3frac (problem 3.3.3)

Percentage Accurate: 69.6% → 99.7%

Time: 9.3s

Alternatives: 7

Speedup: 1.8×

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: 69.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \left({x}^{-3} \cdot \left(\frac{1}{x \cdot x} - -1\right)\right) \cdot \mathsf{fma}\left(\frac{--1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 2, 2\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (pow x -3.0) (- (/ 1.0 (* x x)) -1.0))
  (fma (/ (- -1.0) (* (* x x) (* x x))) 2.0 2.0)))

C

double code(double x) {
	return (pow(x, -3.0) * ((1.0 / (x * x)) - -1.0)) * fma((-(-1.0) / ((x * x) * (x * x))), 2.0, 2.0);
}

Julia

function code(x)
	return Float64(Float64((x ^ -3.0) * Float64(Float64(1.0 / Float64(x * x)) - -1.0)) * fma(Float64(Float64(-(-1.0)) / Float64(Float64(x * x) * Float64(x * x))), 2.0, 2.0))
end

Wolfram

code[x_] := N[(N[(N[Power[x, -3.0], $MachinePrecision] * N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[((--1.0) / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.9%

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

3. Taylor expanded in x around -inf

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

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(2 + 2 \cdot \frac{1}{{x}^{2}}\right)\right)}{{x}^{3}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(2 + 2 \cdot \frac{1}{{x}^{2}}\right)\right)}{{x}^{3}}} \]

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\left(\frac{1}{x \cdot x} - -1\right) \cdot \left(\frac{2}{{x}^{4}} - -2\right)}{{x}^{3}}} \]

7. Applied rewrites 99.9%

   \[\leadsto \mathsf{fma}\left({x}^{-4}, 2, 2\right) \cdot \color{blue}{\left(\left({x}^{-2} - -1\right) \cdot {x}^{-3}\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left({x}^{-4}, 2, 2\right) \cdot \left(\left(1 + \frac{1}{{x}^{2}}\right) \cdot {\color{blue}{x}}^{-3}\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.9%

    \[\leadsto \mathsf{fma}\left({x}^{-4}, 2, 2\right) \cdot \left(\left(\frac{1}{x \cdot x} - -1\right) \cdot {\color{blue}{x}}^{-3}\right) \]
11. Step-by-step derivation

12. Applied rewrites 99.9%

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

13. Final simplification 99.9%

    \[\leadsto \left({x}^{-3} \cdot \left(\frac{1}{x \cdot x} - -1\right)\right) \cdot \mathsf{fma}\left(\frac{--1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 2, 2\right) \]
14. Add Preprocessing

Alternative 2: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-lft-identity N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 72.5

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

4. Applied rewrites 72.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-add N/A

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

   10. *-rgt-identity N/A

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

   11. lift-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-/r* N/A

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

   14. lower-/.f64 N/A

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

6. Applied rewrites 2.7%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.8%

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

    1. lift-/.f64 N/A

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

    2. lift--.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. associate-/l/ N/A

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

    5. associate-/r* N/A

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

    6. lower-/.f64 N/A

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

11. Applied rewrites 99.9%

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

Alternative 3: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-lft-identity N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 72.5

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

4. Applied rewrites 72.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-add N/A

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

   10. *-rgt-identity N/A

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

   11. lift-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-/r* N/A

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

   14. lower-/.f64 N/A

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

6. Applied rewrites 2.7%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.8%

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

Alternative 4: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-lft-identity N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 72.5

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

4. Applied rewrites 72.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-add N/A

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

   10. *-rgt-identity N/A

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

   11. lift-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-/r* N/A

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

   14. lower-/.f64 N/A

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

6. Applied rewrites 2.7%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.8%

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

    1. lift-/.f64 N/A

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

    2. lift--.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. associate-/l/ N/A

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

    5. lower-/.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. lift--.f64 99.4

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

11. Applied rewrites 99.4%

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

12. Final simplification 99.4%

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

Alternative 5: 68.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 72.3

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

5. Applied rewrites 72.3%

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

6. Taylor expanded in x around inf

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

   1. lower-/.f64 72.2

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

8. Applied rewrites 72.2%

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

9. Final simplification 72.2%

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

Alternative 6: 54.0% 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 72.9%

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

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-lft-identity N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 72.5

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

4. Applied rewrites 72.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-add N/A

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

   10. *-rgt-identity N/A

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

   11. lift-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-/r* N/A

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

   14. lower-/.f64 N/A

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

6. Applied rewrites 2.7%

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

7. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 56.9

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

9. Applied rewrites 56.9%

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

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 58.5%

    \[\leadsto \frac{\color{blue}{-2}}{\mathsf{fma}\left(x, x, x\right)} \]
13. 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 72.9%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 5.4

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

5. Applied rewrites 5.4%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]
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 2024288 
(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))))