3frac (problem 3.3.3)

Percentage Accurate: 69.3% → 99.7%

Time: 10.4s

Alternatives: 8

Speedup: 2.1×

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. +-commutative 72.1%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.1%

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

   11. +-commutative 72.1%

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

   12. remove-double-neg 72.1%

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

   13. distribute-neg-frac2 72.1%

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

   14. sub0-neg 72.1%

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

   15. associate-+l- 72.1%

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

   16. neg-sub0 72.1%

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

3. Simplified 72.1%

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

5. Taylor expanded in x around -inf 98.7%

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

   1. mul-1-neg 98.7%

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

   2. distribute-neg-frac 98.7%

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

7. Simplified 98.7%

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

   1. div-inv 98.7%

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

   2. div-inv 98.7%

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

   3. fma-define 98.7%

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

   4. +-commutative 98.7%

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

   5. div-inv 98.7%

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

   6. fma-define 98.7%

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

   7. pow-flip 98.7%

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

   8. metadata-eval 98.7%

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

   9. pow-flip 98.7%

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

   10. metadata-eval 98.7%

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

   11. +-commutative 98.7%

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

   12. div-inv 98.7%

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

   13. fma-define 98.7%

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

   14. pow-flip 98.7%

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

   15. metadata-eval 98.7%

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

   16. pow-flip 99.5%

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

   17. metadata-eval 99.5%

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

9. Applied egg-rr 99.5%

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

    1. fma-undefine 99.5%

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

    2. *-commutative 99.5%

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

    3. distribute-lft1-in 99.5%

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

11. Simplified 99.5%

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

Alternative 2: 99.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. +-commutative 72.1%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.1%

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

   11. +-commutative 72.1%

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

   12. remove-double-neg 72.1%

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

   13. distribute-neg-frac2 72.1%

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

   14. sub0-neg 72.1%

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

   15. associate-+l- 72.1%

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

   16. neg-sub0 72.1%

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

3. Simplified 72.1%

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

5. Taylor expanded in x around inf 98.5%

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

   1. associate-*r/ 98.5%

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

   2. metadata-eval 98.5%

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

7. Simplified 98.5%

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

   1. div-inv 98.5%

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

   2. *-un-lft-identity 98.5%

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

   3. pow-flip 99.2%

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

   4. metadata-eval 99.2%

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

   5. +-commutative 99.2%

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

   6. *-un-lft-identity 99.2%

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

   7. div-inv 99.2%

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

   8. fma-define 99.2%

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

   9. pow-flip 99.2%

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

   10. metadata-eval 99.2%

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

   11. *-un-lft-identity 99.2%

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

   12. associate-*l* 99.2%

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

9. Applied egg-rr 99.2%

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

    1. *-lft-identity 99.2%

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

11. Simplified 99.2%

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

Alternative 3: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. +-commutative 72.1%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.1%

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

   11. +-commutative 72.1%

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

   12. remove-double-neg 72.1%

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

   13. distribute-neg-frac2 72.1%

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

   14. sub0-neg 72.1%

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

   15. associate-+l- 72.1%

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

   16. neg-sub0 72.1%

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

3. Simplified 72.1%

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

5. Taylor expanded in x around -inf 98.7%

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

   1. mul-1-neg 98.7%

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

   2. distribute-neg-frac 98.7%

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

7. Simplified 98.7%

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

   1. +-commutative 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. fma-define 98.7%

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

   4. +-commutative 98.7%

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

   5. div-inv 98.7%

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

   6. fma-define 98.7%

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

   7. pow-flip 98.7%

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

   8. metadata-eval 98.7%

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

   9. div-inv 98.7%

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

   10. +-commutative 98.7%

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

   11. div-inv 98.7%

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

   12. fma-define 98.7%

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

   13. pow-flip 98.7%

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

   14. metadata-eval 98.7%

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

   15. pow-flip 98.7%

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

   16. metadata-eval 98.7%

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

9. Applied egg-rr 98.7%

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

    1. fma-undefine 98.7%

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

    2. *-commutative 98.7%

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

    3. distribute-rgt-out 98.7%

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

11. Simplified 98.7%

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

    1. *-un-lft-identity 98.7%

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

    2. cube-mult 98.7%

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

    3. unpow2 98.7%

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

    4. times-frac 99.3%

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

    5. +-commutative 99.3%

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

    6. *-commutative 99.3%

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

13. Applied egg-rr 99.3%

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

14. Taylor expanded in x around inf 99.1%

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

    1. associate-*r/ 99.1%

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

    2. metadata-eval 99.1%

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

16. Simplified 99.1%

    \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{2 + \frac{2}{{x}^{2}}}{{x}^{2}}} \]
17. Add Preprocessing

Alternative 4: 98.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \cdot \frac{2}{{x}^{2}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. +-commutative 72.1%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.1%

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

   11. +-commutative 72.1%

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

   12. remove-double-neg 72.1%

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

   13. distribute-neg-frac2 72.1%

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

   14. sub0-neg 72.1%

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

   15. associate-+l- 72.1%

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

   16. neg-sub0 72.1%

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

3. Simplified 72.1%

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

5. Taylor expanded in x around -inf 98.7%

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

   1. mul-1-neg 98.7%

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

   2. distribute-neg-frac 98.7%

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

7. Simplified 98.7%

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

   1. +-commutative 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. fma-define 98.7%

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

   4. +-commutative 98.7%

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

   5. div-inv 98.7%

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

   6. fma-define 98.7%

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

   7. pow-flip 98.7%

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

   8. metadata-eval 98.7%

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

   9. div-inv 98.7%

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

   10. +-commutative 98.7%

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

   11. div-inv 98.7%

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

   12. fma-define 98.7%

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

   13. pow-flip 98.7%

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

   14. metadata-eval 98.7%

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

   15. pow-flip 98.7%

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

   16. metadata-eval 98.7%

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

9. Applied egg-rr 98.7%

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

    1. fma-undefine 98.7%

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

    2. *-commutative 98.7%

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

    3. distribute-rgt-out 98.7%

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

11. Simplified 98.7%

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

    1. *-un-lft-identity 98.7%

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

    2. cube-mult 98.7%

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

    3. unpow2 98.7%

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

    4. times-frac 99.3%

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

    5. +-commutative 99.3%

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

    6. *-commutative 99.3%

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

13. Applied egg-rr 99.3%

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

14. Taylor expanded in x around inf 98.7%

    \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{2}{{x}^{2}}} \]
15. Add Preprocessing

Alternative 5: 67.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. +-commutative 72.1%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.1%

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

   11. +-commutative 72.1%

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

   12. remove-double-neg 72.1%

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

   13. distribute-neg-frac2 72.1%

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

   14. sub0-neg 72.1%

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

   15. associate-+l- 72.1%

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

   16. neg-sub0 72.1%

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

3. Simplified 72.1%

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

5. Taylor expanded in x around inf 70.6%

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

   1. frac-add 70.6%

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

   2. *-un-lft-identity 70.6%

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

7. Applied egg-rr 70.6%

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

   1. remove-double-neg 70.6%

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

   2. distribute-frac-neg2 70.6%

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

   3. distribute-frac-neg 70.6%

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

   4. +-commutative 70.6%

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

   5. *-commutative 70.6%

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

   6. neg-mul-1 70.6%

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

   7. distribute-neg-in 70.6%

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

   8. remove-double-neg 70.6%

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

   9. associate-+l+ 70.6%

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

   10. sub-neg 70.6%

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

   11. *-commutative 70.6%

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

   12. distribute-rgt-neg-out 70.6%

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

   13. distribute-neg-in 70.6%

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

   14. metadata-eval 70.6%

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

   15. +-commutative 70.6%

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

   16. unsub-neg 70.6%

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

9. Simplified 70.6%

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

Alternative 6: 67.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. +-commutative 72.1%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.1%

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

   11. +-commutative 72.1%

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

   12. remove-double-neg 72.1%

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

   13. distribute-neg-frac2 72.1%

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

   14. sub0-neg 72.1%

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

   15. associate-+l- 72.1%

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

   16. neg-sub0 72.1%

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

3. Simplified 72.1%

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

5. Taylor expanded in x around inf 70.6%

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

6. Final simplification 70.6%

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

Alternative 7: 98.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. +-commutative 72.1%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.1%

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

   11. +-commutative 72.1%

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

   12. remove-double-neg 72.1%

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

   13. distribute-neg-frac2 72.1%

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

   14. sub0-neg 72.1%

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

   15. associate-+l- 72.1%

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

   16. neg-sub0 72.1%

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

3. Simplified 72.1%

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

   1. inv-pow 72.1%

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

   2. add-sqr-sqrt 24.4%

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

   3. unpow-prod-down 16.8%

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

   4. fma-define 4.0%

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

   5. sub-neg 4.0%

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

   6. distribute-neg-frac 4.0%

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

   7. metadata-eval 4.0%

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

6. Applied egg-rr 4.0%

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

   1. unpow-1 4.0%

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

   2. unpow-1 4.0%

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

8. Simplified 4.0%

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

9. Taylor expanded in x around -inf 0.0%

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

    1. mul-1-neg 0.0%

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

    2. distribute-neg-frac2 0.0%

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

11. Simplified 98.6%

    \[\leadsto \color{blue}{\frac{\frac{\frac{2}{x}}{-x} + 0}{-x}} \]

12. Final simplification 98.6%

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

Alternative 8: 67.7% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 72.1%

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

   1. +-commutative 72.1%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.1%

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

   11. +-commutative 72.1%

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

   12. remove-double-neg 72.1%

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

   13. distribute-neg-frac2 72.1%

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

   14. sub0-neg 72.1%

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

   15. associate-+l- 72.1%

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

   16. neg-sub0 72.1%

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

3. Simplified 72.1%

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

   1. inv-pow 72.1%

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

   2. add-sqr-sqrt 24.4%

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

   3. unpow-prod-down 16.8%

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

   4. fma-define 4.0%

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

   5. sub-neg 4.0%

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

   6. distribute-neg-frac 4.0%

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

   7. metadata-eval 4.0%

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

6. Applied egg-rr 4.0%

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

   1. unpow-1 4.0%

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

   2. unpow-1 4.0%

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

8. Simplified 4.0%

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

9. Taylor expanded in x around -inf 0.0%

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

    1. associate-*r/ 0.0%

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

    2. distribute-rgt-in 0.0%

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

    3. metadata-eval 0.0%

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

    4. unpow2 0.0%

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

    5. rem-square-sqrt 70.5%

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

    6. metadata-eval 70.5%

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

    7. metadata-eval 70.5%

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

    8. metadata-eval 70.5%

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

    9. div0 70.5%

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

11. Simplified 70.5%

    \[\leadsto \color{blue}{0} \]
12. 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 2024106 
(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))))