3frac (problem 3.3.3)

Percentage Accurate: 69.4% → 99.4%

Time: 11.8s

Alternatives: 6

Speedup: 0.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot {x}^{-3} + \frac{2}{{x}^{5}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.9%

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

   1. sub-neg 71.9%

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

   2. distribute-neg-frac 71.9%

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

   3. metadata-eval 71.9%

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

   4. metadata-eval 71.9%

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

   5. metadata-eval 71.9%

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

   6. associate-/r* 71.9%

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

   7. metadata-eval 71.9%

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

   8. neg-mul-1 71.9%

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

   9. +-commutative 71.9%

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

   10. associate-+l+ 71.8%

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

   11. +-commutative 71.8%

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

   12. neg-mul-1 71.8%

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

   13. metadata-eval 71.8%

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

   14. associate-/r* 71.8%

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

   15. metadata-eval 71.8%

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

   16. metadata-eval 71.8%

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

   17. +-commutative 71.8%

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

   18. +-commutative 71.8%

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

   19. sub-neg 71.8%

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

   20. metadata-eval 71.8%

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

3. Simplified 71.8%

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

5. Taylor expanded in x around inf 98.5%

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

   1. associate-*r/ 98.5%

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

   2. metadata-eval 98.5%

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

   3. associate-*r/ 98.5%

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

   4. metadata-eval 98.5%

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

7. Simplified 98.5%

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

   1. div-inv 98.5%

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

   2. pow-flip 99.1%

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

   3. metadata-eval 99.1%

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

9. Applied egg-rr 99.1%

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

10. Final simplification 99.1%

    \[\leadsto 2 \cdot {x}^{-3} + \frac{2}{{x}^{5}} \]
11. Add Preprocessing

Alternative 2: 98.9% 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 71.9%

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

   1. sub-neg 71.9%

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

   2. distribute-neg-frac 71.9%

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

   3. metadata-eval 71.9%

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

   4. metadata-eval 71.9%

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

   5. metadata-eval 71.9%

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

   6. associate-/r* 71.9%

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

   7. metadata-eval 71.9%

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

   8. neg-mul-1 71.9%

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

   9. +-commutative 71.9%

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

   10. associate-+l+ 71.8%

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

   11. +-commutative 71.8%

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

   12. neg-mul-1 71.8%

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

   13. metadata-eval 71.8%

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

   14. associate-/r* 71.8%

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

   15. metadata-eval 71.8%

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

   16. metadata-eval 71.8%

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

   17. +-commutative 71.8%

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

   18. +-commutative 71.8%

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

   19. sub-neg 71.8%

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

   20. metadata-eval 71.8%

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

3. Simplified 71.8%

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

5. Taylor expanded in x around inf 97.8%

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

   1. div-inv 98.5%

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

   2. pow-flip 99.1%

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

   3. metadata-eval 99.1%

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

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

Alternative 3: 44.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+85}:\\ \;\;\;\;\frac{-2}{x} + x \cdot \frac{\frac{2}{x + 1}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} + \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+85)
   (+ (/ -2.0 x) (* x (/ (/ 2.0 (+ x 1.0)) (+ x -1.0))))
   (+ (/ -2.0 x) (/ 2.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 5e+85) {
		tmp = (-2.0 / x) + (x * ((2.0 / (x + 1.0)) / (x + -1.0)));
	} else {
		tmp = (-2.0 / x) + (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5d+85) then
        tmp = ((-2.0d0) / x) + (x * ((2.0d0 / (x + 1.0d0)) / (x + (-1.0d0))))
    else
        tmp = ((-2.0d0) / x) + (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e+85) {
		tmp = (-2.0 / x) + (x * ((2.0 / (x + 1.0)) / (x + -1.0)));
	} else {
		tmp = (-2.0 / x) + (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e+85:
		tmp = (-2.0 / x) + (x * ((2.0 / (x + 1.0)) / (x + -1.0)))
	else:
		tmp = (-2.0 / x) + (2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+85)
		tmp = Float64(Float64(-2.0 / x) + Float64(x * Float64(Float64(2.0 / Float64(x + 1.0)) / Float64(x + -1.0))));
	else
		tmp = Float64(Float64(-2.0 / x) + Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e+85)
		tmp = (-2.0 / x) + (x * ((2.0 / (x + 1.0)) / (x + -1.0)));
	else
		tmp = (-2.0 / x) + (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e+85], N[(N[(-2.0 / x), $MachinePrecision] + N[(x * N[(N[(2.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / x), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e85

   1. Initial program 57.6%

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

      1. sub-neg 57.6%

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

      2. distribute-neg-frac 57.6%

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

      3. metadata-eval 57.6%

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

      4. metadata-eval 57.6%

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

      5. metadata-eval 57.6%

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

      6. associate-/r* 57.6%

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

      7. metadata-eval 57.6%

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

      8. neg-mul-1 57.6%

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

      9. +-commutative 57.6%

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

      10. associate-+l+ 57.5%

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

      11. +-commutative 57.5%

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

      12. neg-mul-1 57.5%

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

      13. metadata-eval 57.5%

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

      14. associate-/r* 57.5%

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

      15. metadata-eval 57.5%

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

      16. metadata-eval 57.5%

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

      17. +-commutative 57.5%

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

      18. +-commutative 57.5%

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

      19. sub-neg 57.5%

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

      20. metadata-eval 57.5%

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

   3. Simplified 57.5%

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

      1. frac-add 19.6%

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

      2. div-inv 18.4%

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

      3. *-un-lft-identity 18.4%

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

      4. *-rgt-identity 18.4%

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

      5. +-commutative 18.4%

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

      6. +-commutative 18.4%

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

      7. metadata-eval 18.4%

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

      8. frac-times 15.8%

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

      9. clear-num 15.8%

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

      10. frac-times 18.4%

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

      11. metadata-eval 18.4%

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

      12. +-commutative 18.4%

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

      13. /-rgt-identity 18.4%

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

   6. Applied egg-rr 18.4%

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

      1. associate-*r/ 19.6%

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

      2. *-rgt-identity 19.6%

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

      3. associate-+l+ 19.6%

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

      4. +-commutative 19.6%

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

      5. associate-+r+ 19.6%

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

      6. metadata-eval 19.6%

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

      7. *-commutative 19.6%

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

   8. Simplified 19.6%

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

   10. Applied egg-rr 54.2%

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

       1. frac-add 57.4%

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

       2. *-commutative 57.4%

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

       3. associate-*l/ 20.3%

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

       4. fma-def 9.1%

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

       5. add-exp-log 2.1%

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

       6. log1p-expm1-u 2.1%

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

       7. expm1-def 2.1%

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

       8. add-exp-log 2.1%

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

       9. *-un-lft-identity 2.1%

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

       10. fma-neg 2.1%

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

       11. metadata-eval 2.1%

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

       12. fma-def 2.1%

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

       13. *-un-lft-identity 2.1%

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

       14. log1p-udef 2.1%

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

       15. add-exp-log 9.1%

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

       16. associate-+l+ 9.1%

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

       17. fma-def 20.3%

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

   12. Applied egg-rr 19.6%

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

       1. associate-/r* 57.5%

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

       2. div-inv 57.5%

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

       3. *-un-lft-identity 57.5%

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

       4. times-frac 57.5%

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

       5. metadata-eval 57.5%

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

   14. Applied egg-rr 57.5%

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

       1. associate-*r/ 57.5%

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

       2. *-rgt-identity 57.5%

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

       3. associate-*r/ 57.5%

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

       4. *-commutative 57.5%

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

       5. associate-*r/ 57.4%

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

       6. *-lft-identity 57.4%

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

       7. times-frac 20.3%

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

       8. /-rgt-identity 20.3%

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

   16. Simplified 20.3%

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

   if 5.0000000000000001e85 < x

   1. Initial program 94.9%

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

      1. sub-neg 94.9%

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

      2. distribute-neg-frac 94.9%

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

      3. metadata-eval 94.9%

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

      4. metadata-eval 94.9%

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

      5. metadata-eval 94.9%

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

      6. associate-/r* 94.9%

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

      7. metadata-eval 94.9%

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

      8. neg-mul-1 94.9%

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

      9. +-commutative 94.9%

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

      10. associate-+l+ 94.9%

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

      11. +-commutative 94.9%

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

      12. neg-mul-1 94.9%

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

      13. metadata-eval 94.9%

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

      14. associate-/r* 94.9%

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

      15. metadata-eval 94.9%

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

      16. metadata-eval 94.9%

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

      17. +-commutative 94.9%

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

      18. +-commutative 94.9%

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

      19. sub-neg 94.9%

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

      20. metadata-eval 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in x around inf 94.9%

      \[\leadsto \frac{-2}{x} + \color{blue}{\frac{2}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+85}:\\ \;\;\;\;\frac{-2}{x} + x \cdot \frac{\frac{2}{x + 1}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} + \frac{2}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.9%

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

   1. sub-neg 71.9%

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

   2. distribute-neg-frac 71.9%

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

   3. metadata-eval 71.9%

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

   4. metadata-eval 71.9%

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

   5. metadata-eval 71.9%

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

   6. associate-/r* 71.9%

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

   7. metadata-eval 71.9%

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

   8. neg-mul-1 71.9%

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

   9. +-commutative 71.9%

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

   10. associate-+l+ 71.8%

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

   11. +-commutative 71.8%

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

   12. neg-mul-1 71.8%

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

   13. metadata-eval 71.8%

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

   14. associate-/r* 71.8%

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

   15. metadata-eval 71.8%

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

   16. metadata-eval 71.8%

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

   17. +-commutative 71.8%

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

   18. +-commutative 71.8%

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

   19. sub-neg 71.8%

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

   20. metadata-eval 71.8%

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

3. Simplified 71.8%

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

   1. frac-add 19.3%

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

   2. div-inv 17.5%

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

   3. *-un-lft-identity 17.5%

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

   4. *-rgt-identity 17.5%

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

   5. +-commutative 17.5%

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

   6. +-commutative 17.5%

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

   7. metadata-eval 17.5%

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

   8. frac-times 16.7%

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

   9. clear-num 16.7%

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

   10. frac-times 17.5%

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

   11. metadata-eval 17.5%

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

   12. +-commutative 17.5%

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

   13. /-rgt-identity 17.5%

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

6. Applied egg-rr 17.5%

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

   1. associate-*r/ 19.3%

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

   2. *-rgt-identity 19.3%

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

   3. associate-+l+ 19.3%

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

   4. +-commutative 19.3%

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

   5. associate-+r+ 19.3%

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

   6. metadata-eval 19.3%

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

   7. *-commutative 19.3%

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

8. Simplified 19.3%

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

10. Applied egg-rr 68.3%

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

    1. frac-add 71.8%

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

    2. *-commutative 71.8%

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

    3. associate-*l/ 21.3%

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

    4. fma-def 8.0%

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

    5. add-exp-log 3.7%

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

    6. log1p-expm1-u 3.7%

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

    7. expm1-def 3.7%

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

    8. add-exp-log 3.7%

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

    9. *-un-lft-identity 3.7%

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

    10. fma-neg 3.7%

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

    11. metadata-eval 3.7%

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

    12. fma-def 3.7%

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

    13. *-un-lft-identity 3.7%

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

    14. log1p-udef 3.7%

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

    15. add-exp-log 8.0%

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

    16. associate-+l+ 8.0%

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

    17. fma-def 21.3%

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

12. Applied egg-rr 19.3%

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

    1. *-commutative 19.3%

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

    2. times-frac 71.8%

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

14. Applied egg-rr 71.8%

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

15. Final simplification 71.8%

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

Alternative 5: 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 71.9%

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

   1. sub-neg 71.9%

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

   2. distribute-neg-frac 71.9%

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

   3. metadata-eval 71.9%

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

   4. metadata-eval 71.9%

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

   5. metadata-eval 71.9%

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

   6. associate-/r* 71.9%

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

   7. metadata-eval 71.9%

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

   8. neg-mul-1 71.9%

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

   9. +-commutative 71.9%

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

   10. associate-+l+ 71.8%

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

   11. +-commutative 71.8%

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

   12. neg-mul-1 71.8%

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

   13. metadata-eval 71.8%

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

   14. associate-/r* 71.8%

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

   15. metadata-eval 71.8%

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

   16. metadata-eval 71.8%

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

   17. +-commutative 71.8%

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

   18. +-commutative 71.8%

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

   19. sub-neg 71.8%

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

   20. metadata-eval 71.8%

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

3. Simplified 71.8%

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

5. Taylor expanded in x around inf 69.5%

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

6. Final simplification 69.5%

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

Alternative 6: 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 71.9%

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

   1. sub-neg 71.9%

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

   2. distribute-neg-frac 71.9%

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

   3. metadata-eval 71.9%

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

   4. metadata-eval 71.9%

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

   5. metadata-eval 71.9%

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

   6. associate-/r* 71.9%

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

   7. metadata-eval 71.9%

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

   8. neg-mul-1 71.9%

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

   9. +-commutative 71.9%

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

   10. associate-+l+ 71.8%

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

   11. +-commutative 71.8%

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

   12. neg-mul-1 71.8%

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

   13. metadata-eval 71.8%

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

   14. associate-/r* 71.8%

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

   15. metadata-eval 71.8%

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

   16. metadata-eval 71.8%

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

   17. +-commutative 71.8%

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

   18. +-commutative 71.8%

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

   19. sub-neg 71.8%

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

   20. metadata-eval 71.8%

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

3. Simplified 71.8%

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

5. Taylor expanded in x around 0 5.0%

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

6. Final simplification 5.0%

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

Developer target: 99.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024039 
(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))))