2frac (problem 3.3.1)

Percentage Accurate: 77.8% → 99.9%

Time: 3.6s

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: 77.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. frac-sub 80.0%

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

   2. *-rgt-identity 80.0%

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

   3. metadata-eval 80.0%

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

   4. div-inv 80.0%

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

   5. associate-/r* 80.0%

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

   6. *-un-lft-identity 80.0%

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

   7. *-rgt-identity 80.0%

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

   8. +-commutative 80.0%

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

   9. div-inv 80.0%

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

   10. metadata-eval 80.0%

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

   11. *-rgt-identity 80.0%

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

   12. +-commutative 80.0%

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

3. Applied egg-rr 80.0%

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

   1. frac-2neg 80.0%

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

   2. div-inv 80.0%

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

   3. +-commutative 80.0%

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

   4. associate--r+ 99.9%

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

   5. distribute-neg-in 99.9%

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

   6. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

   1. distribute-lft-neg-out 99.9%

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

   2. +-inverses 99.9%

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

   3. metadata-eval 99.9%

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

   4. distribute-lft-neg-in 99.9%

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

   5. metadata-eval 99.9%

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

   6. *-lft-identity 99.9%

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

   7. unsub-neg 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 99.6% 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 78.3%

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

   1. frac-sub 80.0%

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

   2. *-rgt-identity 80.0%

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

   3. metadata-eval 80.0%

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

   4. div-inv 80.0%

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

   5. associate-/r* 80.0%

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

   6. *-un-lft-identity 80.0%

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

   7. *-rgt-identity 80.0%

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

   8. +-commutative 80.0%

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

   9. div-inv 80.0%

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

   10. metadata-eval 80.0%

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

   11. *-rgt-identity 80.0%

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

   12. +-commutative 80.0%

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

3. Applied egg-rr 80.0%

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

   1. frac-2neg 80.0%

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

   2. div-inv 80.0%

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

   3. +-commutative 80.0%

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

   4. associate--r+ 99.9%

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

   5. distribute-neg-in 99.9%

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

   6. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

   1. distribute-lft-neg-out 99.9%

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

   2. +-inverses 99.9%

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

   3. metadata-eval 99.9%

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

   4. distribute-lft-neg-in 99.9%

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

   5. metadata-eval 99.9%

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

   6. *-lft-identity 99.9%

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

   7. unsub-neg 99.9%

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

7. Simplified 99.9%

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

   1. add-log-exp 29.4%

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

   2. *-un-lft-identity 29.4%

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

   3. log-prod 29.4%

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

   4. metadata-eval 29.4%

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

   5. associate-/l/ 29.4%

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

   6. add-log-exp 99.4%

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

   7. associate-/r* 99.9%

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

9. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{0 + \frac{\frac{1}{x}}{-1 - x}} \]
10. Step-by-step derivation

    1. +-lft-identity 99.9%

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

    2. associate-/r* 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 3: 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 78.3%

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

   1. sub-neg 78.3%

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

   2. +-commutative 78.3%

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

   3. distribute-neg-frac 78.3%

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

   4. metadata-eval 78.3%

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

3. Applied egg-rr 78.3%

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 4: 51.6% 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 78.3%

   \[\frac{1}{x + 1} - \frac{1}{x} \]

2. Taylor expanded in x around 0 52.6%

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

3. Final simplification 52.6%

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

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 78.3%

   \[\frac{1}{x + 1} - \frac{1}{x} \]

2. Taylor expanded in x around 0 51.9%

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

3. Taylor expanded in x around inf 3.1%

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

4. Final simplification 3.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023333 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))