3frac (problem 3.3.3)

Percentage Accurate: 69.2% → 99.7%

Time: 9.3s

Alternatives: 7

Speedup: 1.6×

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: 69.2% 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: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 2}{{x}^{3}}\right)} \]

   2. div-sub N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} - \frac{2}{{x}^{3}}\right)}\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} + \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)\right)}\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)\right)\right)} \]

   5. unsub-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)} \]

   6. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}}{{x}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   7. distribute-frac-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right)}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}} - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)} \]

5. Applied rewrites 99.3%

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

7. Applied rewrites 99.7%

   \[\leadsto \frac{\frac{2}{x \cdot x} - -2}{{x}^{5}} - {x}^{-3} \cdot \color{blue}{-2} \]
8. Add Preprocessing

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{-3}, 2, 2 \cdot {x}^{-5}\right) \end{array} \]

FPCore

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

C

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

Julia

function code(x)
	return fma((x ^ -3.0), 2.0, Float64(2.0 * (x ^ -5.0)))
end

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 2}{{x}^{3}}\right)} \]

   2. div-sub N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} - \frac{2}{{x}^{3}}\right)}\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} + \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)\right)}\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)\right)\right)} \]

   5. unsub-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)} \]

   6. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}}{{x}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   7. distribute-frac-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right)}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}} - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)} \]

5. Applied rewrites 99.3%

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 99.6%

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

12. Applied rewrites 99.6%

    \[\leadsto \mathsf{fma}\left({x}^{-3}, \color{blue}{2}, {x}^{-5} \cdot 2\right) \]

13. Final simplification 99.6%

    \[\leadsto \mathsf{fma}\left({x}^{-3}, 2, 2 \cdot {x}^{-5}\right) \]
14. Add Preprocessing

Alternative 3: 99.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. lift-/.f64 N/A

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

   7. frac-add N/A

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

   8. clear-num N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

4. Applied rewrites 20.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 4: 99.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-sub N/A

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

   8. associate-/r* N/A

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

   9. frac-add N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 71.5%

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

5. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. neg-sub0 N/A

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

   4. associate-+l- N/A

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

   5. neg-sub0 N/A

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

   6. distribute-neg-frac N/A

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

   7. distribute-neg-frac2 N/A

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

   8. mul-1-neg N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

7. Applied rewrites 98.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 98.6

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

9. Applied rewrites 98.6%

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

10. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

    5. lower-fma.f64 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f64 99.5

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

12. Applied rewrites 99.5%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, x \cdot x, -0.5\right) \cdot x}} \]
13. Add Preprocessing

Alternative 5: 67.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-sub N/A

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

   8. associate-/r* N/A

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

   9. frac-add N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 71.5%

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

5. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. associate-/r/ N/A

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

   10. unpow2 N/A

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

   11. associate-/l* N/A

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

   12. *-inverses N/A

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

   13. *-rgt-identity N/A

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

   14. lower--.f64 N/A

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

   15. associate-*r/ N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f64 70.9

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

7. Applied rewrites 70.9%

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

8. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 71.1

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

10. Applied rewrites 71.1%

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

11. Taylor expanded in x around inf

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

    1. mul-1-neg N/A

       \[\leadsto \frac{x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}{x \cdot x} \]

    2. lower-neg.f64 70.1

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

13. Applied rewrites 70.1%

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

14. Final simplification 70.1%

    \[\leadsto \frac{\left(-x\right) + x}{x \cdot x} \]
15. Add Preprocessing

Alternative 6: 53.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-sub N/A

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

   8. associate-/r* N/A

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

   9. frac-add N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 71.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 54.0%

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

8. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 54.0

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

10. Applied rewrites 54.0%

    \[\leadsto \frac{2}{\color{blue}{x \cdot x}} \]
11. Add Preprocessing

Alternative 7: 5.1% accurate, 3.8× 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 71.6%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 5.0

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

5. Applied rewrites 5.0%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]
6. Add Preprocessing

Developer Target 1: 99.2% accurate, 1.8× 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 2024272 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (! :herbie-platform default (/ 2 (* x (- (* x x) 1))))

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