3frac (problem 3.3.3)

Percentage Accurate: 68.9% → 99.8%

Time: 11.1s

Alternatives: 6

Speedup: 2.1×

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: 68.9% 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: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-sub N/A

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

   13. frac-add N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 19.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. *-commutative N/A

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

   12. associate-/r* N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. sub-neg N/A

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

   17. metadata-eval N/A

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

   18. lower-+.f64 99.8

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

9. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{x + -1}}{x + 1}}{x}} \]
10. Add Preprocessing

Alternative 2: 99.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-sub N/A

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

   13. frac-add N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 19.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. *-commutative N/A

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

   12. associate-/r* N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. sub-neg N/A

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

   17. metadata-eval N/A

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

   18. lower-+.f64 99.8

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

9. Applied rewrites 99.8%

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

    1. lift-+.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. associate-/l/ N/A

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

    4. lift-+.f64 N/A

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

    5. lift-+.f64 N/A

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

    6. metadata-eval N/A

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

    7. sub-neg N/A

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

    8. difference-of-sqr-1 N/A

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

    9. metadata-eval N/A

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

    10. lower-/.f64 N/A

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

    11. metadata-eval N/A

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

    12. difference-of-sqr-1 N/A

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

    13. difference-of-sqr--1 N/A

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

    14. lift-fma.f64 99.8

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

11. Applied rewrites 99.8%

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

Alternative 3: 99.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-sub N/A

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

   13. frac-add N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 19.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. sub-neg N/A

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

   14. metadata-eval N/A

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

   15. lift-+.f64 N/A

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

   16. metadata-eval N/A

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

   17. sub-neg N/A

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

   18. difference-of-sqr-1 N/A

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

   19. metadata-eval N/A

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

   20. lower-*.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. metadata-eval N/A

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

   23. sub-neg N/A

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

   24. lift-*.f64 N/A

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

   25. metadata-eval N/A

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

   26. lower-fma.f64 99.6

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

9. Applied rewrites 99.6%

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

Alternative 4: 98.0% accurate, 2.1× speedup

Math

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

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(2.0 / Float64(x * Float64(x * x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 98.9

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

5. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\frac{2}{x \cdot \left(x \cdot x\right)}} \]
6. Add Preprocessing

Alternative 5: 51.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-sub N/A

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

   13. frac-add N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 19.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.8

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

   8. lift-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. distribute-rgt-in N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 99.8

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

9. Applied rewrites 99.8%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 53.7%

    \[\leadsto \frac{\color{blue}{-2}}{\mathsf{fma}\left(x, x, x\right)} \]
13. Add Preprocessing

Alternative 6: 5.0% 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 69.7%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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 2024216 
(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))))