-commutative 74.0%

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

9. Simplified 74.0%

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

Alternative 6: 68.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. +-commutative 75.1%

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

   2. associate-+r- 75.0%

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

   3. sub-neg 75.0%

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

   4. remove-double-neg 75.0%

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

   5. neg-sub0 75.0%

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

   6. associate-+l- 75.0%

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

   7. neg-sub0 75.0%

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

   8. distribute-neg-frac2 75.0%

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

   9. distribute-frac-neg2 75.0%

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

   10. associate-+r+ 75.1%

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

   11. +-commutative 75.1%

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

   12. remove-double-neg 75.1%

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

   13. distribute-neg-frac2 75.1%

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

   14. sub0-neg 75.1%

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

   15. associate-+l- 75.1%

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

   16. neg-sub0 75.1%

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

3. Simplified 75.1%

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

5. Taylor expanded in x around inf 74.0%

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

Alternative 7: 5.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. +-commutative 75.1%

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

   2. associate-+r- 75.0%

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

   3. sub-neg 75.0%

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

   4. remove-double-neg 75.0%

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

   5. neg-sub0 75.0%

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

   6. associate-+l- 75.0%

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

   7. neg-sub0 75.0%

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

   8. distribute-neg-frac2 75.0%

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

   9. distribute-frac-neg2 75.0%

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

   10. associate-+r+ 75.1%

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

   11. +-commutative 75.1%

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

   12. remove-double-neg 75.1%

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

   13. distribute-neg-frac2 75.1%

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

   14. sub0-neg 75.1%

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

   15. associate-+l- 75.1%

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

   16. neg-sub0 75.1%

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

3. Simplified 75.1%

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

5. Taylor expanded in x around inf 74.0%

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

6. Taylor expanded in x around 0 5.4%

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

Alternative 8: 5.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. +-commutative 75.1%

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

   2. associate-+r- 75.0%

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

   3. sub-neg 75.0%

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

   4. remove-double-neg 75.0%

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

   5. neg-sub0 75.0%

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

   6. associate-+l- 75.0%

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

   7. neg-sub0 75.0%

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

   8. distribute-neg-frac2 75.0%

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

   9. distribute-frac-neg2 75.0%

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

   10. associate-+r+ 75.1%

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

   11. +-commutative 75.1%

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

   12. remove-double-neg 75.1%

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

   13. distribute-neg-frac2 75.1%

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

   14. sub0-neg 75.1%

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

   15. associate-+l- 75.1%

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

   16. neg-sub0 75.1%

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

3. Simplified 75.1%

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

5. Taylor expanded in x around 0 5.3%

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

Developer target: 99.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024107 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

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

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