2frac (problem 3.3.1)

Percentage Accurate: 77.6% → 99.9%

Time: 4.7s

Alternatives: 4

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.3× speedup

Math

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

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 80.8%

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

   1. frac-sub 81.6%

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

   2. *-rgt-identity 81.6%

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

   3. metadata-eval 81.6%

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

   4. div-inv 81.6%

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

   5. associate-/r* 81.6%

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

   6. *-un-lft-identity 81.6%

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

   7. *-rgt-identity 81.6%

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

   8. +-commutative 81.6%

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

   9. div-inv 81.6%

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

   10. metadata-eval 81.6%

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

   11. *-rgt-identity 81.6%

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

   12. +-commutative 81.6%

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

4. Applied egg-rr 81.6%

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

   1. frac-2neg 81.6%

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

   2. div-inv 81.6%

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

   3. +-commutative 81.6%

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

   4. distribute-neg-in 81.6%

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

   5. metadata-eval 81.6%

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

6. Applied egg-rr 81.6%

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

   1. associate-*r/ 81.6%

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

   2. *-rgt-identity 81.6%

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

   3. associate--r+ 99.9%

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

   4. +-inverses 99.9%

      \[\leadsto \frac{\frac{-\left(\color{blue}{0} - 1\right)}{-1 + \left(-x\right)}}{x} \]

   5. metadata-eval 99.9%

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

   6. metadata-eval 99.9%

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

   7. unsub-neg 99.9%

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

8. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. associate-/l/ 99.1%

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

   3. associate-/r* 99.9%

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

10. Applied egg-rr 99.9%

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

    1. *-lft-identity 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 99.4% accurate, 1.3× speedup

Math

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

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(-1.0 / Float64(x * Float64(1.0 + x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

   1. sub-neg 80.8%

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

   2. +-commutative 80.8%

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

   3. distribute-neg-frac 80.8%

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

   4. metadata-eval 80.8%

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

4. Applied egg-rr 80.8%

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

   1. *-rgt-identity 80.8%

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

   2. fma-undefine 80.8%

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

   3. *-inverses 80.8%

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

   4. metadata-eval 80.8%

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

   5. distribute-neg-frac 80.8%

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

   6. fma-neg 80.8%

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

   7. *-inverses 80.8%

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

   8. *-rgt-identity 80.8%

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

   9. *-inverses 80.8%

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

   10. associate-/r* 56.1%

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

   11. *-commutative 56.1%

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

   12. associate-/r* 80.8%

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

   13. div-sub 80.8%

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

   14. *-inverses 80.8%

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

   15. div-sub 81.6%

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

   16. associate-/l/ 81.6%

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

   17. +-commutative 81.6%

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

   18. associate--r+ 99.1%

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

   19. div-sub 99.1%

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

   20. +-inverses 99.1%

       \[\leadsto \frac{\color{blue}{0}}{x \cdot \left(1 + x\right)} - \frac{1}{x \cdot \left(1 + x\right)} \]

   21. div0 99.1%

       \[\leadsto \color{blue}{0} - \frac{1}{x \cdot \left(1 + x\right)} \]

   22. associate-/r* 99.9%

       \[\leadsto 0 - \color{blue}{\frac{\frac{1}{x}}{1 + x}} \]

6. Simplified 99.1%

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

7. Final simplification 99.1%

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

Alternative 3: 52.1% accurate, 3.0× 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 80.8%

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

3. Taylor expanded in x around 0 53.6%

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

4. Final simplification 53.6%

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

Alternative 4: 3.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 80.8%

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

3. Taylor expanded in x around 0 52.4%

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

4. Taylor expanded in x around inf 3.0%

   \[\leadsto \color{blue}{1} \]

5. Final simplification 3.0%

   \[\leadsto 1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))