3frac (problem 3.3.3)

Percentage Accurate: 69.4% → 98.9%

Time: 10.4s

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: 69.4% 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: 98.9% accurate, 1.6× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.4%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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 98.3

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

5. Simplified 98.3%

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

   1. associate-*r* N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f64 N/A

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

   5. lower-/.f64 99.1

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

7. Applied egg-rr 99.1%

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

Alternative 2: 98.9% 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 70.4%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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 98.3

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

5. Simplified 98.3%

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

   1. lift-*.f64 N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 99.1

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

7. Applied egg-rr 99.1%

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

Alternative 3: 98.2% 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 70.4%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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 98.3

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

5. Simplified 98.3%

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

Alternative 4: 5.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.4%

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

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

4. Applied egg-rr 6.6%

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

5. Taylor expanded in x around 0

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

7. Simplified 2.8%

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

8. Taylor expanded in x around inf

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

   1. lower-/.f64 2.8

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

10. Simplified 2.8%

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

11. Taylor expanded in x around 0

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

    1. lower-/.f64 5.1

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

13. Simplified 5.1%

    \[\leadsto \color{blue}{\frac{-1}{x}} \]
14. Add Preprocessing

Alternative 5: 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 70.4%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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. Simplified 5.1%

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

Alternative 6: 2.9% accurate, 11.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x + -1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.4%

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

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

4. Applied egg-rr 6.6%

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

5. Taylor expanded in x around 0

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

7. Simplified 2.8%

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

8. Taylor expanded in x around 0

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

   1. lower-/.f64 2.7

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

10. Simplified 2.7%

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

11. Taylor expanded in x around inf

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

    1. sub-neg N/A

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

    2. distribute-rgt-in N/A

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

    3. *-lft-identity N/A

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

    4. distribute-lft-neg-out N/A

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

    5. lft-mult-inverse N/A

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

    6. metadata-eval N/A

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

    7. lower-+.f64 2.8

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

13. Simplified 2.8%

    \[\leadsto \color{blue}{x + -1} \]
14. 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 2024215 
(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))))