3frac (problem 3.3.3)

Percentage Accurate: 68.9% → 99.5%

Time: 8.6s

Alternatives: 4

Speedup: 1.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left({x}^{-5} + {x}^{-3}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (+ (pow x -5.0) (pow x -3.0))))

C

double code(double x) {
	return 2.0 * (pow(x, -5.0) + pow(x, -3.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * ((x ** (-5.0d0)) + (x ** (-3.0d0)))
end function

Java

public static double code(double x) {
	return 2.0 * (Math.pow(x, -5.0) + Math.pow(x, -3.0));
}

Python

def code(x):
	return 2.0 * (math.pow(x, -5.0) + math.pow(x, -3.0))

Julia

function code(x)
	return Float64(2.0 * Float64((x ^ -5.0) + (x ^ -3.0)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * ((x ^ -5.0) + (x ^ -3.0));
end

Wolfram

code[x_] := N[(2.0 * N[(N[Power[x, -5.0], $MachinePrecision] + N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.9%

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

   1. sub-neg 72.9%

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

   2. distribute-neg-frac 72.9%

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

   3. metadata-eval 72.9%

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

   4. metadata-eval 72.9%

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

   5. metadata-eval 72.9%

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

   6. associate-/r* 72.9%

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

   7. metadata-eval 72.9%

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

   8. neg-mul-1 72.9%

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

   9. +-commutative 72.9%

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

   10. associate-+l+ 73.0%

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

   11. neg-mul-1 73.0%

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

   12. metadata-eval 73.0%

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

   13. associate-/r* 73.0%

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

   14. metadata-eval 73.0%

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

   15. metadata-eval 73.0%

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

   16. +-commutative 73.0%

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

   17. sub-neg 73.0%

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

   18. metadata-eval 73.0%

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

3. Simplified 73.0%

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

5. Taylor expanded in x around inf 99.1%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.1%

      \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{3}}} + 2 \cdot \frac{1}{{x}^{5}} \]

   2. metadata-eval 99.1%

      \[\leadsto \frac{\color{blue}{2}}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}} \]

   3. associate-*r/ 99.1%

      \[\leadsto \frac{2}{{x}^{3}} + \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} \]

   4. metadata-eval 99.1%

      \[\leadsto \frac{2}{{x}^{3}} + \frac{\color{blue}{2}}{{x}^{5}} \]

7. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}} \]
8. Step-by-step derivation

   1. *-un-lft-identity 99.1%

      \[\leadsto \color{blue}{1 \cdot \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\right)} \]

   2. +-commutative 99.1%

      \[\leadsto 1 \cdot \color{blue}{\left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)} \]

   3. div-inv 99.1%

      \[\leadsto 1 \cdot \left(\color{blue}{2 \cdot \frac{1}{{x}^{5}}} + \frac{2}{{x}^{3}}\right) \]

   4. fma-define 99.1%

      \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)} \]

   5. pow-flip 99.1%

      \[\leadsto 1 \cdot \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-5\right)}}, \frac{2}{{x}^{3}}\right) \]

   6. metadata-eval 99.1%

      \[\leadsto 1 \cdot \mathsf{fma}\left(2, {x}^{\color{blue}{-5}}, \frac{2}{{x}^{3}}\right) \]

   7. div-inv 99.1%

      \[\leadsto 1 \cdot \mathsf{fma}\left(2, {x}^{-5}, \color{blue}{2 \cdot \frac{1}{{x}^{3}}}\right) \]

   8. pow-flip 99.7%

      \[\leadsto 1 \cdot \mathsf{fma}\left(2, {x}^{-5}, 2 \cdot \color{blue}{{x}^{\left(-3\right)}}\right) \]

   9. metadata-eval 99.7%

      \[\leadsto 1 \cdot \mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{\color{blue}{-3}}\right) \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{1 \cdot \mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{-3}\right)} \]
10. Step-by-step derivation

    1. *-lft-identity 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{-3}\right)} \]

    2. fma-undefine 99.7%

       \[\leadsto \color{blue}{2 \cdot {x}^{-5} + 2 \cdot {x}^{-3}} \]

    3. distribute-lft-out 99.7%

       \[\leadsto \color{blue}{2 \cdot \left({x}^{-5} + {x}^{-3}\right)} \]

11. Simplified 99.7%

    \[\leadsto \color{blue}{2 \cdot \left({x}^{-5} + {x}^{-3}\right)} \]

12. Final simplification 99.7%

    \[\leadsto 2 \cdot \left({x}^{-5} + {x}^{-3}\right) \]
13. Add Preprocessing

Alternative 2: 99.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. sub-neg 72.9%

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

   2. distribute-neg-frac 72.9%

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

   3. metadata-eval 72.9%

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

   4. metadata-eval 72.9%

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

   5. metadata-eval 72.9%

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

   6. associate-/r* 72.9%

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

   7. metadata-eval 72.9%

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

   8. neg-mul-1 72.9%

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

   9. +-commutative 72.9%

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

   10. associate-+l+ 73.0%

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

   11. neg-mul-1 73.0%

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

   12. metadata-eval 73.0%

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

   13. associate-/r* 73.0%

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

   14. metadata-eval 73.0%

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

   15. metadata-eval 73.0%

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

   16. +-commutative 73.0%

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

   17. sub-neg 73.0%

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

   18. metadata-eval 73.0%

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

3. Simplified 73.0%

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

   1. frac-add 15.2%

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

   2. clear-num 19.7%

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

   3. +-commutative 19.7%

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

   4. *-un-lft-identity 19.7%

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

   5. *-rgt-identity 19.7%

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

   6. +-commutative 19.7%

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

   7. +-commutative 19.7%

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

6. Applied egg-rr 19.7%

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

   1. associate-/r/ 14.7%

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

   2. *-commutative 14.7%

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

   3. associate-+l+ 14.7%

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

   4. add-exp-log 3.4%

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

   5. log1p-undefine 3.4%

      \[\leadsto \frac{-2}{x} + \frac{1}{\left(x + -1\right) \cdot \left(x + 1\right)} \cdot \left(x + e^{\color{blue}{\mathsf{log1p}\left(x + -1\right)}}\right) \]

   6. *-un-lft-identity 3.4%

      \[\leadsto \frac{-2}{x} + \frac{1}{\left(x + -1\right) \cdot \left(x + 1\right)} \cdot \left(x + e^{\mathsf{log1p}\left(\color{blue}{1 \cdot x} + -1\right)}\right) \]

   7. fma-define 3.4%

      \[\leadsto \frac{-2}{x} + \frac{1}{\left(x + -1\right) \cdot \left(x + 1\right)} \cdot \left(x + e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(1, x, -1\right)}\right)}\right) \]

   8. metadata-eval 3.4%

      \[\leadsto \frac{-2}{x} + \frac{1}{\left(x + -1\right) \cdot \left(x + 1\right)} \cdot \left(x + e^{\mathsf{log1p}\left(\mathsf{fma}\left(1, x, \color{blue}{-1}\right)\right)}\right) \]

   9. fma-neg 3.4%

      \[\leadsto \frac{-2}{x} + \frac{1}{\left(x + -1\right) \cdot \left(x + 1\right)} \cdot \left(x + e^{\mathsf{log1p}\left(\color{blue}{1 \cdot x - 1}\right)}\right) \]

   10. *-un-lft-identity 3.4%

       \[\leadsto \frac{-2}{x} + \frac{1}{\left(x + -1\right) \cdot \left(x + 1\right)} \cdot \left(x + e^{\mathsf{log1p}\left(\color{blue}{x} - 1\right)}\right) \]

   11. add-exp-log 3.4%

       \[\leadsto \frac{-2}{x} + \frac{1}{\left(x + -1\right) \cdot \left(x + 1\right)} \cdot \left(x + e^{\mathsf{log1p}\left(\color{blue}{e^{\log x}} - 1\right)}\right) \]

   12. expm1-define 3.4%

       \[\leadsto \frac{-2}{x} + \frac{1}{\left(x + -1\right) \cdot \left(x + 1\right)} \cdot \left(x + e^{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log x\right)}\right)}\right) \]

   13. log1p-expm1-u 3.4%

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

   14. add-sqr-sqrt 3.4%

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

   15. sqrt-unprod 2.2%

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

   16. sqr-neg 2.2%

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

   17. sqrt-unprod 2.6%

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

   18. add-sqr-sqrt 2.6%

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

   19. add-exp-log 5.0%

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

   20. add-sqr-sqrt 2.6%

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

   21. sqrt-unprod 5.7%

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

   22. sqr-neg 5.7%

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

   23. sqrt-unprod 6.4%

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

   24. add-sqr-sqrt 14.7%

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

8. Applied egg-rr 14.7%

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

   1. +-commutative 14.7%

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

   2. associate-*l/ 15.2%

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

   3. *-un-lft-identity 15.2%

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

   4. frac-add 17.8%

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

   5. count-2 17.8%

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

10. Applied egg-rr 17.8%

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

11. Taylor expanded in x around 0 99.4%

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

12. Final simplification 99.4%

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

Alternative 3: 67.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. sub-neg 72.9%

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

   2. distribute-neg-frac 72.9%

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

   3. metadata-eval 72.9%

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

   4. metadata-eval 72.9%

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

   5. metadata-eval 72.9%

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

   6. associate-/r* 72.9%

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

   7. metadata-eval 72.9%

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

   8. neg-mul-1 72.9%

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

   9. +-commutative 72.9%

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

   10. associate-+l+ 73.0%

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

   11. neg-mul-1 73.0%

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

   12. metadata-eval 73.0%

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

   13. associate-/r* 73.0%

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

   14. metadata-eval 73.0%

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

   15. metadata-eval 73.0%

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

   16. +-commutative 73.0%

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

   17. sub-neg 73.0%

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

   18. metadata-eval 73.0%

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

3. Simplified 73.0%

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

5. Taylor expanded in x around inf 71.7%

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

6. Final simplification 71.7%

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

Alternative 4: 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 72.9%

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

   1. sub-neg 72.9%

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

   2. distribute-neg-frac 72.9%

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

   3. metadata-eval 72.9%

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

   4. metadata-eval 72.9%

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

   5. metadata-eval 72.9%

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

   6. associate-/r* 72.9%

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

   7. metadata-eval 72.9%

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

   8. neg-mul-1 72.9%

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

   9. +-commutative 72.9%

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

   10. associate-+l+ 73.0%

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

   11. neg-mul-1 73.0%

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

   12. metadata-eval 73.0%

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

   13. associate-/r* 73.0%

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

   14. metadata-eval 73.0%

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

   15. metadata-eval 73.0%

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

   16. +-commutative 73.0%

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

   17. sub-neg 73.0%

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

   18. metadata-eval 73.0%

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

3. Simplified 73.0%

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

5. Taylor expanded in x around 0 5.0%

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

6. Final simplification 5.0%

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

Developer target: 99.2% 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 2024036 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

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