2frac (problem 3.3.1)

Percentage Accurate: 78.1% → 99.9%

Time: 4.7s

Alternatives: 5

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: 78.1% 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 / Float64(x + 1.0)) / x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.2%

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

   1. frac-sub 76.4%

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

   2. *-rgt-identity 76.4%

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

   3. metadata-eval 76.4%

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

   4. div-inv 76.4%

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

   5. associate-/r* 76.4%

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

   6. *-un-lft-identity 76.4%

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

   7. *-rgt-identity 76.4%

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

   8. +-commutative 76.4%

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

   9. div-inv 76.4%

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

   10. metadata-eval 76.4%

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

   11. *-rgt-identity 76.4%

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

   12. +-commutative 76.4%

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

4. Applied egg-rr 76.4%

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

   1. div-inv 76.4%

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

   2. +-commutative 76.4%

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

   3. +-commutative 76.4%

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

6. Applied egg-rr 76.4%

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

   1. associate--r+ 99.9%

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

   2. +-inverses 99.9%

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

   3. metadata-eval 99.9%

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

   4. associate-*r/ 99.9%

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

   5. metadata-eval 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.2%

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

   1. sub-neg 75.2%

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

   2. +-commutative 75.2%

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

   3. distribute-neg-frac 75.2%

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

   4. metadata-eval 75.2%

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

4. Applied egg-rr 75.2%

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

6. Simplified 99.9%

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

7. Final simplification 99.9%

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

Alternative 3: 51.7% 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 75.2%

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

3. Taylor expanded in x around 0 50.2%

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

4. Final simplification 50.2%

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

Alternative 4: 3.1% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_] := (-x)

Derivation

1. Initial program 75.2%

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

3. Taylor expanded in x around 0 49.5%

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

   1. neg-mul-1 49.5%

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

   2. sub-neg 49.5%

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

5. Simplified 49.5%

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

6. Taylor expanded in x around inf 3.1%

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

   1. neg-mul-1 3.1%

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

8. Simplified 3.1%

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

9. Final simplification 3.1%

   \[\leadsto -x \]
10. Add Preprocessing

Alternative 5: 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 75.2%

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

   1. frac-sub 76.4%

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

   2. *-rgt-identity 76.4%

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

   3. metadata-eval 76.4%

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

   4. div-inv 76.4%

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

   5. associate-/r* 76.4%

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

   6. *-un-lft-identity 76.4%

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

   7. *-rgt-identity 76.4%

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

   8. +-commutative 76.4%

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

   9. div-inv 76.4%

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

   10. metadata-eval 76.4%

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

   11. *-rgt-identity 76.4%

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

   12. +-commutative 76.4%

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

4. Applied egg-rr 76.4%

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

5. Taylor expanded in x around 0 49.2%

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

6. Taylor expanded in x around inf 3.1%

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

7. Final simplification 3.1%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))