2frac (problem 3.3.1)

Percentage Accurate: 77.1% → 99.9%

Time: 4.3s

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: 77.1% 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

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

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 77.8%

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

   1. sub-neg 77.8%

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

   2. +-commutative 77.8%

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

   3. distribute-neg-frac 77.8%

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

   4. metadata-eval 77.8%

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

4. Applied egg-rr 77.8%

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

   1. *-rgt-identity 77.8%

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

   2. *-inverses 77.8%

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

   3. associate-/r* 56.0%

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

   4. *-rgt-identity 56.0%

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

   5. metadata-eval 56.0%

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

   6. distribute-neg-frac 56.0%

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

   7. sub-neg 56.0%

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

   8. *-inverses 56.0%

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

   9. associate-/r* 77.9%

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

   10. *-commutative 77.9%

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

   11. div-sub 79.0%

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

   12. +-commutative 79.0%

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

   13. associate--r+ 99.2%

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

   14. +-inverses 99.2%

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

   15. metadata-eval 99.2%

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

   16. *-commutative 99.2%

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

   17. associate-/l/ 99.9%

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

   18. +-commutative 99.9%

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

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 2: 99.3% accurate, 1.3× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. sub-neg 77.8%

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

   2. +-commutative 77.8%

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

   3. distribute-neg-frac 77.8%

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

   4. metadata-eval 77.8%

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

4. Applied egg-rr 77.8%

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

   1. *-rgt-identity 77.8%

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

   2. *-inverses 77.8%

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

   3. associate-/r* 56.0%

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

   4. *-rgt-identity 56.0%

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

   5. metadata-eval 56.0%

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

   6. distribute-neg-frac 56.0%

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

   7. sub-neg 56.0%

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

   8. *-inverses 56.0%

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

   9. associate-/r* 77.9%

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

   10. *-commutative 77.9%

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

   11. div-sub 79.0%

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

   12. +-commutative 79.0%

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

   13. associate--r+ 99.2%

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

   14. +-inverses 99.2%

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

   15. metadata-eval 99.2%

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

   16. *-commutative 99.2%

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

   17. associate-/l/ 99.9%

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

   18. +-commutative 99.9%

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

6. Simplified 99.9%

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

   1. add0 99.9%

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

   2. associate-/l/ 99.2%

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

   3. +-commutative 99.2%

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

   4. *-commutative 99.2%

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

   5. distribute-lft-in 99.2%

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

   6. *-rgt-identity 99.2%

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

   7. pow2 99.2%

      \[\leadsto \frac{-1}{x + \color{blue}{{x}^{2}}} + 0 \]

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{-1}{x + {x}^{2}} + 0} \]
9. Step-by-step derivation

   1. add0 99.2%

      \[\leadsto \color{blue}{\frac{-1}{x + {x}^{2}}} \]

10. Simplified 99.2%

    \[\leadsto \color{blue}{\frac{-1}{x + {x}^{2}}} \]
11. Step-by-step derivation

    1. unpow2 99.2%

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

    2. distribute-rgt1-in 99.2%

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

12. Applied egg-rr 99.2%

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

13. Final simplification 99.2%

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

Alternative 3: 50.4% 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 77.8%

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

3. Taylor expanded in x around 0 53.0%

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

4. Final simplification 53.0%

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

Alternative 4: 2.9% 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 77.8%

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

3. Taylor expanded in x around 0 52.2%

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

4. Taylor expanded in x around inf 2.9%

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

5. Final simplification 2.9%

   \[\leadsto 1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))