3frac (problem 3.3.3)

Percentage Accurate: 68.9% → 99.6%

Time: 9.0s

Alternatives: 6

Speedup: 1.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.9% 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: 99.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot \left(1 + {x}^{-4}\right)\right) \cdot {x}^{-3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* (fma 2.0 (pow x -2.0) 2.0) (+ 1.0 (pow x -4.0))) (pow x -3.0)))

C

double code(double x) {
	return (fma(2.0, pow(x, -2.0), 2.0) * (1.0 + pow(x, -4.0))) * pow(x, -3.0);
}

Julia

function code(x)
	return Float64(Float64(fma(2.0, (x ^ -2.0), 2.0) * Float64(1.0 + (x ^ -4.0))) * (x ^ -3.0))
end

Wolfram

code[x_] := N[(N[(N[(2.0 * N[Power[x, -2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[(1.0 + N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.3%

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

   1. +-commutative 69.3%

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

   2. associate-+r- 69.3%

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

   3. sub-neg 69.3%

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

   4. remove-double-neg 69.3%

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

   5. neg-sub0 69.3%

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

   6. associate-+l- 69.3%

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

   7. neg-sub0 69.3%

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

   8. distribute-neg-frac2 69.3%

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

   9. distribute-frac-neg2 69.3%

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

   10. associate-+r+ 69.3%

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

   11. +-commutative 69.3%

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

   12. remove-double-neg 69.3%

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

   13. distribute-neg-frac2 69.3%

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

   14. sub0-neg 69.3%

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

   15. associate-+l- 69.3%

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

   16. neg-sub0 69.3%

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

3. Simplified 69.3%

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

5. Taylor expanded in x around -inf 98.9%

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

   1. mul-1-neg 98.9%

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

   2. distribute-neg-frac 98.9%

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

7. Simplified 98.9%

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

   1. div-inv 98.9%

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

   2. div-inv 98.9%

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

   3. fma-define 98.9%

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

   4. +-commutative 98.9%

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

   5. div-inv 98.9%

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

   6. fma-define 98.9%

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

   7. pow-flip 98.9%

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

   8. metadata-eval 98.9%

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

   9. pow-flip 98.9%

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

   10. metadata-eval 98.9%

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

   11. +-commutative 98.9%

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

   12. div-inv 98.9%

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

   13. fma-define 98.9%

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

   14. pow-flip 98.9%

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

   15. metadata-eval 98.9%

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

   16. pow-flip 99.6%

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

   17. metadata-eval 99.6%

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

9. Applied egg-rr 99.6%

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

    1. fma-undefine 99.6%

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

    2. +-commutative 99.6%

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

    3. *-lft-identity 99.6%

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

    4. *-commutative 99.6%

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

    5. distribute-rgt-out 99.6%

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

11. Simplified 99.6%

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

Alternative 2: 99.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.3%

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

   1. +-commutative 69.3%

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

   2. associate-+r- 69.3%

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

   3. sub-neg 69.3%

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

   4. remove-double-neg 69.3%

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

   5. neg-sub0 69.3%

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

   6. associate-+l- 69.3%

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

   7. neg-sub0 69.3%

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

   8. distribute-neg-frac2 69.3%

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

   9. distribute-frac-neg2 69.3%

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

   10. associate-+r+ 69.3%

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

   11. +-commutative 69.3%

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

   12. remove-double-neg 69.3%

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

   13. distribute-neg-frac2 69.3%

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

   14. sub0-neg 69.3%

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

   15. associate-+l- 69.3%

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

   16. neg-sub0 69.3%

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

3. Simplified 69.3%

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

   1. frac-sub 19.0%

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

   2. frac-add 17.8%

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

   3. *-un-lft-identity 17.8%

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

   4. fma-define 17.1%

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

   5. *-rgt-identity 17.1%

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

   6. fma-neg 17.1%

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

6. Applied egg-rr 17.1%

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

   1. fma-neg 17.1%

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

   2. *-commutative 17.1%

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

   3. associate-*l* 17.1%

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

8. Simplified 17.1%

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

9. Taylor expanded in x around 0 99.2%

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

10. Taylor expanded in x around inf 99.2%

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

    1. metadata-eval 99.2%

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

    2. unpow2 99.2%

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

    3. frac-times 99.2%

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

12. Applied egg-rr 99.2%

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

13. Final simplification 99.2%

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

Alternative 3: 99.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(-2.0 / Float64(x * Float64(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[(x * N[(N[(-1.0 - x), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.3%

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

   1. +-commutative 69.3%

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

   2. associate-+r- 69.3%

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

   3. sub-neg 69.3%

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

   4. remove-double-neg 69.3%

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

   5. neg-sub0 69.3%

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

   6. associate-+l- 69.3%

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

   7. neg-sub0 69.3%

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

   8. distribute-neg-frac2 69.3%

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

   9. distribute-frac-neg2 69.3%

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

   10. associate-+r+ 69.3%

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

   11. +-commutative 69.3%

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

   12. remove-double-neg 69.3%

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

   13. distribute-neg-frac2 69.3%

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

   14. sub0-neg 69.3%

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

   15. associate-+l- 69.3%

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

   16. neg-sub0 69.3%

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

3. Simplified 69.3%

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

   1. frac-sub 19.0%

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

   2. frac-add 17.8%

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

   3. *-un-lft-identity 17.8%

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

   4. fma-define 17.1%

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

   5. *-rgt-identity 17.1%

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

   6. fma-neg 17.1%

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

6. Applied egg-rr 17.1%

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

   1. fma-neg 17.1%

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

   2. *-commutative 17.1%

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

   3. associate-*l* 17.1%

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

8. Simplified 17.1%

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

9. Taylor expanded in x around 0 99.2%

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

Alternative 4: 67.2% 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 69.3%

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

   1. +-commutative 69.3%

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

   2. associate-+r- 69.3%

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

   3. sub-neg 69.3%

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

   4. remove-double-neg 69.3%

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

   5. neg-sub0 69.3%

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

   6. associate-+l- 69.3%

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

   7. neg-sub0 69.3%

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

   8. distribute-neg-frac2 69.3%

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

   9. distribute-frac-neg2 69.3%

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

   10. associate-+r+ 69.3%

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

   11. +-commutative 69.3%

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

   12. remove-double-neg 69.3%

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

   13. distribute-neg-frac2 69.3%

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

   14. sub0-neg 69.3%

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

   15. associate-+l- 69.3%

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

   16. neg-sub0 69.3%

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

3. Simplified 69.3%

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

5. Taylor expanded in x around inf 68.4%

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

Alternative 5: 67.0% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 0.0 x))

C

double code(double x) {
	return 0.0 / x;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0 / x

Julia

function code(x)
	return Float64(0.0 / x)
end

MATLAB

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

Wolfram

code[x_] := N[(0.0 / x), $MachinePrecision]

Derivation

1. Initial program 69.3%

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

   1. +-commutative 69.3%

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

   2. associate-+r- 69.3%

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

   3. sub-neg 69.3%

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

   4. remove-double-neg 69.3%

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

   5. neg-sub0 69.3%

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

   6. associate-+l- 69.3%

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

   7. neg-sub0 69.3%

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

   8. distribute-neg-frac2 69.3%

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

   9. distribute-frac-neg2 69.3%

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

   10. associate-+r+ 69.3%

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

   11. +-commutative 69.3%

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

   12. remove-double-neg 69.3%

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

   13. distribute-neg-frac2 69.3%

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

   14. sub0-neg 69.3%

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

   15. associate-+l- 69.3%

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

   16. neg-sub0 69.3%

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

3. Simplified 69.3%

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

   1. add-sqr-sqrt 21.3%

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

   2. fma-define 4.8%

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

   3. inv-pow 4.8%

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

   4. sqrt-pow1 4.8%

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

   5. metadata-eval 4.8%

      \[\leadsto \mathsf{fma}\left({\left(x + -1\right)}^{\color{blue}{-0.5}}, \sqrt{\frac{1}{x + -1}}, \frac{-2}{x} - \frac{1}{-1 - x}\right) \]

   6. inv-pow 4.8%

      \[\leadsto \mathsf{fma}\left({\left(x + -1\right)}^{-0.5}, \sqrt{\color{blue}{{\left(x + -1\right)}^{-1}}}, \frac{-2}{x} - \frac{1}{-1 - x}\right) \]

   7. sqrt-pow1 4.8%

      \[\leadsto \mathsf{fma}\left({\left(x + -1\right)}^{-0.5}, \color{blue}{{\left(x + -1\right)}^{\left(\frac{-1}{2}\right)}}, \frac{-2}{x} - \frac{1}{-1 - x}\right) \]

   8. metadata-eval 4.8%

      \[\leadsto \mathsf{fma}\left({\left(x + -1\right)}^{-0.5}, {\left(x + -1\right)}^{\color{blue}{-0.5}}, \frac{-2}{x} - \frac{1}{-1 - x}\right) \]

   9. sub-neg 4.8%

      \[\leadsto \mathsf{fma}\left({\left(x + -1\right)}^{-0.5}, {\left(x + -1\right)}^{-0.5}, \color{blue}{\frac{-2}{x} + \left(-\frac{1}{-1 - x}\right)}\right) \]

   10. distribute-neg-frac 4.8%

       \[\leadsto \mathsf{fma}\left({\left(x + -1\right)}^{-0.5}, {\left(x + -1\right)}^{-0.5}, \frac{-2}{x} + \color{blue}{\frac{-1}{-1 - x}}\right) \]

   11. metadata-eval 4.8%

       \[\leadsto \mathsf{fma}\left({\left(x + -1\right)}^{-0.5}, {\left(x + -1\right)}^{-0.5}, \frac{-2}{x} + \frac{\color{blue}{-1}}{-1 - x}\right) \]

6. Applied egg-rr 4.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x + -1\right)}^{-0.5}, {\left(x + -1\right)}^{-0.5}, \frac{-2}{x} + \frac{-1}{-1 - x}\right)} \]

7. Taylor expanded in x around -inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 68.2%

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

   4. metadata-eval 68.2%

      \[\leadsto \frac{-1 \cdot \color{blue}{0}}{x} \]

   5. metadata-eval 68.2%

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

9. Simplified 68.2%

   \[\leadsto \color{blue}{\frac{0}{x}} \]
10. Add Preprocessing

Alternative 6: 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 69.3%

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

   1. +-commutative 69.3%

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

   2. associate-+r- 69.3%

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

   3. sub-neg 69.3%

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

   4. remove-double-neg 69.3%

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

   5. neg-sub0 69.3%

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

   6. associate-+l- 69.3%

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

   7. neg-sub0 69.3%

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

   8. distribute-neg-frac2 69.3%

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

   9. distribute-frac-neg2 69.3%

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

   10. associate-+r+ 69.3%

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

   11. +-commutative 69.3%

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

   12. remove-double-neg 69.3%

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

   13. distribute-neg-frac2 69.3%

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

   14. sub0-neg 69.3%

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

   15. associate-+l- 69.3%

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

   16. neg-sub0 69.3%

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

3. Simplified 69.3%

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

5. Taylor expanded in x around 0 5.2%

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

Developer target: 99.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024097 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

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

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