3frac (problem 3.3.3)

Percentage Accurate: 84.6% → 99.9%

Time: 7.2s

Alternatives: 5

Speedup: 1.7×

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: 84.6% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

   1. associate-+l- 87.4%

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

   2. sub-neg 87.4%

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

   3. neg-mul-1 87.4%

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

   4. metadata-eval 87.4%

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

   5. cancel-sign-sub-inv 87.4%

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

   6. +-commutative 87.4%

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

   7. *-lft-identity 87.4%

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

   8. sub-neg 87.4%

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

   9. metadata-eval 87.4%

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

3. Simplified 87.4%

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

   1. frac-sub 63.7%

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

   2. associate-/r* 87.4%

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

   3. *-rgt-identity 87.4%

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

   4. distribute-rgt-in 87.4%

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

   5. metadata-eval 87.4%

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

   6. metadata-eval 87.4%

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

   7. fma-def 87.4%

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

   8. metadata-eval 87.4%

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

5. Applied egg-rr 87.4%

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

   1. frac-sub 87.5%

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

   2. *-un-lft-identity 87.5%

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

7. Applied egg-rr 87.5%

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

8. Taylor expanded in x around 0 99.9%

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

   1. expm1-log1p-u 70.2%

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

   2. expm1-udef 57.7%

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

   3. +-commutative 57.7%

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

10. Applied egg-rr 57.7%

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

    1. expm1-def 70.2%

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

    2. expm1-log1p 99.9%

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

    3. associate-/r* 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{2}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (/ (/ 2.0 x) (* x x))
   (- (* x -2.0) (/ 2.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (2.0 / x) / (x * x);
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (2.0d0 / x) / (x * x)
    else
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (2.0 / x) / (x * x);
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = (2.0 / x) / (x * x)
	else:
		tmp = (x * -2.0) - (2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(2.0 / x) / Float64(x * x));
	else
		tmp = Float64(Float64(x * -2.0) - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(2.0 / x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.2%

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

      1. associate-+l- 73.2%

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

      2. sub-neg 73.2%

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

      3. neg-mul-1 73.2%

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

      4. metadata-eval 73.2%

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

      5. cancel-sign-sub-inv 73.2%

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

      6. +-commutative 73.2%

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

      7. *-lft-identity 73.2%

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

      8. sub-neg 73.2%

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

      9. metadata-eval 73.2%

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

   3. Simplified 73.2%

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

      1. frac-sub 22.6%

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

      2. associate-/r* 73.2%

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

      3. *-rgt-identity 73.2%

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

      4. distribute-rgt-in 73.2%

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

      5. metadata-eval 73.2%

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

      6. metadata-eval 73.2%

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

      7. fma-def 73.2%

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

      8. metadata-eval 73.2%

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

   5. Applied egg-rr 73.2%

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

      1. frac-sub 73.3%

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

      2. *-un-lft-identity 73.3%

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

   7. Applied egg-rr 73.3%

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

   8. Taylor expanded in x around 0 99.8%

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

   9. Taylor expanded in x around inf 99.3%

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

       1. unpow2 99.3%

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

   11. Simplified 99.3%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. metadata-eval 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. *-lft-identity 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{2}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

   1. associate-+l- 87.4%

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

   2. sub-neg 87.4%

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

   3. neg-mul-1 87.4%

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

   4. metadata-eval 87.4%

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

   5. cancel-sign-sub-inv 87.4%

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

   6. +-commutative 87.4%

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

   7. *-lft-identity 87.4%

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

   8. sub-neg 87.4%

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

   9. metadata-eval 87.4%

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

3. Simplified 87.4%

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

   1. frac-sub 63.7%

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

   2. associate-/r* 87.4%

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

   3. *-rgt-identity 87.4%

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

   4. distribute-rgt-in 87.4%

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

   5. metadata-eval 87.4%

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

   6. metadata-eval 87.4%

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

   7. fma-def 87.4%

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

   8. metadata-eval 87.4%

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

5. Applied egg-rr 87.4%

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

   1. frac-sub 87.5%

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

   2. *-un-lft-identity 87.5%

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

7. Applied egg-rr 87.5%

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

8. Taylor expanded in x around 0 99.9%

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

9. Taylor expanded in x around 0 99.9%

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

    1. sub-neg 99.9%

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

    2. unpow2 99.9%

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

    3. metadata-eval 99.9%

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

11. Simplified 99.9%

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

12. Final simplification 99.9%

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

Alternative 4: 83.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

   1. associate-+l- 87.4%

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

   2. sub-neg 87.4%

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

   3. neg-mul-1 87.4%

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

   4. metadata-eval 87.4%

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

   5. cancel-sign-sub-inv 87.4%

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

   6. +-commutative 87.4%

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

   7. *-lft-identity 87.4%

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

   8. sub-neg 87.4%

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

   9. metadata-eval 87.4%

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

3. Simplified 87.4%

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

4. Taylor expanded in x around 0 54.0%

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

5. Taylor expanded in x around 0 86.4%

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

6. Final simplification 86.4%

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

Alternative 5: 51.2% 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 87.4%

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

   1. associate-+l- 87.4%

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

   2. sub-neg 87.4%

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

   3. neg-mul-1 87.4%

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

   4. metadata-eval 87.4%

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

   5. cancel-sign-sub-inv 87.4%

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

   6. +-commutative 87.4%

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

   7. *-lft-identity 87.4%

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

   8. sub-neg 87.4%

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

   9. metadata-eval 87.4%

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

3. Simplified 87.4%

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

4. Taylor expanded in x around 0 54.9%

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

5. Final simplification 54.9%

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

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 2023207 
(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))))