3frac (problem 3.3.3)

Percentage Accurate: 69.3% → 99.5%

Time: 11.7s

Alternatives: 10

Speedup: 1.0×

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: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. sub-neg 62.9%

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

   2. distribute-neg-frac 62.9%

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

   3. metadata-eval 62.9%

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

   4. metadata-eval 62.9%

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

   5. metadata-eval 62.9%

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

   6. associate-/r* 62.9%

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

   7. metadata-eval 62.9%

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

   8. neg-mul-1 62.9%

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

   9. +-commutative 62.9%

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

   10. associate-+l+ 62.9%

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

   11. +-commutative 62.9%

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

   12. neg-mul-1 62.9%

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

   13. metadata-eval 62.9%

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

   14. associate-/r* 62.9%

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

   15. metadata-eval 62.9%

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

   16. metadata-eval 62.9%

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

   17. +-commutative 62.9%

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

   18. +-commutative 62.9%

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

   19. sub-neg 62.9%

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

   20. metadata-eval 62.9%

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

3. Simplified 62.9%

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

5. Taylor expanded in x around inf 98.8%

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

   1. associate-*r/ 98.8%

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

   2. metadata-eval 98.8%

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

   3. associate-*r/ 98.8%

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

   4. metadata-eval 98.8%

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

7. Simplified 98.8%

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

   1. expm1-log1p-u 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\right)\right)} \]

   2. expm1-udef 62.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\right)} - 1} \]

   3. +-commutative 62.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}}\right)} - 1 \]

   4. div-inv 62.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{5}}} + \frac{2}{{x}^{3}}\right)} - 1 \]

   5. fma-def 62.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)}\right)} - 1 \]

   6. pow-flip 62.4%

      \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, \color{blue}{{x}^{\left(-5\right)}}, \frac{2}{{x}^{3}}\right)\right)} - 1 \]

   7. metadata-eval 62.4%

      \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{\color{blue}{-5}}, \frac{2}{{x}^{3}}\right)\right)} - 1 \]

   8. div-inv 62.4%

      \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, \color{blue}{2 \cdot \frac{1}{{x}^{3}}}\right)\right)} - 1 \]

   9. pow-flip 62.4%

      \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot \color{blue}{{x}^{\left(-3\right)}}\right)\right)} - 1 \]

   10. metadata-eval 62.4%

       \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{\color{blue}{-3}}\right)\right)} - 1 \]

9. Applied egg-rr 62.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{-3}\right)\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 100.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot {x}^{-3}\right)\right)\right)} \]

    2. expm1-log1p 100.0%

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

    3. fma-udef 100.0%

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

    4. +-commutative 100.0%

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

    5. distribute-lft-out 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot {x}^{-3} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. sub-neg 62.9%

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

   2. distribute-neg-frac 62.9%

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

   3. metadata-eval 62.9%

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

   4. metadata-eval 62.9%

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

   5. metadata-eval 62.9%

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

   6. associate-/r* 62.9%

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

   7. metadata-eval 62.9%

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

   8. neg-mul-1 62.9%

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

   9. +-commutative 62.9%

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

   10. associate-+l+ 62.9%

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

   11. +-commutative 62.9%

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

   12. neg-mul-1 62.9%

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

   13. metadata-eval 62.9%

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

   14. associate-/r* 62.9%

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

   15. metadata-eval 62.9%

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

   16. metadata-eval 62.9%

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

   17. +-commutative 62.9%

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

   18. +-commutative 62.9%

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

   19. sub-neg 62.9%

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

   20. metadata-eval 62.9%

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

3. Simplified 62.9%

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

5. Taylor expanded in x around inf 98.5%

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

   1. expm1-log1p-u 98.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)\right)} \]

   2. expm1-udef 62.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)} - 1} \]

   3. div-inv 62.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{3}}}\right)} - 1 \]

   4. pow-flip 62.4%

      \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{x}^{\left(-3\right)}}\right)} - 1 \]

   5. metadata-eval 62.4%

      \[\leadsto e^{\mathsf{log1p}\left(2 \cdot {x}^{\color{blue}{-3}}\right)} - 1 \]

7. Applied egg-rr 62.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 99.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)\right)} \]

   2. expm1-log1p 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

    \[\leadsto 2 \cdot {x}^{-3} \]
11. Add Preprocessing

Alternative 3: 69.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. associate-+l- 62.9%

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

   2. sub-neg 62.9%

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

   3. +-commutative 62.9%

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

   4. neg-sub0 62.9%

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

   5. associate-+l- 62.9%

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

   6. neg-sub0 62.9%

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

   7. distribute-neg-frac 62.9%

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

   8. metadata-eval 62.9%

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

   9. sub-neg 62.9%

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

   10. metadata-eval 62.9%

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

3. Simplified 62.9%

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

   1. clear-num 62.9%

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

   2. frac-2neg 62.9%

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

   3. metadata-eval 62.9%

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

   4. frac-add 16.1%

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

   5. *-un-lft-identity 16.1%

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

   6. +-commutative 16.1%

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

   7. distribute-neg-in 16.1%

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

   8. metadata-eval 16.1%

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

   9. sub-neg 16.1%

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

   10. div-inv 16.1%

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

   11. metadata-eval 16.1%

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

   12. div-inv 16.1%

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

   13. metadata-eval 16.1%

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

   14. +-commutative 16.1%

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

   15. distribute-neg-in 16.1%

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

   16. metadata-eval 16.1%

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

   17. sub-neg 16.1%

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

6. Applied egg-rr 16.1%

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

   1. associate-*l* 16.1%

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

   2. associate-+l- 16.1%

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

   3. *-commutative 16.1%

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

   4. associate-*r* 16.1%

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

   5. neg-mul-1 16.1%

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

   6. cancel-sign-sub 16.1%

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

   7. +-commutative 16.1%

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

   8. *-commutative 16.1%

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

   9. distribute-lft1-in 16.1%

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

   10. metadata-eval 16.1%

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

   11. metadata-eval 16.1%

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

   12. *-commutative 16.1%

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

   13. metadata-eval 16.1%

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

8. Simplified 16.1%

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

   1. expm1-log1p-u 16.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + x} + \frac{1 - x \cdot 0.5}{x \cdot \left(-0.5 \cdot \left(1 - x\right)\right)}\right)\right)} \]

   2. expm1-udef 62.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{1 + x} + \frac{1 - x \cdot 0.5}{x \cdot \left(-0.5 \cdot \left(1 - x\right)\right)}\right)} - 1} \]

   3. +-commutative 62.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 - x \cdot 0.5}{x \cdot \left(-0.5 \cdot \left(1 - x\right)\right)} + \frac{1}{1 + x}}\right)} - 1 \]

   4. +-commutative 62.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1 - x \cdot 0.5}{x \cdot \left(-0.5 \cdot \left(1 - x\right)\right)} + \frac{1}{\color{blue}{x + 1}}\right)} - 1 \]

10. Applied egg-rr 62.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 - x \cdot 0.5}{x \cdot \left(-0.5 \cdot \left(1 - x\right)\right)} + \frac{1}{x + 1}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 16.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 - x \cdot 0.5}{x \cdot \left(-0.5 \cdot \left(1 - x\right)\right)} + \frac{1}{x + 1}\right)\right)} \]

    2. expm1-log1p 16.1%

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

    3. associate-/r* 63.0%

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

    4. div-sub 63.0%

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

    5. *-commutative 63.0%

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

    6. associate-/l* 63.0%

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

    7. *-inverses 63.0%

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

    8. metadata-eval 63.0%

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

12. Simplified 63.0%

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

13. Final simplification 63.0%

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

Alternative 4: 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]

Derivation

1. Initial program 62.9%

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

3. Final simplification 62.9%

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

Alternative 5: 69.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{x} + \frac{1}{x \cdot 0.5 - \frac{0.5}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (/ -2.0 x) (/ 1.0 (- (* x 0.5) (/ 0.5 x)))))

C

double code(double x) {
	return (-2.0 / x) + (1.0 / ((x * 0.5) - (0.5 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-2.0d0) / x) + (1.0d0 / ((x * 0.5d0) - (0.5d0 / x)))
end function

Java

public static double code(double x) {
	return (-2.0 / x) + (1.0 / ((x * 0.5) - (0.5 / x)));
}

Python

def code(x):
	return (-2.0 / x) + (1.0 / ((x * 0.5) - (0.5 / x)))

Julia

function code(x)
	return Float64(Float64(-2.0 / x) + Float64(1.0 / Float64(Float64(x * 0.5) - Float64(0.5 / x))))
end

MATLAB

function tmp = code(x)
	tmp = (-2.0 / x) + (1.0 / ((x * 0.5) - (0.5 / x)));
end

Wolfram

code[x_] := N[(N[(-2.0 / x), $MachinePrecision] + N[(1.0 / N[(N[(x * 0.5), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

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

   1. sub-neg 62.9%

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

   2. distribute-neg-frac 62.9%

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

   3. metadata-eval 62.9%

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

   4. metadata-eval 62.9%

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

   5. metadata-eval 62.9%

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

   6. associate-/r* 62.9%

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

   7. metadata-eval 62.9%

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

   8. neg-mul-1 62.9%

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

   9. +-commutative 62.9%

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

   10. associate-+l+ 62.9%

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

   11. +-commutative 62.9%

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

   12. neg-mul-1 62.9%

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

   13. metadata-eval 62.9%

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

   14. associate-/r* 62.9%

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

   15. metadata-eval 62.9%

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

   16. metadata-eval 62.9%

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

   17. +-commutative 62.9%

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

   18. +-commutative 62.9%

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

   19. sub-neg 62.9%

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

   20. metadata-eval 62.9%

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

3. Simplified 62.9%

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

   1. frac-add 15.6%

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

   2. clear-num 19.1%

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

   3. +-commutative 19.1%

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

   4. *-un-lft-identity 19.1%

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

   5. *-rgt-identity 19.1%

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

   6. +-commutative 19.1%

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

   7. +-commutative 19.1%

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

6. Applied egg-rr 19.1%

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

7. Taylor expanded in x around 0 62.9%

   \[\leadsto \frac{-2}{x} + \frac{1}{\color{blue}{0.5 \cdot x - 0.5 \cdot \frac{1}{x}}} \]
8. Step-by-step derivation

   1. *-commutative 62.9%

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

   2. associate-*r/ 62.9%

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

   3. metadata-eval 62.9%

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

9. Simplified 62.9%

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

10. Final simplification 62.9%

    \[\leadsto \frac{-2}{x} + \frac{1}{x \cdot 0.5 - \frac{0.5}{x}} \]
11. Add Preprocessing

Alternative 6: 67.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{x} + \frac{1}{0.5 \cdot \left(x + -1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (/ -2.0 x) (/ 1.0 (* 0.5 (+ x -1.0)))))

C

double code(double x) {
	return (-2.0 / x) + (1.0 / (0.5 * (x + -1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-2.0d0) / x) + (1.0d0 / (0.5d0 * (x + (-1.0d0))))
end function

Java

public static double code(double x) {
	return (-2.0 / x) + (1.0 / (0.5 * (x + -1.0)));
}

Python

def code(x):
	return (-2.0 / x) + (1.0 / (0.5 * (x + -1.0)))

Julia

function code(x)
	return Float64(Float64(-2.0 / x) + Float64(1.0 / Float64(0.5 * Float64(x + -1.0))))
end

MATLAB

function tmp = code(x)
	tmp = (-2.0 / x) + (1.0 / (0.5 * (x + -1.0)));
end

Wolfram

code[x_] := N[(N[(-2.0 / x), $MachinePrecision] + N[(1.0 / N[(0.5 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

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

   1. sub-neg 62.9%

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

   2. distribute-neg-frac 62.9%

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

   3. metadata-eval 62.9%

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

   4. metadata-eval 62.9%

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

   5. metadata-eval 62.9%

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

   6. associate-/r* 62.9%

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

   7. metadata-eval 62.9%

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

   8. neg-mul-1 62.9%

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

   9. +-commutative 62.9%

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

   10. associate-+l+ 62.9%

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

   11. +-commutative 62.9%

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

   12. neg-mul-1 62.9%

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

   13. metadata-eval 62.9%

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

   14. associate-/r* 62.9%

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

   15. metadata-eval 62.9%

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

   16. metadata-eval 62.9%

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

   17. +-commutative 62.9%

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

   18. +-commutative 62.9%

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

   19. sub-neg 62.9%

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

   20. metadata-eval 62.9%

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

3. Simplified 62.9%

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

   1. frac-add 15.6%

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

   2. clear-num 19.1%

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

   3. +-commutative 19.1%

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

   4. *-un-lft-identity 19.1%

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

   5. *-rgt-identity 19.1%

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

   6. +-commutative 19.1%

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

   7. +-commutative 19.1%

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

6. Applied egg-rr 19.1%

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

   1. expm1-log1p-u 3.4%

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

   2. expm1-udef 3.4%

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

8. Applied egg-rr 4.5%

   \[\leadsto \frac{-2}{x} + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\frac{x + 1}{\frac{x + x}{x + -1}}\right)} - 1}} \]
9. Step-by-step derivation

   1. expm1-def 4.5%

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

   2. expm1-log1p 62.8%

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

   3. associate-/r/ 62.8%

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

   4. count-2 62.8%

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

10. Simplified 62.8%

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

11. Taylor expanded in x around inf 62.3%

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

12. Final simplification 62.3%

    \[\leadsto \frac{-2}{x} + \frac{1}{0.5 \cdot \left(x + -1\right)} \]
13. Add Preprocessing

Alternative 7: 67.8% 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 62.9%

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

   1. sub-neg 62.9%

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

   2. distribute-neg-frac 62.9%

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

   3. metadata-eval 62.9%

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

   4. metadata-eval 62.9%

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

   5. metadata-eval 62.9%

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

   6. associate-/r* 62.9%

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

   7. metadata-eval 62.9%

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

   8. neg-mul-1 62.9%

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

   9. +-commutative 62.9%

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

   10. associate-+l+ 62.9%

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

   11. +-commutative 62.9%

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

   12. neg-mul-1 62.9%

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

   13. metadata-eval 62.9%

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

   14. associate-/r* 62.9%

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

   15. metadata-eval 62.9%

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

   16. metadata-eval 62.9%

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

   17. +-commutative 62.9%

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

   18. +-commutative 62.9%

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

   19. sub-neg 62.9%

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

   20. metadata-eval 62.9%

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

3. Simplified 62.9%

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

5. Taylor expanded in x around inf 62.1%

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

6. Final simplification 62.1%

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

Alternative 8: 2.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x + -2.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. sub-neg 62.9%

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

   2. distribute-neg-frac 62.9%

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

   3. metadata-eval 62.9%

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

   4. metadata-eval 62.9%

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

   5. metadata-eval 62.9%

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

   6. associate-/r* 62.9%

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

   7. metadata-eval 62.9%

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

   8. neg-mul-1 62.9%

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

   9. +-commutative 62.9%

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

   10. associate-+l+ 62.9%

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

   11. +-commutative 62.9%

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

   12. neg-mul-1 62.9%

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

   13. metadata-eval 62.9%

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

   14. associate-/r* 62.9%

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

   15. metadata-eval 62.9%

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

   16. metadata-eval 62.9%

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

   17. +-commutative 62.9%

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

   18. +-commutative 62.9%

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

   19. sub-neg 62.9%

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

   20. metadata-eval 62.9%

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

3. Simplified 62.9%

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

   1. frac-add 15.6%

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

   2. div-inv 15.4%

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

   3. *-un-lft-identity 15.4%

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

   4. *-rgt-identity 15.4%

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

   5. +-commutative 15.4%

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

   6. +-commutative 15.4%

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

   7. metadata-eval 15.4%

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

   8. frac-times 15.7%

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

   9. clear-num 15.7%

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

   10. frac-times 15.4%

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

   11. metadata-eval 15.4%

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

   12. +-commutative 15.4%

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

   13. /-rgt-identity 15.4%

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

6. Applied egg-rr 15.4%

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

   1. associate-*r/ 15.6%

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

   2. *-rgt-identity 15.6%

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

   3. associate-+l+ 15.6%

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

   4. +-commutative 15.6%

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

   5. associate-+r+ 15.6%

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

   6. metadata-eval 15.6%

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

   7. *-commutative 15.6%

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

8. Simplified 15.6%

   \[\leadsto \frac{-2}{x} + \color{blue}{\frac{x + \left(0 + x\right)}{\left(x + -1\right) \cdot \left(x + 1\right)}} \]
9. Step-by-step derivation

   1. clear-num 19.1%

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

   2. *-commutative 19.1%

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

   3. add-exp-log 3.2%

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

   4. +-lft-identity 3.2%

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

   5. log1p-expm1-u 3.2%

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

   6. expm1-def 3.2%

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

   7. add-exp-log 3.2%

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

   8. *-un-lft-identity 3.2%

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

   9. fma-neg 3.2%

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

   10. metadata-eval 3.2%

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

   11. fma-def 3.2%

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

   12. *-un-lft-identity 3.2%

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

   13. log1p-udef 3.2%

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

   14. add-exp-log 19.1%

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

   15. associate-+l+ 19.1%

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

   16. associate-/r/ 15.4%

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

10. Applied egg-rr 15.4%

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

    1. *-commutative 15.4%

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

    2. div-inv 15.6%

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

    3. associate-/r* 62.9%

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

    4. clear-num 62.8%

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

    5. *-un-lft-identity 62.8%

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

    6. frac-add 62.9%

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

12. Applied egg-rr 0.0%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{0}{0} \cdot \mathsf{fma}\left(x, x, -1\right), x\right)}{x \cdot \left(\frac{0}{0} \cdot \mathsf{fma}\left(x, x, -1\right)\right)}} \]

14. Simplified 3.0%

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

15. Final simplification 3.0%

    \[\leadsto x + -2 \]
16. Add Preprocessing

Alternative 9: 5.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. sub-neg 62.9%

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

   2. distribute-neg-frac 62.9%

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

   3. metadata-eval 62.9%

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

   4. metadata-eval 62.9%

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

   5. metadata-eval 62.9%

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

   6. associate-/r* 62.9%

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

   7. metadata-eval 62.9%

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

   8. neg-mul-1 62.9%

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

   9. +-commutative 62.9%

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

   10. associate-+l+ 62.9%

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

   11. +-commutative 62.9%

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

   12. neg-mul-1 62.9%

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

   13. metadata-eval 62.9%

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

   14. associate-/r* 62.9%

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

   15. metadata-eval 62.9%

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

   16. metadata-eval 62.9%

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

   17. +-commutative 62.9%

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

   18. +-commutative 62.9%

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

   19. sub-neg 62.9%

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

   20. metadata-eval 62.9%

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

3. Simplified 62.9%

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

5. Taylor expanded in x around 0 4.8%

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

6. Final simplification 4.8%

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

Alternative 10: 3.3% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -7 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -7.0)

C

double code(double x) {
	return -7.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -7.0d0
end function

Java

public static double code(double x) {
	return -7.0;
}

Python

def code(x):
	return -7.0

Julia

function code(x)
	return -7.0
end

MATLAB

function tmp = code(x)
	tmp = -7.0;
end

Wolfram

code[x_] := -7.0

Derivation

1. Initial program 62.9%

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

   1. sub-neg 62.9%

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

   2. distribute-neg-frac 62.9%

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

   3. metadata-eval 62.9%

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

   4. metadata-eval 62.9%

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

   5. metadata-eval 62.9%

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

   6. associate-/r* 62.9%

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

   7. metadata-eval 62.9%

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

   8. neg-mul-1 62.9%

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

   9. +-commutative 62.9%

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

   10. associate-+l+ 62.9%

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

   11. +-commutative 62.9%

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

   12. neg-mul-1 62.9%

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

   13. metadata-eval 62.9%

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

   14. associate-/r* 62.9%

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

   15. metadata-eval 62.9%

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

   16. metadata-eval 62.9%

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

   17. +-commutative 62.9%

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

   18. +-commutative 62.9%

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

   19. sub-neg 62.9%

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

   20. metadata-eval 62.9%

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

3. Simplified 62.9%

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

5. Taylor expanded in x around inf 98.5%

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

   1. expm1-log1p-u 98.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)\right)} \]

   2. expm1-udef 62.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)} - 1} \]

   3. div-inv 62.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{3}}}\right)} - 1 \]

   4. pow-flip 62.4%

      \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{x}^{\left(-3\right)}}\right)} - 1 \]

   5. metadata-eval 62.4%

      \[\leadsto e^{\mathsf{log1p}\left(2 \cdot {x}^{\color{blue}{-3}}\right)} - 1 \]

7. Applied egg-rr 62.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 99.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)\right)} \]

   2. expm1-log1p 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in x around 0 98.5%

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

12. Simplified 3.4%

    \[\leadsto \color{blue}{-7} \]

13. Final simplification 3.4%

    \[\leadsto -7 \]
14. 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 2024024 
(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))))