3frac (problem 3.3.3)

Percentage Accurate: 84.4% → 99.9%

Time: 6.2s

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: 84.4% 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: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.6%

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

   1. associate-+l- 81.6%

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

   2. sub-neg 81.6%

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

   3. neg-mul-1 81.6%

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

   4. metadata-eval 81.6%

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

   5. cancel-sign-sub-inv 81.6%

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

   6. +-commutative 81.6%

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

   7. *-lft-identity 81.6%

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

   8. sub-neg 81.6%

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

   9. metadata-eval 81.6%

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

3. Simplified 81.6%

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

   1. frac-2neg 81.6%

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

   2. metadata-eval 81.6%

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

   3. frac-sub 57.5%

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

   4. frac-sub 59.5%

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

5. Applied egg-rr 59.2%

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

7. Simplified 59.5%

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

8. Taylor expanded in x around 0 99.7%

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

   1. expm1-log1p-u 73.1%

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

   2. expm1-udef 54.8%

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

10. Applied egg-rr 54.8%

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

    1. expm1-def 73.1%

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

    2. expm1-log1p 99.7%

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

    3. associate-/r* 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.6%

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

   1. associate-+l- 81.6%

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

   2. sub-neg 81.6%

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

   3. neg-mul-1 81.6%

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

   4. metadata-eval 81.6%

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

   5. cancel-sign-sub-inv 81.6%

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

   6. +-commutative 81.6%

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

   7. *-lft-identity 81.6%

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

   8. sub-neg 81.6%

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

   9. metadata-eval 81.6%

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

3. Simplified 81.6%

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

   1. frac-2neg 81.6%

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

   2. metadata-eval 81.6%

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

   3. frac-sub 57.5%

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

   4. frac-sub 59.5%

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

5. Applied egg-rr 59.2%

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

7. Simplified 59.5%

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

8. Taylor expanded in x around 0 99.7%

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

   1. pow1 99.7%

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

10. Applied egg-rr 99.7%

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

    1. unpow1 99.7%

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

    2. fma-udef 99.7%

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

    3. *-commutative 99.7%

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

    4. +-commutative 99.7%

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

    5. neg-mul-1 99.7%

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

    6. unsub-neg 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 3: 82.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.6%

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

   1. associate-+l- 81.6%

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

   2. sub-neg 81.6%

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

   3. neg-mul-1 81.6%

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

   4. metadata-eval 81.6%

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

   5. cancel-sign-sub-inv 81.6%

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

   6. +-commutative 81.6%

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

   7. *-lft-identity 81.6%

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

   8. sub-neg 81.6%

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

   9. metadata-eval 81.6%

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

3. Simplified 81.6%

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

4. Taylor expanded in x around 0 47.7%

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

5. Taylor expanded in x around 0 79.2%

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

6. Final simplification 79.2%

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

Alternative 4: 51.4% 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 81.6%

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

   1. associate-+l- 81.6%

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

   2. sub-neg 81.6%

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

   3. neg-mul-1 81.6%

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

   4. metadata-eval 81.6%

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

   5. cancel-sign-sub-inv 81.6%

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

   6. +-commutative 81.6%

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

   7. *-lft-identity 81.6%

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

   8. sub-neg 81.6%

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

   9. metadata-eval 81.6%

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

3. Simplified 81.6%

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

4. Taylor expanded in x around 0 48.3%

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

5. Final simplification 48.3%

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

Alternative 5: 3.3% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 81.6%

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

   1. associate-+l- 81.6%

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

   2. sub-neg 81.6%

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

   3. neg-mul-1 81.6%

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

   4. metadata-eval 81.6%

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

   5. cancel-sign-sub-inv 81.6%

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

   6. +-commutative 81.6%

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

   7. *-lft-identity 81.6%

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

   8. sub-neg 81.6%

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

   9. metadata-eval 81.6%

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

3. Simplified 81.6%

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

4. Taylor expanded in x around 0 47.7%

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

5. Taylor expanded in x around inf 3.4%

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

6. Final simplification 3.4%

   \[\leadsto -1 \]

Developer target: 99.6% 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 2023229 
(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))))