3frac (problem 3.3.3)

Percentage Accurate: 69.2% → 99.5%

Time: 9.5s

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: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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 98.7%

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

7. Applied rewrites 100.0%

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

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 100.0%

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

12. Applied rewrites 100.0%

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

Alternative 2: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 72.0%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-in N/A

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

   6. lift-*.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. sub-neg N/A

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

   9. lower--.f64 99.8

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

9. Applied rewrites 99.8%

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

Alternative 3: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 72.0%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 72.0%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 98.7

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

7. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 98.7

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

9. Applied rewrites 98.7%

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

10. Taylor expanded in x around inf

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

    1. lower-/.f64 N/A

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

    2. associate-*r/ N/A

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

    3. metadata-eval N/A

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

    4. +-commutative N/A

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

    5. metadata-eval N/A

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

    6. sub-neg N/A

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

    7. lower--.f64 N/A

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

    8. unpow2 N/A

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

    9. associate-/r* N/A

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

    10. metadata-eval N/A

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

    11. associate-*r/ N/A

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

    12. lower-/.f64 N/A

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

    13. associate-*r/ N/A

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

    14. metadata-eval N/A

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

    15. lower-/.f64 N/A

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

    16. unpow2 N/A

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

    17. lower-*.f64 99.8

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

12. Applied rewrites 99.8%

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

Alternative 5: 98.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 72.0%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 6: 52.6% 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 72.0%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 72.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 53.5

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

7. Applied rewrites 53.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 55.1%

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

11. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 55.1

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

13. Applied rewrites 55.1%

    \[\leadsto \frac{2}{\color{blue}{x \cdot x}} \]
14. 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 72.0%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 5.1

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

5. Applied rewrites 5.1%

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

Developer Target 1: 99.1% 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 2024332 
(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))))