3frac (problem 3.3.3)

Percentage Accurate: 69.2% → 99.4%

Time: 9.4s

Alternatives: 10

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: 69.2% 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: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \left(2 - 2 \cdot \frac{-1}{x \cdot x}\right) \cdot {x}^{-3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (- 2.0 (* 2.0 (/ -1.0 (* x x)))) (pow x -3.0)))

C

double code(double x) {
	return (2.0 - (2.0 * (-1.0 / (x * x)))) * pow(x, -3.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 - (2.0d0 * ((-1.0d0) / (x * x)))) * (x ** (-3.0d0))
end function

Java

public static double code(double x) {
	return (2.0 - (2.0 * (-1.0 / (x * x)))) * Math.pow(x, -3.0);
}

Python

def code(x):
	return (2.0 - (2.0 * (-1.0 / (x * x)))) * math.pow(x, -3.0)

Julia

function code(x)
	return Float64(Float64(2.0 - Float64(2.0 * Float64(-1.0 / Float64(x * x)))) * (x ^ -3.0))
end

MATLAB

function tmp = code(x)
	tmp = (2.0 - (2.0 * (-1.0 / (x * x)))) * (x ^ -3.0);
end

Wolfram

code[x_] := N[(N[(2.0 - N[(2.0 * N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

   1. +-commutative 65.7%

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

   2. associate-+r- 65.6%

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

   3. sub-neg 65.6%

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

   4. remove-double-neg 65.6%

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

   5. neg-sub0 65.6%

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

   6. associate-+l- 65.6%

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

   7. neg-sub0 65.6%

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

   8. distribute-neg-frac2 65.6%

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

   9. distribute-frac-neg2 65.6%

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

   10. associate-+r+ 65.7%

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

   11. +-commutative 65.7%

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

   12. remove-double-neg 65.7%

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

   13. distribute-neg-frac2 65.7%

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

   14. sub0-neg 65.7%

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

   15. associate-+l- 65.7%

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

   16. neg-sub0 65.7%

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

3. Simplified 65.7%

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

5. Taylor expanded in x around inf 99.4%

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

   1. associate-*r/ 99.4%

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

   2. metadata-eval 99.4%

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

7. Simplified 99.4%

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

   1. div-inv 99.4%

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

   2. +-commutative 99.4%

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

   3. div-inv 99.4%

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

   4. fma-define 99.4%

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

   5. pow-flip 99.4%

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

   6. metadata-eval 99.4%

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

   7. pow-flip 100.0%

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

   8. metadata-eval 100.0%

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

9. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot {x}^{-3}} \]
10. Step-by-step derivation

    1. fma-undefine 100.0%

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

11. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{\left(2 \cdot {x}^{-2} + 2\right)} \cdot {x}^{-3} \]
12. Step-by-step derivation

    1. metadata-eval 100.0%

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

    2. pow-prod-up 100.0%

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

    3. inv-pow 100.0%

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

    4. inv-pow 100.0%

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

    5. frac-2neg 100.0%

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

    6. metadata-eval 100.0%

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

    7. frac-times 100.0%

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

    8. metadata-eval 100.0%

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

13. Applied egg-rr 100.0%

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

14. Final simplification 100.0%

    \[\leadsto \left(2 - 2 \cdot \frac{-1}{x \cdot x}\right) \cdot {x}^{-3} \]
15. Add Preprocessing

Alternative 2: 98.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. +-commutative 65.7%

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

   2. associate-+r- 65.6%

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

   3. sub-neg 65.6%

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

   4. remove-double-neg 65.6%

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

   5. neg-sub0 65.6%

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

   6. associate-+l- 65.6%

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

   7. neg-sub0 65.6%

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

   8. distribute-neg-frac2 65.6%

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

   9. distribute-frac-neg2 65.6%

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

   10. associate-+r+ 65.7%

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

   11. +-commutative 65.7%

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

   12. remove-double-neg 65.7%

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

   13. distribute-neg-frac2 65.7%

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

   14. sub0-neg 65.7%

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

   15. associate-+l- 65.7%

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

   16. neg-sub0 65.7%

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

3. Simplified 65.7%

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

5. Taylor expanded in x around inf 99.4%

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

   1. associate-*r/ 99.4%

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

   2. metadata-eval 99.4%

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

7. Simplified 99.4%

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

   1. div-inv 99.4%

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

   2. +-commutative 99.4%

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

   3. div-inv 99.4%

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

   4. fma-define 99.4%

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

   5. pow-flip 99.4%

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

   6. metadata-eval 99.4%

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

   7. pow-flip 100.0%

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

   8. metadata-eval 100.0%

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

9. Applied egg-rr 100.0%

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

10. Taylor expanded in x around inf 99.4%

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

Alternative 3: 99.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. +-commutative 65.7%

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

   2. associate-+r- 65.6%

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

   3. sub-neg 65.6%

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

   4. remove-double-neg 65.6%

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

   5. neg-sub0 65.6%

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

   6. associate-+l- 65.6%

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

   7. neg-sub0 65.6%

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

   8. distribute-neg-frac2 65.6%

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

   9. distribute-frac-neg2 65.6%

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

   10. associate-+r+ 65.7%

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

   11. +-commutative 65.7%

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

   12. remove-double-neg 65.7%

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

   13. distribute-neg-frac2 65.7%

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

   14. sub0-neg 65.7%

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

   15. associate-+l- 65.7%

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

   16. neg-sub0 65.7%

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

3. Simplified 65.7%

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

   1. +-commutative 65.7%

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

   2. associate-+l- 65.6%

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

6. Applied egg-rr 65.6%

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

   1. frac-sub 17.1%

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

   2. *-un-lft-identity 17.1%

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

8. Applied egg-rr 17.1%

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

   1. *-rgt-identity 17.1%

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

   2. associate--l+ 17.1%

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

   3. *-commutative 17.1%

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

10. Simplified 17.1%

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

    1. frac-sub 19.1%

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

    2. fmm-def 19.1%

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

    3. associate--r- 19.1%

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

    4. metadata-eval 19.1%

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

12. Applied egg-rr 19.1%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \left(x + -1\right) \cdot \left(-1 - x\right), -x \cdot \left(x + \left(0 + x\right)\right)\right)}{x \cdot \left(\left(x + -1\right) \cdot \left(-1 - x\right)\right)}} \]
13. Step-by-step derivation

    1. fmm-undef 19.1%

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

    2. +-lft-identity 19.1%

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

14. Simplified 19.1%

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

15. Taylor expanded in x around 0 99.4%

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

16. Final simplification 99.4%

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

Alternative 4: 67.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. +-commutative 65.7%

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

   2. associate-+r- 65.6%

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

   3. sub-neg 65.6%

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

   4. remove-double-neg 65.6%

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

   5. neg-sub0 65.6%

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

   6. associate-+l- 65.6%

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

   7. neg-sub0 65.6%

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

   8. distribute-neg-frac2 65.6%

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

   9. distribute-frac-neg2 65.6%

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

   10. associate-+r+ 65.7%

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

   11. +-commutative 65.7%

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

   12. remove-double-neg 65.7%

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

   13. distribute-neg-frac2 65.7%

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

   14. sub0-neg 65.7%

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

   15. associate-+l- 65.7%

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

   16. neg-sub0 65.7%

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

3. Simplified 65.7%

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

5. Taylor expanded in x around inf 65.1%

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

6. Taylor expanded in x around inf 64.8%

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

Alternative 5: 67.9% accurate, 1.7× speedup

Math

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

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

Derivation

1. Initial program 65.7%

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

   1. +-commutative 65.7%

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

   2. associate-+r- 65.6%

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

   3. sub-neg 65.6%

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

   4. remove-double-neg 65.6%

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

   5. neg-sub0 65.6%

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

   6. associate-+l- 65.6%

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

   7. neg-sub0 65.6%

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

   8. distribute-neg-frac2 65.6%

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

   9. distribute-frac-neg2 65.6%

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

   10. associate-+r+ 65.7%

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

   11. +-commutative 65.7%

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

   12. remove-double-neg 65.7%

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

   13. distribute-neg-frac2 65.7%

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

   14. sub0-neg 65.7%

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

   15. associate-+l- 65.7%

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

   16. neg-sub0 65.7%

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

3. Simplified 65.7%

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

5. Taylor expanded in x around inf 64.8%

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

6. Final simplification 64.8%

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

Alternative 6: 67.7% 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 65.7%

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

   1. +-commutative 65.7%

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

   2. associate-+r- 65.6%

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

   3. sub-neg 65.6%

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

   4. remove-double-neg 65.6%

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

   5. neg-sub0 65.6%

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

   6. associate-+l- 65.6%

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

   7. neg-sub0 65.6%

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

   8. distribute-neg-frac2 65.6%

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

   9. distribute-frac-neg2 65.6%

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

   10. associate-+r+ 65.7%

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

   11. +-commutative 65.7%

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

   12. remove-double-neg 65.7%

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

   13. distribute-neg-frac2 65.7%

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

   14. sub0-neg 65.7%

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

   15. associate-+l- 65.7%

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

   16. neg-sub0 65.7%

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

3. Simplified 65.7%

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

   1. +-commutative 65.7%

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

   2. associate-+l- 65.6%

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

6. Applied egg-rr 65.6%

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

   1. frac-sub 17.1%

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

   2. *-un-lft-identity 17.1%

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

8. Applied egg-rr 17.1%

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

   1. *-rgt-identity 17.1%

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

   2. associate--l+ 17.1%

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

   3. *-commutative 17.1%

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

10. Simplified 17.1%

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

11. Taylor expanded in x around inf 64.5%

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

Alternative 7: 67.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. +-commutative 65.7%

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

   2. associate-+r- 65.6%

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

   3. sub-neg 65.6%

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

   4. remove-double-neg 65.6%

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

   5. neg-sub0 65.6%

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

   6. associate-+l- 65.6%

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

   7. neg-sub0 65.6%

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

   8. distribute-neg-frac2 65.6%

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

   9. distribute-frac-neg2 65.6%

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

   10. associate-+r+ 65.7%

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

   11. +-commutative 65.7%

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

   12. remove-double-neg 65.7%

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

   13. distribute-neg-frac2 65.7%

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

   14. sub0-neg 65.7%

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

   15. associate-+l- 65.7%

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

   16. neg-sub0 65.7%

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

3. Simplified 65.7%

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

5. Taylor expanded in x around 0 3.4%

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

6. Taylor expanded in x around 0 64.4%

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

Alternative 8: 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 65.7%

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

   1. +-commutative 65.7%

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

   2. associate-+r- 65.6%

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

   3. sub-neg 65.6%

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

   4. remove-double-neg 65.6%

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

   5. neg-sub0 65.6%

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

   6. associate-+l- 65.6%

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

   7. neg-sub0 65.6%

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

   8. distribute-neg-frac2 65.6%

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

   9. distribute-frac-neg2 65.6%

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

   10. associate-+r+ 65.7%

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

   11. +-commutative 65.7%

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

   12. remove-double-neg 65.7%

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

   13. distribute-neg-frac2 65.7%

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

   14. sub0-neg 65.7%

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

   15. associate-+l- 65.7%

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

   16. neg-sub0 65.7%

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

3. Simplified 65.7%

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

5. Taylor expanded in x around 0 3.4%

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

6. Taylor expanded in x around inf 3.4%

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

7. Taylor expanded in x around 0 5.0%

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

Alternative 9: 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 65.7%

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

   1. +-commutative 65.7%

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

   2. associate-+r- 65.6%

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

   3. sub-neg 65.6%

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

   4. remove-double-neg 65.6%

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

   5. neg-sub0 65.6%

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

   6. associate-+l- 65.6%

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

   7. neg-sub0 65.6%

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

   8. distribute-neg-frac2 65.6%

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

   9. distribute-frac-neg2 65.6%

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

   10. associate-+r+ 65.7%

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

   11. +-commutative 65.7%

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

   12. remove-double-neg 65.7%

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

   13. distribute-neg-frac2 65.7%

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

   14. sub0-neg 65.7%

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

   15. associate-+l- 65.7%

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

   16. neg-sub0 65.7%

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

3. Simplified 65.7%

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

5. Taylor expanded in x around 0 5.0%

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

Alternative 10: 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 65.7%

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

   1. +-commutative 65.7%

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

   2. associate-+r- 65.6%

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

   3. sub-neg 65.6%

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

   4. remove-double-neg 65.6%

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

   5. neg-sub0 65.6%

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

   6. associate-+l- 65.6%

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

   7. neg-sub0 65.6%

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

   8. distribute-neg-frac2 65.6%

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

   9. distribute-frac-neg2 65.6%

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

   10. associate-+r+ 65.7%

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

   11. +-commutative 65.7%

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

   12. remove-double-neg 65.7%

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

   13. distribute-neg-frac2 65.7%

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

   14. sub0-neg 65.7%

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

   15. associate-+l- 65.7%

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

   16. neg-sub0 65.7%

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

3. Simplified 65.7%

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

5. Taylor expanded in x around 0 3.4%

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

6. Taylor expanded in x around inf 3.4%

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

Developer Target 1: 99.1% 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 2024151 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

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

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