3frac (problem 3.3.3)

Percentage Accurate: 85.0% → 99.9%

Time: 5.5s

Alternatives: 5

Speedup: 1.4×

Specification

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: 85.0% 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: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

   1. sub-neg 87.0%

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

   2. distribute-neg-frac 87.0%

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

   3. metadata-eval 87.0%

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

   4. metadata-eval 87.0%

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

   5. metadata-eval 87.0%

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

   6. associate-/r* 87.0%

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

   7. metadata-eval 87.0%

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

   8. neg-mul-1 87.0%

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

   9. +-commutative 87.0%

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

   10. associate-+l+ 86.9%

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

   11. +-commutative 86.9%

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

   12. neg-mul-1 86.9%

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

   13. metadata-eval 86.9%

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

   14. associate-/r* 86.9%

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

   15. metadata-eval 86.9%

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

   16. metadata-eval 86.9%

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

   17. +-commutative 86.9%

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

   18. +-commutative 86.9%

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

3. Simplified 86.9%

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

   1. +-commutative 86.9%

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

   2. frac-add 57.4%

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

   3. frac-add 57.9%

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

   4. *-un-lft-identity 57.9%

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

   5. *-commutative 57.9%

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

   6. neg-mul-1 57.9%

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

   7. distribute-neg-in 57.9%

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

   8. metadata-eval 57.9%

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

5. Applied egg-rr 57.9%

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

6. Taylor expanded in x around 0 99.0%

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

   1. associate-/r* 99.9%

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

   2. div-inv 99.9%

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

8. Applied egg-rr 99.9%

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

   1. un-div-inv 99.9%

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

   2. associate-/r* 99.9%

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

   3. associate-/l/ 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

   1. sub-neg 87.0%

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

   2. distribute-neg-frac 87.0%

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

   3. metadata-eval 87.0%

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

   4. metadata-eval 87.0%

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

   5. metadata-eval 87.0%

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

   6. associate-/r* 87.0%

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

   7. metadata-eval 87.0%

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

   8. neg-mul-1 87.0%

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

   9. +-commutative 87.0%

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

   10. associate-+l+ 86.9%

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

   11. +-commutative 86.9%

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

   12. neg-mul-1 86.9%

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

   13. metadata-eval 86.9%

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

   14. associate-/r* 86.9%

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

   15. metadata-eval 86.9%

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

   16. metadata-eval 86.9%

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

   17. +-commutative 86.9%

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

   18. +-commutative 86.9%

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

3. Simplified 86.9%

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

   1. +-commutative 86.9%

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

   2. frac-add 57.4%

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

   3. frac-add 57.9%

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

   4. *-un-lft-identity 57.9%

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

   5. *-commutative 57.9%

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

   6. neg-mul-1 57.9%

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

   7. distribute-neg-in 57.9%

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

   8. metadata-eval 57.9%

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

5. Applied egg-rr 57.9%

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

6. Taylor expanded in x around 0 99.0%

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

7. Final simplification 99.0%

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

Alternative 3: 68.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -2.0 / x;
	} 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 <= 1.0d0) then
        tmp = (-2.0d0) / x
    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 <= 1.0) {
		tmp = -2.0 / x;
	} else {
		tmp = (2.0 / x) + (-2.0 / x);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-2.0 / x);
	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 <= 1.0)
		tmp = -2.0 / x;
	else
		tmp = (2.0 / x) + (-2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 90.7%

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

      1. sub-neg 90.7%

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

      2. distribute-neg-frac 90.7%

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

      3. metadata-eval 90.7%

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

      4. metadata-eval 90.7%

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

      5. metadata-eval 90.7%

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

      6. associate-/r* 90.7%

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

      7. metadata-eval 90.7%

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

      8. neg-mul-1 90.7%

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

      9. +-commutative 90.7%

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

      10. associate-+l+ 90.7%

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

      11. +-commutative 90.7%

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

      12. neg-mul-1 90.7%

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

      13. metadata-eval 90.7%

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

      14. associate-/r* 90.7%

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

      15. metadata-eval 90.7%

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

      16. metadata-eval 90.7%

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

      17. +-commutative 90.7%

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

      18. +-commutative 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in x around 0 67.5%

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

   if 1 < x

   1. Initial program 76.9%

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

      1. sub-neg 76.9%

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

      2. distribute-neg-frac 76.9%

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

      3. metadata-eval 76.9%

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

      4. metadata-eval 76.9%

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

      5. metadata-eval 76.9%

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

      6. associate-/r* 76.9%

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

      7. metadata-eval 76.9%

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

      8. neg-mul-1 76.9%

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

      9. +-commutative 76.9%

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

      10. associate-+l+ 76.8%

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

      11. +-commutative 76.8%

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

      12. neg-mul-1 76.8%

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

      13. metadata-eval 76.8%

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

      14. associate-/r* 76.8%

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

      15. metadata-eval 76.8%

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

      16. metadata-eval 76.8%

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

      17. +-commutative 76.8%

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

      18. +-commutative 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in x around inf 76.2%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x} + \frac{-2}{x}\\ \end{array} \]

Alternative 4: 67.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (-2.0 * x) - (2.0 / x);
	} 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 <= 1.0d0) then
        tmp = ((-2.0d0) * x) - (2.0d0 / x)
    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 <= 1.0) {
		tmp = (-2.0 * x) - (2.0 / x);
	} else {
		tmp = (2.0 / x) + (-2.0 / x);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(-2.0 * x) - Float64(2.0 / x));
	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 <= 1.0)
		tmp = (-2.0 * x) - (2.0 / x);
	else
		tmp = (2.0 / x) + (-2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 90.7%

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

      1. sub-neg 90.7%

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

      2. distribute-neg-frac 90.7%

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

      3. metadata-eval 90.7%

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

      4. metadata-eval 90.7%

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

      5. metadata-eval 90.7%

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

      6. associate-/r* 90.7%

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

      7. metadata-eval 90.7%

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

      8. neg-mul-1 90.7%

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

      9. +-commutative 90.7%

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

      10. associate-+l+ 90.7%

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

      11. +-commutative 90.7%

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

      12. neg-mul-1 90.7%

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

      13. metadata-eval 90.7%

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

      14. associate-/r* 90.7%

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

      15. metadata-eval 90.7%

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

      16. metadata-eval 90.7%

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

      17. +-commutative 90.7%

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

      18. +-commutative 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in x around 0 66.5%

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

      1. associate-*r/ 66.5%

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

      2. metadata-eval 66.5%

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

   6. Simplified 66.5%

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

   if 1 < x

   1. Initial program 76.9%

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

      1. sub-neg 76.9%

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

      2. distribute-neg-frac 76.9%

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

      3. metadata-eval 76.9%

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

      4. metadata-eval 76.9%

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

      5. metadata-eval 76.9%

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

      6. associate-/r* 76.9%

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

      7. metadata-eval 76.9%

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

      8. neg-mul-1 76.9%

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

      9. +-commutative 76.9%

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

      10. associate-+l+ 76.8%

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

      11. +-commutative 76.8%

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

      12. neg-mul-1 76.8%

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

      13. metadata-eval 76.8%

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

      14. associate-/r* 76.8%

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

      15. metadata-eval 76.8%

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

      16. metadata-eval 76.8%

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

      17. +-commutative 76.8%

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

      18. +-commutative 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in x around inf 76.2%

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

4. Final simplification 69.1%

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

Alternative 5: 51.9% 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 87.0%

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

   1. sub-neg 87.0%

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

   2. distribute-neg-frac 87.0%

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

   3. metadata-eval 87.0%

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

   4. metadata-eval 87.0%

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

   5. metadata-eval 87.0%

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

   6. associate-/r* 87.0%

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

   7. metadata-eval 87.0%

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

   8. neg-mul-1 87.0%

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

   9. +-commutative 87.0%

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

   10. associate-+l+ 86.9%

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

   11. +-commutative 86.9%

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

   12. neg-mul-1 86.9%

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

   13. metadata-eval 86.9%

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

   14. associate-/r* 86.9%

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

   15. metadata-eval 86.9%

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

   16. metadata-eval 86.9%

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

   17. +-commutative 86.9%

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

   18. +-commutative 86.9%

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

3. Simplified 86.9%

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

4. Taylor expanded in x around 0 50.8%

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

5. Final simplification 50.8%

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

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 2023318 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

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

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