3frac (problem 3.3.3)

Percentage Accurate: 68.4% → 99.5%

Time: 8.6s

Alternatives: 7

Speedup: 1.7×

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: 68.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.3%

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

   1. +-commutative 66.3%

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

   2. associate-+r- 66.3%

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

   3. sub-neg 66.3%

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

   4. remove-double-neg 66.3%

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

   5. neg-sub0 66.3%

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

   6. associate-+l- 66.3%

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

   7. neg-sub0 66.3%

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

   8. distribute-neg-frac2 66.3%

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

   9. distribute-frac-neg2 66.3%

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

   10. associate-+r+ 66.3%

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

   11. +-commutative 66.3%

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

   12. remove-double-neg 66.3%

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

   13. distribute-neg-frac2 66.3%

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

   14. sub0-neg 66.3%

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

   15. associate-+l- 66.3%

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

   16. neg-sub0 66.3%

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

3. Simplified 66.3%

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

5. Taylor expanded in x around inf 98.9%

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

   1. associate-*r/ 98.9%

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

   2. metadata-eval 98.9%

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

7. Simplified 98.9%

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

   1. div-inv 98.9%

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

   2. +-commutative 98.9%

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

   3. div-inv 98.9%

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

   4. fma-define 98.9%

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

   5. pow-flip 98.9%

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

   6. metadata-eval 98.9%

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

   7. pow-flip 99.4%

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

   8. metadata-eval 99.4%

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

9. Applied egg-rr 99.4%

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

Alternative 2: 99.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(-0.5 \cdot \left(x \cdot \left(-1 - x\right)\right)\right) \cdot \left(x + -1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (* (* -0.5 (* x (- -1.0 x))) (+ x -1.0))))

C

double code(double x) {
	return 1.0 / ((-0.5 * (x * (-1.0 - x))) * (x + -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (((-0.5d0) * (x * ((-1.0d0) - x))) * (x + (-1.0d0)))
end function

Java

public static double code(double x) {
	return 1.0 / ((-0.5 * (x * (-1.0 - x))) * (x + -1.0));
}

Python

def code(x):
	return 1.0 / ((-0.5 * (x * (-1.0 - x))) * (x + -1.0))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(-0.5 * Float64(x * Float64(-1.0 - x))) * Float64(x + -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((-0.5 * (x * (-1.0 - x))) * (x + -1.0));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(-0.5 * N[(x * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

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

   1. +-commutative 66.3%

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

   2. associate-+r- 66.3%

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

   3. sub-neg 66.3%

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

   4. remove-double-neg 66.3%

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

   5. neg-sub0 66.3%

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

   6. associate-+l- 66.3%

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

   7. neg-sub0 66.3%

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

   8. distribute-neg-frac2 66.3%

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

   9. distribute-frac-neg2 66.3%

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

   10. associate-+r+ 66.3%

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

   11. +-commutative 66.3%

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

   12. remove-double-neg 66.3%

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

   13. distribute-neg-frac2 66.3%

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

   14. sub0-neg 66.3%

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

   15. associate-+l- 66.3%

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

   16. neg-sub0 66.3%

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

3. Simplified 66.3%

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

   1. clear-num 66.3%

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

   2. frac-sub 17.3%

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

   3. *-un-lft-identity 17.3%

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

   4. div-inv 17.3%

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

   5. metadata-eval 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) - \left(x \cdot \color{blue}{-0.5}\right) \cdot 1}{\frac{x}{-2} \cdot \left(-1 - x\right)} \]

   6. div-inv 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) - \left(x \cdot -0.5\right) \cdot 1}{\color{blue}{\left(x \cdot \frac{1}{-2}\right)} \cdot \left(-1 - x\right)} \]

   7. metadata-eval 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) - \left(x \cdot -0.5\right) \cdot 1}{\left(x \cdot \color{blue}{-0.5}\right) \cdot \left(-1 - x\right)} \]

6. Applied egg-rr 17.3%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{\left(-1 - x\right) - \left(x \cdot -0.5\right) \cdot 1}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)}} \]
7. Step-by-step derivation

   1. sub-neg 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{\left(-1 - x\right) + \left(-\left(x \cdot -0.5\right) \cdot 1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   2. *-rgt-identity 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) + \left(-\color{blue}{x \cdot -0.5}\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   3. sub-neg 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{\left(-1 + \left(-x\right)\right)} + \left(-x \cdot -0.5\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   4. neg-mul-1 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 + \left(-x\right)\right) + \color{blue}{-1 \cdot \left(x \cdot -0.5\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   5. *-commutative 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 + \left(-x\right)\right) + \color{blue}{\left(x \cdot -0.5\right) \cdot -1}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   6. associate-+l+ 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{-1 + \left(\left(-x\right) + \left(x \cdot -0.5\right) \cdot -1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   7. neg-mul-1 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \left(\color{blue}{-1 \cdot x} + \left(x \cdot -0.5\right) \cdot -1\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   8. *-commutative 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \left(\color{blue}{x \cdot -1} + \left(x \cdot -0.5\right) \cdot -1\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   9. associate-*l* 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \left(x \cdot -1 + \color{blue}{x \cdot \left(-0.5 \cdot -1\right)}\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   10. distribute-lft-out 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + \color{blue}{x \cdot \left(-1 + -0.5 \cdot -1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   11. metadata-eval 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot \left(-1 + \color{blue}{0.5}\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   12. metadata-eval 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot \color{blue}{-0.5}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   13. *-commutative 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{\color{blue}{\left(-1 - x\right) \cdot \left(x \cdot -0.5\right)}} \]

   14. associate-*r* 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{\color{blue}{\left(\left(-1 - x\right) \cdot x\right) \cdot -0.5}} \]

   15. *-commutative 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{\color{blue}{\left(x \cdot \left(-1 - x\right)\right)} \cdot -0.5} \]

8. Simplified 17.3%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{-1 + x \cdot -0.5}{\left(x \cdot \left(-1 - x\right)\right) \cdot -0.5}} \]
9. Step-by-step derivation

   1. +-commutative 17.3%

      \[\leadsto \color{blue}{\frac{-1 + x \cdot -0.5}{\left(x \cdot \left(-1 - x\right)\right) \cdot -0.5} + \frac{1}{x + -1}} \]

   2. frac-add 22.8%

      \[\leadsto \color{blue}{\frac{\left(-1 + x \cdot -0.5\right) \cdot \left(x + -1\right) + \left(\left(x \cdot \left(-1 - x\right)\right) \cdot -0.5\right) \cdot 1}{\left(\left(x \cdot \left(-1 - x\right)\right) \cdot -0.5\right) \cdot \left(x + -1\right)}} \]

   3. +-commutative 22.8%

      \[\leadsto \frac{\color{blue}{\left(x \cdot -0.5 + -1\right)} \cdot \left(x + -1\right) + \left(\left(x \cdot \left(-1 - x\right)\right) \cdot -0.5\right) \cdot 1}{\left(\left(x \cdot \left(-1 - x\right)\right) \cdot -0.5\right) \cdot \left(x + -1\right)} \]

   4. fma-define 22.8%

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

   5. *-commutative 22.8%

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

   6. *-commutative 22.8%

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

10. Applied egg-rr 22.8%

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

11. Taylor expanded in x around 0 22.8%

    \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(-0.5 \cdot x - 0.5\right)\right)} + \left(-0.5 \cdot \left(x \cdot \left(-1 - x\right)\right)\right) \cdot 1}{\left(-0.5 \cdot \left(x \cdot \left(-1 - x\right)\right)\right) \cdot \left(x + -1\right)} \]

12. Taylor expanded in x around 0 99.4%

    \[\leadsto \frac{\color{blue}{1}}{\left(-0.5 \cdot \left(x \cdot \left(-1 - x\right)\right)\right) \cdot \left(x + -1\right)} \]
13. Add Preprocessing

Alternative 3: 71.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.3%

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

   1. +-commutative 66.3%

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

   2. associate-+r- 66.3%

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

   3. sub-neg 66.3%

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

   4. remove-double-neg 66.3%

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

   5. neg-sub0 66.3%

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

   6. associate-+l- 66.3%

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

   7. neg-sub0 66.3%

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

   8. distribute-neg-frac2 66.3%

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

   9. distribute-frac-neg2 66.3%

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

   10. associate-+r+ 66.3%

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

   11. +-commutative 66.3%

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

   12. remove-double-neg 66.3%

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

   13. distribute-neg-frac2 66.3%

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

   14. sub0-neg 66.3%

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

   15. associate-+l- 66.3%

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

   16. neg-sub0 66.3%

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

3. Simplified 66.3%

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

5. Taylor expanded in x around inf 65.1%

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

   1. associate-*r/ 65.1%

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

   2. neg-mul-1 65.1%

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

   3. distribute-neg-in 65.1%

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

   4. metadata-eval 65.1%

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

   5. distribute-neg-frac 65.1%

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

   6. metadata-eval 65.1%

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

7. Simplified 65.1%

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

   1. frac-add 65.1%

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

   2. *-un-lft-identity 65.1%

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

9. Applied egg-rr 65.1%

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

10. Taylor expanded in x around 0 69.5%

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

11. Final simplification 69.5%

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

Alternative 4: 66.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.3%

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

   1. +-commutative 66.3%

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

   2. associate-+r- 66.3%

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

   3. sub-neg 66.3%

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

   4. remove-double-neg 66.3%

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

   5. neg-sub0 66.3%

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

   6. associate-+l- 66.3%

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

   7. neg-sub0 66.3%

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

   8. distribute-neg-frac2 66.3%

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

   9. distribute-frac-neg2 66.3%

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

   10. associate-+r+ 66.3%

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

   11. +-commutative 66.3%

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

   12. remove-double-neg 66.3%

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

   13. distribute-neg-frac2 66.3%

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

   14. sub0-neg 66.3%

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

   15. associate-+l- 66.3%

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

   16. neg-sub0 66.3%

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

3. Simplified 66.3%

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

5. Taylor expanded in x around inf 64.5%

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

Alternative 5: 6.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.3%

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

   1. +-commutative 66.3%

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

   2. associate-+r- 66.3%

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

   3. sub-neg 66.3%

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

   4. remove-double-neg 66.3%

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

   5. neg-sub0 66.3%

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

   6. associate-+l- 66.3%

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

   7. neg-sub0 66.3%

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

   8. distribute-neg-frac2 66.3%

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

   9. distribute-frac-neg2 66.3%

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

   10. associate-+r+ 66.3%

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

   11. +-commutative 66.3%

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

   12. remove-double-neg 66.3%

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

   13. distribute-neg-frac2 66.3%

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

   14. sub0-neg 66.3%

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

   15. associate-+l- 66.3%

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

   16. neg-sub0 66.3%

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

3. Simplified 66.3%

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

   1. clear-num 66.3%

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

   2. frac-sub 17.3%

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

   3. *-un-lft-identity 17.3%

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

   4. div-inv 17.3%

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

   5. metadata-eval 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) - \left(x \cdot \color{blue}{-0.5}\right) \cdot 1}{\frac{x}{-2} \cdot \left(-1 - x\right)} \]

   6. div-inv 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) - \left(x \cdot -0.5\right) \cdot 1}{\color{blue}{\left(x \cdot \frac{1}{-2}\right)} \cdot \left(-1 - x\right)} \]

   7. metadata-eval 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) - \left(x \cdot -0.5\right) \cdot 1}{\left(x \cdot \color{blue}{-0.5}\right) \cdot \left(-1 - x\right)} \]

6. Applied egg-rr 17.3%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{\left(-1 - x\right) - \left(x \cdot -0.5\right) \cdot 1}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)}} \]
7. Step-by-step derivation

   1. sub-neg 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{\left(-1 - x\right) + \left(-\left(x \cdot -0.5\right) \cdot 1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   2. *-rgt-identity 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) + \left(-\color{blue}{x \cdot -0.5}\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   3. sub-neg 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{\left(-1 + \left(-x\right)\right)} + \left(-x \cdot -0.5\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   4. neg-mul-1 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 + \left(-x\right)\right) + \color{blue}{-1 \cdot \left(x \cdot -0.5\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   5. *-commutative 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 + \left(-x\right)\right) + \color{blue}{\left(x \cdot -0.5\right) \cdot -1}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   6. associate-+l+ 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{-1 + \left(\left(-x\right) + \left(x \cdot -0.5\right) \cdot -1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   7. neg-mul-1 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \left(\color{blue}{-1 \cdot x} + \left(x \cdot -0.5\right) \cdot -1\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   8. *-commutative 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \left(\color{blue}{x \cdot -1} + \left(x \cdot -0.5\right) \cdot -1\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   9. associate-*l* 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \left(x \cdot -1 + \color{blue}{x \cdot \left(-0.5 \cdot -1\right)}\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   10. distribute-lft-out 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + \color{blue}{x \cdot \left(-1 + -0.5 \cdot -1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   11. metadata-eval 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot \left(-1 + \color{blue}{0.5}\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   12. metadata-eval 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot \color{blue}{-0.5}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   13. *-commutative 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{\color{blue}{\left(-1 - x\right) \cdot \left(x \cdot -0.5\right)}} \]

   14. associate-*r* 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{\color{blue}{\left(\left(-1 - x\right) \cdot x\right) \cdot -0.5}} \]

   15. *-commutative 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{\color{blue}{\left(x \cdot \left(-1 - x\right)\right)} \cdot -0.5} \]

8. Simplified 17.3%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{-1 + x \cdot -0.5}{\left(x \cdot \left(-1 - x\right)\right) \cdot -0.5}} \]

9. Taylor expanded in x around inf 15.7%

   \[\leadsto \color{blue}{\frac{1}{x}} + \frac{-1 + x \cdot -0.5}{\left(x \cdot \left(-1 - x\right)\right) \cdot -0.5} \]

10. Taylor expanded in x around 0 4.8%

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

    1. div-inv 4.8%

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

    2. metadata-eval 4.8%

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

    3. add-sqr-sqrt 2.4%

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

    4. sqrt-prod 48.2%

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

    5. sqr-neg 48.2%

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

    6. pow2 48.2%

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

    7. sqrt-pow1 6.3%

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

    8. metadata-eval 6.3%

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

    9. pow1 6.3%

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

    10. frac-2neg 6.3%

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

12. Applied egg-rr 6.3%

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

    1. associate-*r/ 6.3%

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

    2. metadata-eval 6.3%

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

14. Simplified 6.3%

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

Alternative 6: 5.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.3%

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

   1. +-commutative 66.3%

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

   2. associate-+r- 66.3%

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

   3. sub-neg 66.3%

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

   4. remove-double-neg 66.3%

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

   5. neg-sub0 66.3%

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

   6. associate-+l- 66.3%

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

   7. neg-sub0 66.3%

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

   8. distribute-neg-frac2 66.3%

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

   9. distribute-frac-neg2 66.3%

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

   10. associate-+r+ 66.3%

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

   11. +-commutative 66.3%

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

   12. remove-double-neg 66.3%

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

   13. distribute-neg-frac2 66.3%

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

   14. sub0-neg 66.3%

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

   15. associate-+l- 66.3%

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

   16. neg-sub0 66.3%

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

3. Simplified 66.3%

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

   1. clear-num 66.3%

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

   2. frac-sub 17.3%

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

   3. *-un-lft-identity 17.3%

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

   4. div-inv 17.3%

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

   5. metadata-eval 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) - \left(x \cdot \color{blue}{-0.5}\right) \cdot 1}{\frac{x}{-2} \cdot \left(-1 - x\right)} \]

   6. div-inv 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) - \left(x \cdot -0.5\right) \cdot 1}{\color{blue}{\left(x \cdot \frac{1}{-2}\right)} \cdot \left(-1 - x\right)} \]

   7. metadata-eval 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) - \left(x \cdot -0.5\right) \cdot 1}{\left(x \cdot \color{blue}{-0.5}\right) \cdot \left(-1 - x\right)} \]

6. Applied egg-rr 17.3%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{\left(-1 - x\right) - \left(x \cdot -0.5\right) \cdot 1}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)}} \]
7. Step-by-step derivation

   1. sub-neg 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{\left(-1 - x\right) + \left(-\left(x \cdot -0.5\right) \cdot 1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   2. *-rgt-identity 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 - x\right) + \left(-\color{blue}{x \cdot -0.5}\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   3. sub-neg 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{\left(-1 + \left(-x\right)\right)} + \left(-x \cdot -0.5\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   4. neg-mul-1 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 + \left(-x\right)\right) + \color{blue}{-1 \cdot \left(x \cdot -0.5\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   5. *-commutative 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\left(-1 + \left(-x\right)\right) + \color{blue}{\left(x \cdot -0.5\right) \cdot -1}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   6. associate-+l+ 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{-1 + \left(\left(-x\right) + \left(x \cdot -0.5\right) \cdot -1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   7. neg-mul-1 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \left(\color{blue}{-1 \cdot x} + \left(x \cdot -0.5\right) \cdot -1\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   8. *-commutative 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \left(\color{blue}{x \cdot -1} + \left(x \cdot -0.5\right) \cdot -1\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   9. associate-*l* 17.3%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \left(x \cdot -1 + \color{blue}{x \cdot \left(-0.5 \cdot -1\right)}\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   10. distribute-lft-out 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + \color{blue}{x \cdot \left(-1 + -0.5 \cdot -1\right)}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   11. metadata-eval 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot \left(-1 + \color{blue}{0.5}\right)}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   12. metadata-eval 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot \color{blue}{-0.5}}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)} \]

   13. *-commutative 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{\color{blue}{\left(-1 - x\right) \cdot \left(x \cdot -0.5\right)}} \]

   14. associate-*r* 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{\color{blue}{\left(\left(-1 - x\right) \cdot x\right) \cdot -0.5}} \]

   15. *-commutative 17.3%

       \[\leadsto \frac{1}{x + -1} + \frac{-1 + x \cdot -0.5}{\color{blue}{\left(x \cdot \left(-1 - x\right)\right)} \cdot -0.5} \]

8. Simplified 17.3%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{-1 + x \cdot -0.5}{\left(x \cdot \left(-1 - x\right)\right) \cdot -0.5}} \]

9. Taylor expanded in x around inf 15.7%

   \[\leadsto \color{blue}{\frac{1}{x}} + \frac{-1 + x \cdot -0.5}{\left(x \cdot \left(-1 - x\right)\right) \cdot -0.5} \]

10. Taylor expanded in x around 0 4.8%

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

Alternative 7: 5.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.3%

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

   1. +-commutative 66.3%

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

   2. associate-+r- 66.3%

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

   3. sub-neg 66.3%

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

   4. remove-double-neg 66.3%

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

   5. neg-sub0 66.3%

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

   6. associate-+l- 66.3%

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

   7. neg-sub0 66.3%

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

   8. distribute-neg-frac2 66.3%

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

   9. distribute-frac-neg2 66.3%

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

   10. associate-+r+ 66.3%

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

   11. +-commutative 66.3%

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

   12. remove-double-neg 66.3%

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

   13. distribute-neg-frac2 66.3%

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

   14. sub0-neg 66.3%

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

   15. associate-+l- 66.3%

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

   16. neg-sub0 66.3%

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

3. Simplified 66.3%

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

5. Taylor expanded in x around 0 4.7%

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

Developer Target 1: 99.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024148 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

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

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