3frac (problem 3.3.3)

Percentage Accurate: 68.9% → 99.7%

Time: 8.7s

Alternatives: 4

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: 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: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \left(-1 - {x}^{-4}\right) \cdot \left({\left(-x\right)}^{-3} \cdot \left(\frac{2}{x \cdot x} + 2\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (- -1.0 (pow x -4.0)) (* (pow (- x) -3.0) (+ (/ 2.0 (* x x)) 2.0))))

C

double code(double x) {
	return (-1.0 - pow(x, -4.0)) * (pow(-x, -3.0) * ((2.0 / (x * x)) + 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-1.0d0) - (x ** (-4.0d0))) * ((-x ** (-3.0d0)) * ((2.0d0 / (x * x)) + 2.0d0))
end function

Java

public static double code(double x) {
	return (-1.0 - Math.pow(x, -4.0)) * (Math.pow(-x, -3.0) * ((2.0 / (x * x)) + 2.0));
}

Python

def code(x):
	return (-1.0 - math.pow(x, -4.0)) * (math.pow(-x, -3.0) * ((2.0 / (x * x)) + 2.0))

Julia

function code(x)
	return Float64(Float64(-1.0 - (x ^ -4.0)) * Float64((Float64(-x) ^ -3.0) * Float64(Float64(2.0 / Float64(x * x)) + 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = (-1.0 - (x ^ -4.0)) * ((-x ^ -3.0) * ((2.0 / (x * x)) + 2.0));
end

Wolfram

code[x_] := N[(N[(-1.0 - N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[(-x), -3.0], $MachinePrecision] * N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

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

3. Taylor expanded in x around -inf

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

5. Applied rewrites 98.7%

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

11. Applied rewrites 99.5%

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

12. Final simplification 99.5%

    \[\leadsto \left(-1 - {x}^{-4}\right) \cdot \left({\left(-x\right)}^{-3} \cdot \left(\frac{2}{x \cdot x} + 2\right)\right) \]
13. Add Preprocessing

Alternative 2: 99.2% accurate, 1.8× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 68.5%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. lift-/.f64 N/A

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

   7. frac-add N/A

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

   8. lower-/.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. *-lft-identity N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 16.7

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

4. Applied rewrites 16.7%

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

6. Applied rewrites 18.2%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.2%

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

10. Final simplification 99.2%

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

Alternative 3: 52.8% 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 68.5%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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 68.6%

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

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 53.0

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

7. Applied rewrites 53.0%

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

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 54.5%

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

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 54.5

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

13. Applied rewrites 54.5%

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

Alternative 4: 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 68.5%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 5.2

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

5. Applied rewrites 5.2%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]
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 2024296 
(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))))