3frac (problem 3.3.3)

Percentage Accurate: 69.3% → 99.7%

Time: 8.5s

Alternatives: 7

Speedup: 1.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.7% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

   1. +-commutative 65.3%

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

   2. associate-+r- 65.3%

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

   3. sub-neg 65.3%

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

   4. remove-double-neg 65.3%

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

   5. neg-sub0 65.3%

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

   6. associate-+l- 65.3%

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

   7. neg-sub0 65.3%

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

   8. distribute-neg-frac2 65.3%

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

   9. distribute-frac-neg2 65.3%

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

   10. associate-+r+ 65.3%

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

   11. +-commutative 65.3%

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

   12. remove-double-neg 65.3%

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

   13. distribute-neg-frac2 65.3%

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

   14. sub0-neg 65.3%

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

   15. associate-+l- 65.3%

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

   16. neg-sub0 65.3%

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

3. Simplified 65.3%

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

5. Taylor expanded in x around inf 99.1%

   \[\leadsto \color{blue}{\frac{2 + \left(2 \cdot \frac{1}{{x}^{2}} + \frac{2}{{x}^{4}}\right)}{{x}^{3}}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.1%

      \[\leadsto \frac{2 + \left(\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

   2. metadata-eval 99.1%

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

7. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{2 + \left(\frac{2}{{x}^{2}} + \frac{2}{{x}^{4}}\right)}{{x}^{3}}} \]
8. Step-by-step derivation

   1. div-inv 99.1%

      \[\leadsto \color{blue}{\left(2 + \left(\frac{2}{{x}^{2}} + \frac{2}{{x}^{4}}\right)\right) \cdot \frac{1}{{x}^{3}}} \]

   2. +-commutative 99.1%

      \[\leadsto \left(2 + \color{blue}{\left(\frac{2}{{x}^{4}} + \frac{2}{{x}^{2}}\right)}\right) \cdot \frac{1}{{x}^{3}} \]

   3. div-inv 99.1%

      \[\leadsto \left(2 + \left(\color{blue}{2 \cdot \frac{1}{{x}^{4}}} + \frac{2}{{x}^{2}}\right)\right) \cdot \frac{1}{{x}^{3}} \]

   4. fma-define 99.1%

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

   5. pow-flip 99.1%

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

   6. metadata-eval 99.1%

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

   7. pow-flip 99.6%

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

   8. metadata-eval 99.6%

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

9. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\left(2 + \mathsf{fma}\left(2, {x}^{-4}, \frac{2}{{x}^{2}}\right)\right) \cdot {x}^{-3}} \]
10. Step-by-step derivation

    1. pow1 99.6%

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

    2. div-inv 99.6%

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

    3. pow-flip 99.6%

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

    4. metadata-eval 99.6%

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

11. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{{\left(\left(2 + \mathsf{fma}\left(2, {x}^{-4}, 2 \cdot {x}^{-2}\right)\right) \cdot {x}^{-3}\right)}^{1}} \]
12. Step-by-step derivation

    1. unpow1 99.6%

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

    2. *-lft-identity 99.6%

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

    3. *-lft-identity 99.6%

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

    4. fma-define 99.6%

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

    5. distribute-lft-out 99.6%

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

13. Simplified 99.6%

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

Alternative 2: 99.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

   1. +-commutative 65.3%

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

   2. associate-+r- 65.3%

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

   3. sub-neg 65.3%

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

   4. remove-double-neg 65.3%

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

   5. neg-sub0 65.3%

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

   6. associate-+l- 65.3%

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

   7. neg-sub0 65.3%

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

   8. distribute-neg-frac2 65.3%

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

   9. distribute-frac-neg2 65.3%

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

   10. associate-+r+ 65.3%

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

   11. +-commutative 65.3%

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

   12. remove-double-neg 65.3%

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

   13. distribute-neg-frac2 65.3%

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

   14. sub0-neg 65.3%

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

   15. associate-+l- 65.3%

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

   16. neg-sub0 65.3%

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

3. Simplified 65.3%

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

   1. frac-2neg 65.3%

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

   2. frac-2neg 65.3%

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

   3. metadata-eval 65.3%

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

   4. frac-sub 18.0%

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

   5. metadata-eval 18.0%

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

   6. sub-neg 18.0%

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

   7. distribute-neg-in 18.0%

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

   8. metadata-eval 18.0%

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

   9. neg-mul-1 18.0%

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

   10. *-commutative 18.0%

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

   11. sub-neg 18.0%

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

   12. *-commutative 18.0%

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

   13. neg-mul-1 18.0%

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

   14. sub-neg 18.0%

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

   15. distribute-neg-in 18.0%

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

   16. metadata-eval 18.0%

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

   17. neg-mul-1 18.0%

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

   18. *-commutative 18.0%

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

   19. sub-neg 18.0%

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

   20. *-commutative 18.0%

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

6. Applied egg-rr 18.0%

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

   1. *-commutative 18.0%

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

   2. sub-neg 18.0%

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

   3. remove-double-neg 18.0%

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

   4. distribute-lft-in 18.0%

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

   5. metadata-eval 18.0%

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

   6. neg-mul-1 18.0%

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

   7. remove-double-neg 18.0%

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

   8. sub-neg 18.0%

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

   9. remove-double-neg 18.0%

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

   10. distribute-lft-in 17.9%

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

   11. *-rgt-identity 17.9%

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

   12. neg-mul-1 17.9%

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

   13. distribute-rgt-in 18.0%

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

   14. sub-neg 18.0%

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

8. Simplified 18.0%

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

9. Taylor expanded in x around 0 18.1%

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

    1. +-commutative 18.1%

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

    2. frac-add 18.7%

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

11. Applied egg-rr 18.7%

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

12. Taylor expanded in x around 0 99.4%

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

Alternative 3: 67.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

   1. +-commutative 65.3%

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

   2. associate-+r- 65.3%

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

   3. sub-neg 65.3%

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

   4. remove-double-neg 65.3%

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

   5. neg-sub0 65.3%

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

   6. associate-+l- 65.3%

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

   7. neg-sub0 65.3%

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

   8. distribute-neg-frac2 65.3%

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

   9. distribute-frac-neg2 65.3%

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

   10. associate-+r+ 65.3%

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

   11. +-commutative 65.3%

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

   12. remove-double-neg 65.3%

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

   13. distribute-neg-frac2 65.3%

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

   14. sub0-neg 65.3%

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

   15. associate-+l- 65.3%

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

   16. neg-sub0 65.3%

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

3. Simplified 65.3%

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

5. Taylor expanded in x around inf 63.8%

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

   1. frac-add 63.8%

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

   2. *-un-lft-identity 63.8%

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

7. Applied egg-rr 63.8%

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

   1. *-commutative 63.8%

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

   2. neg-mul-1 63.8%

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

   3. unsub-neg 63.8%

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

   4. *-commutative 63.8%

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

9. Simplified 63.8%

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

10. Taylor expanded in x around inf 63.8%

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

Alternative 4: 67.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

   1. +-commutative 65.3%

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

   2. associate-+r- 65.3%

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

   3. sub-neg 65.3%

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

   4. remove-double-neg 65.3%

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

   5. neg-sub0 65.3%

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

   6. associate-+l- 65.3%

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

   7. neg-sub0 65.3%

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

   8. distribute-neg-frac2 65.3%

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

   9. distribute-frac-neg2 65.3%

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

   10. associate-+r+ 65.3%

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

   11. +-commutative 65.3%

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

   12. remove-double-neg 65.3%

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

   13. distribute-neg-frac2 65.3%

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

   14. sub0-neg 65.3%

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

   15. associate-+l- 65.3%

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

   16. neg-sub0 65.3%

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

3. Simplified 65.3%

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

5. Taylor expanded in x around inf 63.8%

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

Alternative 5: 67.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

   1. +-commutative 65.3%

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

   2. associate-+r- 65.3%

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

   3. sub-neg 65.3%

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

   4. remove-double-neg 65.3%

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

   5. neg-sub0 65.3%

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

   6. associate-+l- 65.3%

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

   7. neg-sub0 65.3%

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

   8. distribute-neg-frac2 65.3%

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

   9. distribute-frac-neg2 65.3%

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

   10. associate-+r+ 65.3%

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

   11. +-commutative 65.3%

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

   12. remove-double-neg 65.3%

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

   13. distribute-neg-frac2 65.3%

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

   14. sub0-neg 65.3%

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

   15. associate-+l- 65.3%

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

   16. neg-sub0 65.3%

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

3. Simplified 65.3%

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

5. Taylor expanded in x around inf 63.8%

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

6. Taylor expanded in x around inf 63.7%

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

7. Final simplification 63.7%

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

Alternative 6: 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 65.3%

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

   1. +-commutative 65.3%

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

   2. associate-+r- 65.3%

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

   3. sub-neg 65.3%

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

   4. remove-double-neg 65.3%

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

   5. neg-sub0 65.3%

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

   6. associate-+l- 65.3%

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

   7. neg-sub0 65.3%

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

   8. distribute-neg-frac2 65.3%

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

   9. distribute-frac-neg2 65.3%

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

   10. associate-+r+ 65.3%

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

   11. +-commutative 65.3%

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

   12. remove-double-neg 65.3%

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

   13. distribute-neg-frac2 65.3%

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

   14. sub0-neg 65.3%

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

   15. associate-+l- 65.3%

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

   16. neg-sub0 65.3%

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

3. Simplified 65.3%

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

5. Taylor expanded in x around inf 63.8%

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

6. Taylor expanded in x around 0 4.9%

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

Alternative 7: 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 65.3%

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

   1. +-commutative 65.3%

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

   2. associate-+r- 65.3%

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

   3. sub-neg 65.3%

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

   4. remove-double-neg 65.3%

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

   5. neg-sub0 65.3%

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

   6. associate-+l- 65.3%

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

   7. neg-sub0 65.3%

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

   8. distribute-neg-frac2 65.3%

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

   9. distribute-frac-neg2 65.3%

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

   10. associate-+r+ 65.3%

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

   11. +-commutative 65.3%

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

   12. remove-double-neg 65.3%

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

   13. distribute-neg-frac2 65.3%

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

   14. sub0-neg 65.3%

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

   15. associate-+l- 65.3%

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

   16. neg-sub0 65.3%

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

3. Simplified 65.3%

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

5. Taylor expanded in x around 0 4.9%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]
6. 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 2024172 
(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))))