3frac (problem 3.3.3)

Percentage Accurate: 9.9% → 99.2%

Time: 10.4s

Alternatives: 12

Speedup: 1.0×

Specification

\[\left|x\right| > 1 \land \left|x\right| < 10^{+100}\]

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: 9.9% 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: 99.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(2 \cdot {x}^{-3} + \frac{2}{{x}^{9}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (/ 2.0 (pow x 5.0))
  (+ (/ 2.0 (pow x 7.0)) (+ (* 2.0 (pow x -3.0)) (/ 2.0 (pow x 9.0))))))

C

double code(double x) {
	return (2.0 / pow(x, 5.0)) + ((2.0 / pow(x, 7.0)) + ((2.0 * pow(x, -3.0)) + (2.0 / pow(x, 9.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 / (x ** 5.0d0)) + ((2.0d0 / (x ** 7.0d0)) + ((2.0d0 * (x ** (-3.0d0))) + (2.0d0 / (x ** 9.0d0))))
end function

Java

public static double code(double x) {
	return (2.0 / Math.pow(x, 5.0)) + ((2.0 / Math.pow(x, 7.0)) + ((2.0 * Math.pow(x, -3.0)) + (2.0 / Math.pow(x, 9.0))));
}

Python

def code(x):
	return (2.0 / math.pow(x, 5.0)) + ((2.0 / math.pow(x, 7.0)) + ((2.0 * math.pow(x, -3.0)) + (2.0 / math.pow(x, 9.0))))

Julia

function code(x)
	return Float64(Float64(2.0 / (x ^ 5.0)) + Float64(Float64(2.0 / (x ^ 7.0)) + Float64(Float64(2.0 * (x ^ -3.0)) + Float64(2.0 / (x ^ 9.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (2.0 / (x ^ 5.0)) + ((2.0 / (x ^ 7.0)) + ((2.0 * (x ^ -3.0)) + (2.0 / (x ^ 9.0))));
end

Wolfram

code[x_] := N[(N[(2.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x, 9.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. sub-neg 7.7%

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

   2. distribute-neg-frac 7.7%

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

   3. metadata-eval 7.7%

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

   4. metadata-eval 7.7%

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

   5. metadata-eval 7.7%

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

   6. associate-/r* 7.7%

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

   7. metadata-eval 7.7%

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

   8. neg-mul-1 7.7%

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

   9. +-commutative 7.7%

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

   10. associate-+l+ 7.5%

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

   11. +-commutative 7.5%

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

   12. neg-mul-1 7.5%

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

   13. metadata-eval 7.5%

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

   14. associate-/r* 7.5%

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

   15. metadata-eval 7.5%

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

   16. metadata-eval 7.5%

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

   17. +-commutative 7.5%

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

   18. +-commutative 7.5%

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

3. Simplified 7.5%

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

4. Taylor expanded in x around inf 99.6%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{9}} + 2 \cdot \frac{1}{{x}^{3}}\right)\right)} \]
5. Step-by-step derivation

   1. associate-*r/ 99.6%

      \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} + \left(2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{9}} + 2 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   2. metadata-eval 99.6%

      \[\leadsto \frac{\color{blue}{2}}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{9}} + 2 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   3. associate-*r/ 99.6%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\color{blue}{\frac{2 \cdot 1}{{x}^{7}}} + \left(2 \cdot \frac{1}{{x}^{9}} + 2 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   4. metadata-eval 99.6%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{\color{blue}{2}}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{9}} + 2 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   5. +-commutative 99.6%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \color{blue}{\left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{9}}\right)}\right) \]

   6. associate-*r/ 99.6%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\color{blue}{\frac{2 \cdot 1}{{x}^{3}}} + 2 \cdot \frac{1}{{x}^{9}}\right)\right) \]

   7. metadata-eval 99.6%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\frac{\color{blue}{2}}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{9}}\right)\right) \]

   8. associate-*r/ 99.6%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{3}} + \color{blue}{\frac{2 \cdot 1}{{x}^{9}}}\right)\right) \]

   9. metadata-eval 99.6%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{3}} + \frac{\color{blue}{2}}{{x}^{9}}\right)\right) \]

6. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{9}}\right)\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 99.6%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)\right)} + \frac{2}{{x}^{9}}\right)\right) \]

   2. expm1-udef 8.1%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)} - 1\right)} + \frac{2}{{x}^{9}}\right)\right) \]

   3. div-inv 8.1%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{3}}}\right)} - 1\right) + \frac{2}{{x}^{9}}\right)\right) \]

   4. pow-flip 8.1%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\left(e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{x}^{\left(-3\right)}}\right)} - 1\right) + \frac{2}{{x}^{9}}\right)\right) \]

   5. metadata-eval 8.1%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\left(e^{\mathsf{log1p}\left(2 \cdot {x}^{\color{blue}{-3}}\right)} - 1\right) + \frac{2}{{x}^{9}}\right)\right) \]

8. Applied egg-rr 8.1%

   \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)} - 1\right)} + \frac{2}{{x}^{9}}\right)\right) \]
9. Step-by-step derivation

   1. expm1-def 99.9%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)\right)} + \frac{2}{{x}^{9}}\right)\right) \]

   2. expm1-log1p 99.9%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\color{blue}{2 \cdot {x}^{-3}} + \frac{2}{{x}^{9}}\right)\right) \]

10. Simplified 99.9%

    \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\color{blue}{2 \cdot {x}^{-3}} + \frac{2}{{x}^{9}}\right)\right) \]

11. Final simplification 99.9%

    \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(2 \cdot {x}^{-3} + \frac{2}{{x}^{9}}\right)\right) \]

Alternative 2: 96.8% 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 7.7%

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

   1. sub-neg 7.7%

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

   2. distribute-neg-frac 7.7%

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

   3. metadata-eval 7.7%

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

   4. metadata-eval 7.7%

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

   5. metadata-eval 7.7%

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

   6. associate-/r* 7.7%

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

   7. metadata-eval 7.7%

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

   8. neg-mul-1 7.7%

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

   9. +-commutative 7.7%

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

   10. associate-+l+ 7.5%

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

   11. +-commutative 7.5%

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

   12. neg-mul-1 7.5%

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

   13. metadata-eval 7.5%

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

   14. associate-/r* 7.5%

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

   15. metadata-eval 7.5%

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

   16. metadata-eval 7.5%

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

   17. +-commutative 7.5%

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

   18. +-commutative 7.5%

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

3. Simplified 7.5%

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

4. Taylor expanded in x around inf 98.1%

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

   1. expm1-log1p-u 99.6%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)\right)} + \frac{2}{{x}^{9}}\right)\right) \]

   2. expm1-udef 8.1%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)} - 1\right)} + \frac{2}{{x}^{9}}\right)\right) \]

   3. div-inv 8.1%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{3}}}\right)} - 1\right) + \frac{2}{{x}^{9}}\right)\right) \]

   4. pow-flip 8.1%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\left(e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{x}^{\left(-3\right)}}\right)} - 1\right) + \frac{2}{{x}^{9}}\right)\right) \]

   5. metadata-eval 8.1%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\left(e^{\mathsf{log1p}\left(2 \cdot {x}^{\color{blue}{-3}}\right)} - 1\right) + \frac{2}{{x}^{9}}\right)\right) \]

6. Applied egg-rr 5.8%

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

   1. expm1-def 99.9%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)\right)} + \frac{2}{{x}^{9}}\right)\right) \]

   2. expm1-log1p 99.9%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\color{blue}{2 \cdot {x}^{-3}} + \frac{2}{{x}^{9}}\right)\right) \]

8. Simplified 98.5%

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

9. Final simplification 98.5%

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

Alternative 3: 12.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + 1\right) \cdot \left(1 - x\right)\\ \frac{x \cdot \left(\left(1 - x\right) + \left(-1 - x\right)\right) + t_0 \cdot -2}{x \cdot t_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (+ x 1.0) (- 1.0 x))))
   (/ (+ (* x (+ (- 1.0 x) (- -1.0 x))) (* t_0 -2.0)) (* x t_0))))

C

double code(double x) {
	double t_0 = (x + 1.0) * (1.0 - x);
	return ((x * ((1.0 - x) + (-1.0 - x))) + (t_0 * -2.0)) / (x * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (x + 1.0d0) * (1.0d0 - x)
    code = ((x * ((1.0d0 - x) + ((-1.0d0) - x))) + (t_0 * (-2.0d0))) / (x * t_0)
end function

Java

public static double code(double x) {
	double t_0 = (x + 1.0) * (1.0 - x);
	return ((x * ((1.0 - x) + (-1.0 - x))) + (t_0 * -2.0)) / (x * t_0);
}

Python

def code(x):
	t_0 = (x + 1.0) * (1.0 - x)
	return ((x * ((1.0 - x) + (-1.0 - x))) + (t_0 * -2.0)) / (x * t_0)

Julia

function code(x)
	t_0 = Float64(Float64(x + 1.0) * Float64(1.0 - x))
	return Float64(Float64(Float64(x * Float64(Float64(1.0 - x) + Float64(-1.0 - x))) + Float64(t_0 * -2.0)) / Float64(x * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = (x + 1.0) * (1.0 - x);
	tmp = ((x * ((1.0 - x) + (-1.0 - x))) + (t_0 * -2.0)) / (x * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(x * N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.7%

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

   1. sub-neg 7.7%

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

   2. distribute-neg-frac 7.7%

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

   3. metadata-eval 7.7%

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

   4. metadata-eval 7.7%

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

   5. metadata-eval 7.7%

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

   6. associate-/r* 7.7%

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

   7. metadata-eval 7.7%

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

   8. neg-mul-1 7.7%

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

   9. +-commutative 7.7%

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

   10. associate-+l+ 7.5%

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

   11. +-commutative 7.5%

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

   12. neg-mul-1 7.5%

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

   13. metadata-eval 7.5%

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

   14. associate-/r* 7.5%

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

   15. metadata-eval 7.5%

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

   16. metadata-eval 7.5%

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

   17. +-commutative 7.5%

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

   18. +-commutative 7.5%

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

3. Simplified 7.5%

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

   1. +-commutative 7.5%

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

   2. frac-add 7.2%

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

   3. frac-add 10.0%

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

   4. *-un-lft-identity 10.0%

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

   5. *-commutative 10.0%

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

   6. neg-mul-1 10.0%

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

   7. distribute-neg-in 10.0%

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

   8. metadata-eval 10.0%

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

   9. +-commutative 10.0%

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

   10. +-commutative 10.0%

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

5. Applied egg-rr 10.0%

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

6. Final simplification 10.0%

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

Alternative 4: 12.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - x\right) \cdot \left(x \cdot -0.5\right)\\ \frac{\left(1 - x \cdot 0.5\right) \cdot \left(x + 1\right) + t_0}{\left(x + 1\right) \cdot t_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (- 1.0 x) (* x -0.5))))
   (/ (+ (* (- 1.0 (* x 0.5)) (+ x 1.0)) t_0) (* (+ x 1.0) t_0))))

C

double code(double x) {
	double t_0 = (1.0 - x) * (x * -0.5);
	return (((1.0 - (x * 0.5)) * (x + 1.0)) + t_0) / ((x + 1.0) * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (1.0d0 - x) * (x * (-0.5d0))
    code = (((1.0d0 - (x * 0.5d0)) * (x + 1.0d0)) + t_0) / ((x + 1.0d0) * t_0)
end function

Java

public static double code(double x) {
	double t_0 = (1.0 - x) * (x * -0.5);
	return (((1.0 - (x * 0.5)) * (x + 1.0)) + t_0) / ((x + 1.0) * t_0);
}

Python

def code(x):
	t_0 = (1.0 - x) * (x * -0.5)
	return (((1.0 - (x * 0.5)) * (x + 1.0)) + t_0) / ((x + 1.0) * t_0)

Julia

function code(x)
	t_0 = Float64(Float64(1.0 - x) * Float64(x * -0.5))
	return Float64(Float64(Float64(Float64(1.0 - Float64(x * 0.5)) * Float64(x + 1.0)) + t_0) / Float64(Float64(x + 1.0) * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = (1.0 - x) * (x * -0.5);
	tmp = (((1.0 - (x * 0.5)) * (x + 1.0)) + t_0) / ((x + 1.0) * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(1.0 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.7%

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

   1. associate-+l- 7.7%

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

   2. sub-neg 7.7%

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

   3. +-commutative 7.7%

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

   4. sub-neg 7.7%

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

   5. distribute-neg-in 7.7%

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

   6. distribute-neg-frac 7.7%

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

   7. metadata-eval 7.7%

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

   8. remove-double-neg 7.7%

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

   9. sub-neg 7.7%

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

   10. metadata-eval 7.7%

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

3. Simplified 7.7%

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

   1. clear-num 7.7%

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

   2. frac-2neg 7.7%

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

   3. metadata-eval 7.7%

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

   4. frac-add 7.5%

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

   5. *-un-lft-identity 7.5%

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

   6. +-commutative 7.5%

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

   7. distribute-neg-in 7.5%

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

   8. metadata-eval 7.5%

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

   9. sub-neg 7.5%

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

   10. div-inv 7.5%

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

   11. metadata-eval 7.5%

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

   12. div-inv 7.5%

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

   13. metadata-eval 7.5%

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

   14. +-commutative 7.5%

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

   15. distribute-neg-in 7.5%

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

   16. metadata-eval 7.5%

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

   17. sub-neg 7.5%

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

5. Applied egg-rr 7.5%

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

   1. +-commutative 7.5%

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

   2. *-commutative 7.5%

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

   3. associate-/l/ 7.5%

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

   4. *-commutative 7.5%

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

   5. +-commutative 7.5%

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

   6. associate-+l- 7.5%

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

   7. *-commutative 7.5%

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

   8. associate-*r* 7.5%

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

   9. neg-mul-1 7.5%

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

   10. cancel-sign-sub 7.5%

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

   11. +-commutative 7.5%

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

   12. *-commutative 7.5%

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

   13. distribute-lft1-in 7.5%

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

   14. metadata-eval 7.5%

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

   15. metadata-eval 7.5%

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

   16. *-commutative 7.5%

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

   17. metadata-eval 7.5%

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

7. Simplified 7.5%

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

   1. expm1-log1p-u 7.5%

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

   2. expm1-udef 6.9%

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

   3. associate-/l/ 6.9%

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

9. Applied egg-rr 6.9%

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

    1. expm1-def 7.6%

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

    2. expm1-log1p 7.6%

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

    3. *-commutative 7.6%

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

11. Simplified 7.6%

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

    1. +-commutative 7.6%

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

    2. frac-add 10.7%

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

    3. +-commutative 10.7%

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

    4. +-commutative 10.7%

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

13. Applied egg-rr 10.7%

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

14. Final simplification 10.7%

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

Alternative 5: 9.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

   1. associate-+l- 7.7%

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

   2. sub-neg 7.7%

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

   3. +-commutative 7.7%

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

   4. sub-neg 7.7%

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

   5. distribute-neg-in 7.7%

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

   6. distribute-neg-frac 7.7%

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

   7. metadata-eval 7.7%

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

   8. remove-double-neg 7.7%

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

   9. sub-neg 7.7%

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

   10. metadata-eval 7.7%

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

3. Simplified 7.7%

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

   1. clear-num 7.7%

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

   2. frac-2neg 7.7%

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

   3. metadata-eval 7.7%

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

   4. frac-add 7.5%

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

   5. *-un-lft-identity 7.5%

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

   6. +-commutative 7.5%

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

   7. distribute-neg-in 7.5%

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

   8. metadata-eval 7.5%

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

   9. sub-neg 7.5%

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

   10. div-inv 7.5%

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

   11. metadata-eval 7.5%

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

   12. div-inv 7.5%

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

   13. metadata-eval 7.5%

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

   14. +-commutative 7.5%

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

   15. distribute-neg-in 7.5%

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

   16. metadata-eval 7.5%

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

   17. sub-neg 7.5%

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

5. Applied egg-rr 7.5%

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

   1. +-commutative 7.5%

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

   2. *-commutative 7.5%

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

   3. associate-/l/ 7.5%

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

   4. *-commutative 7.5%

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

   5. +-commutative 7.5%

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

   6. associate-+l- 7.5%

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

   7. *-commutative 7.5%

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

   8. associate-*r* 7.5%

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

   9. neg-mul-1 7.5%

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

   10. cancel-sign-sub 7.5%

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

   11. +-commutative 7.5%

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

   12. *-commutative 7.5%

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

   13. distribute-lft1-in 7.5%

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

   14. metadata-eval 7.5%

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

   15. metadata-eval 7.5%

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

   16. *-commutative 7.5%

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

   17. metadata-eval 7.5%

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

7. Simplified 7.5%

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

   1. expm1-log1p-u 7.5%

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

   2. expm1-udef 6.9%

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

   3. associate-/l/ 6.9%

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

9. Applied egg-rr 6.9%

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

    1. expm1-def 7.6%

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

    2. expm1-log1p 7.6%

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

    3. *-commutative 7.6%

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

11. Simplified 7.6%

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

    1. associate-/r* 7.5%

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

    2. frac-add 7.8%

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

    3. *-un-lft-identity 7.8%

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

    4. +-commutative 7.8%

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

    5. +-commutative 7.8%

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

13. Applied egg-rr 7.8%

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

14. Final simplification 7.8%

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

Alternative 6: 9.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

   1. associate-+l- 7.7%

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

   2. sub-neg 7.7%

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

   3. +-commutative 7.7%

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

   4. sub-neg 7.7%

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

   5. distribute-neg-in 7.7%

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

   6. distribute-neg-frac 7.7%

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

   7. metadata-eval 7.7%

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

   8. remove-double-neg 7.7%

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

   9. sub-neg 7.7%

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

   10. metadata-eval 7.7%

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

3. Simplified 7.7%

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

   1. clear-num 7.7%

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

   2. frac-2neg 7.7%

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

   3. metadata-eval 7.7%

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

   4. frac-add 7.5%

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

   5. *-un-lft-identity 7.5%

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

   6. +-commutative 7.5%

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

   7. distribute-neg-in 7.5%

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

   8. metadata-eval 7.5%

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

   9. sub-neg 7.5%

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

   10. div-inv 7.5%

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

   11. metadata-eval 7.5%

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

   12. div-inv 7.5%

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

   13. metadata-eval 7.5%

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

   14. +-commutative 7.5%

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

   15. distribute-neg-in 7.5%

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

   16. metadata-eval 7.5%

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

   17. sub-neg 7.5%

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

5. Applied egg-rr 7.5%

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

   1. +-commutative 7.5%

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

   2. *-commutative 7.5%

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

   3. associate-/l/ 7.5%

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

   4. *-commutative 7.5%

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

   5. +-commutative 7.5%

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

   6. associate-+l- 7.5%

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

   7. *-commutative 7.5%

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

   8. associate-*r* 7.5%

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

   9. neg-mul-1 7.5%

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

   10. cancel-sign-sub 7.5%

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

   11. +-commutative 7.5%

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

   12. *-commutative 7.5%

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

   13. distribute-lft1-in 7.5%

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

   14. metadata-eval 7.5%

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

   15. metadata-eval 7.5%

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

   16. *-commutative 7.5%

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

   17. metadata-eval 7.5%

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

7. Simplified 7.5%

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

   1. div-inv 7.6%

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

9. Applied egg-rr 7.6%

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

10. Final simplification 7.6%

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

Alternative 7: 9.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

   1. associate-+l- 7.7%

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

   2. sub-neg 7.7%

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

   3. +-commutative 7.7%

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

   4. sub-neg 7.7%

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

   5. distribute-neg-in 7.7%

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

   6. distribute-neg-frac 7.7%

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

   7. metadata-eval 7.7%

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

   8. remove-double-neg 7.7%

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

   9. sub-neg 7.7%

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

   10. metadata-eval 7.7%

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

3. Simplified 7.7%

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

   1. clear-num 7.7%

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

   2. frac-2neg 7.7%

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

   3. metadata-eval 7.7%

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

   4. frac-add 7.5%

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

   5. *-un-lft-identity 7.5%

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

   6. +-commutative 7.5%

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

   7. distribute-neg-in 7.5%

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

   8. metadata-eval 7.5%

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

   9. sub-neg 7.5%

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

   10. div-inv 7.5%

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

   11. metadata-eval 7.5%

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

   12. div-inv 7.5%

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

   13. metadata-eval 7.5%

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

   14. +-commutative 7.5%

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

   15. distribute-neg-in 7.5%

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

   16. metadata-eval 7.5%

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

   17. sub-neg 7.5%

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

5. Applied egg-rr 7.5%

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

   1. +-commutative 7.5%

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

   2. *-commutative 7.5%

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

   3. associate-/l/ 7.5%

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

   4. *-commutative 7.5%

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

   5. +-commutative 7.5%

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

   6. associate-+l- 7.5%

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

   7. *-commutative 7.5%

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

   8. associate-*r* 7.5%

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

   9. neg-mul-1 7.5%

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

   10. cancel-sign-sub 7.5%

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

   11. +-commutative 7.5%

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

   12. *-commutative 7.5%

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

   13. distribute-lft1-in 7.5%

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

   14. metadata-eval 7.5%

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

   15. metadata-eval 7.5%

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

   16. *-commutative 7.5%

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

   17. metadata-eval 7.5%

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

7. Simplified 7.5%

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

   1. expm1-log1p-u 7.5%

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

   2. expm1-udef 6.9%

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

   3. associate-/l/ 6.9%

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

9. Applied egg-rr 6.9%

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

    1. expm1-def 7.6%

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

    2. expm1-log1p 7.6%

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

    3. *-lft-identity 7.6%

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

    4. times-frac 7.7%

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

    5. *-commutative 7.7%

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

    6. associate-/r* 7.7%

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

    7. metadata-eval 7.7%

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

    8. *-commutative 7.7%

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

    9. cancel-sign-sub-inv 7.7%

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

    10. metadata-eval 7.7%

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

    11. *-commutative 7.7%

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

11. Simplified 7.7%

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

12. Final simplification 7.7%

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

Alternative 8: 9.9% 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]

Derivation

1. Initial program 7.7%

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

2. Final simplification 7.7%

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

Alternative 9: 9.7% 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 7.7%

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

   1. sub-neg 7.7%

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

   2. distribute-neg-frac 7.7%

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

   3. metadata-eval 7.7%

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

   4. metadata-eval 7.7%

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

   5. metadata-eval 7.7%

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

   6. associate-/r* 7.7%

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

   7. metadata-eval 7.7%

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

   8. neg-mul-1 7.7%

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

   9. +-commutative 7.7%

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

   10. associate-+l+ 7.5%

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

   11. +-commutative 7.5%

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

   12. neg-mul-1 7.5%

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

   13. metadata-eval 7.5%

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

   14. associate-/r* 7.5%

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

   15. metadata-eval 7.5%

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

   16. metadata-eval 7.5%

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

   17. +-commutative 7.5%

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

   18. +-commutative 7.5%

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

3. Simplified 7.5%

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

   1. frac-add 7.2%

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

   2. clear-num 7.4%

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

   3. +-commutative 7.4%

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

   4. *-un-lft-identity 7.4%

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

   5. *-commutative 7.4%

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

   6. neg-mul-1 7.4%

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

   7. distribute-neg-in 7.4%

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

   8. metadata-eval 7.4%

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

5. Applied egg-rr 7.4%

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

6. Taylor expanded in x around 0 7.4%

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

   1. *-commutative 7.4%

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

   2. associate-*r/ 7.4%

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

   3. metadata-eval 7.4%

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

8. Simplified 7.4%

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

9. Final simplification 7.4%

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

Alternative 10: 5.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

   1. sub-neg 7.7%

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

   2. distribute-neg-frac 7.7%

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

   3. metadata-eval 7.7%

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

   4. metadata-eval 7.7%

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

   5. metadata-eval 7.7%

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

   6. associate-/r* 7.7%

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

   7. metadata-eval 7.7%

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

   8. neg-mul-1 7.7%

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

   9. +-commutative 7.7%

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

   10. associate-+l+ 7.5%

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

   11. +-commutative 7.5%

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

   12. neg-mul-1 7.5%

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

   13. metadata-eval 7.5%

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

   14. associate-/r* 7.5%

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

   15. metadata-eval 7.5%

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

   16. metadata-eval 7.5%

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

   17. +-commutative 7.5%

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

   18. +-commutative 7.5%

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

3. Simplified 7.5%

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

   1. frac-add 7.2%

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

   2. associate-/r* 7.5%

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

   3. *-un-lft-identity 7.5%

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

   4. *-commutative 7.5%

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

   5. neg-mul-1 7.5%

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

   6. distribute-neg-in 7.5%

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

   7. metadata-eval 7.5%

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

   8. +-commutative 7.5%

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

5. Applied egg-rr 7.5%

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

6. Taylor expanded in x around inf 5.3%

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

7. Final simplification 5.3%

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

Alternative 11: 3.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

   1. sub-neg 7.7%

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

   2. distribute-neg-frac 7.7%

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

   3. metadata-eval 7.7%

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

   4. metadata-eval 7.7%

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

   5. metadata-eval 7.7%

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

   6. associate-/r* 7.7%

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

   7. metadata-eval 7.7%

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

   8. neg-mul-1 7.7%

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

   9. +-commutative 7.7%

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

   10. associate-+l+ 7.5%

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

   11. +-commutative 7.5%

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

   12. neg-mul-1 7.5%

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

   13. metadata-eval 7.5%

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

   14. associate-/r* 7.5%

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

   15. metadata-eval 7.5%

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

   16. metadata-eval 7.5%

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

   17. +-commutative 7.5%

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

   18. +-commutative 7.5%

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

3. Simplified 7.5%

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

4. Taylor expanded in x around 0 3.7%

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

5. Taylor expanded in x around inf 3.7%

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

6. Final simplification 3.7%

   \[\leadsto 1 - \frac{1}{x} \]

Alternative 12: 3.9% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 7.7%

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

   1. sub-neg 7.7%

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

   2. distribute-neg-frac 7.7%

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

   3. metadata-eval 7.7%

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

   4. metadata-eval 7.7%

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

   5. metadata-eval 7.7%

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

   6. associate-/r* 7.7%

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

   7. metadata-eval 7.7%

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

   8. neg-mul-1 7.7%

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

   9. +-commutative 7.7%

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

   10. associate-+l+ 7.5%

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

   11. +-commutative 7.5%

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

   12. neg-mul-1 7.5%

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

   13. metadata-eval 7.5%

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

   14. associate-/r* 7.5%

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

   15. metadata-eval 7.5%

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

   16. metadata-eval 7.5%

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

   17. +-commutative 7.5%

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

   18. +-commutative 7.5%

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

3. Simplified 7.5%

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

4. Taylor expanded in x around 0 3.7%

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

5. Taylor expanded in x around inf 3.7%

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

6. Final simplification 3.7%

   \[\leadsto 1 \]

Developer target: 99.5% 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 2023339 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (and (> (fabs x) 1.0) (< (fabs x) 1e+100))

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

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