2frac (problem 3.3.1)

Percentage Accurate: 77.7% → 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: 77.7% 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 81.0%

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

   1. frac-sub 81.4%

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

   2. *-rgt-identity 81.4%

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

   3. metadata-eval 81.4%

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

   4. div-inv 81.4%

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

   5. associate-/r* 81.4%

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

   6. *-un-lft-identity 81.4%

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

   7. *-rgt-identity 81.4%

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

   8. +-commutative 81.4%

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

   9. div-inv 81.4%

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

   10. metadata-eval 81.4%

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

   11. *-rgt-identity 81.4%

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

   12. +-commutative 81.4%

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

4. Applied egg-rr 81.4%

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

   1. div-inv 81.4%

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

   2. +-commutative 81.4%

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

   3. +-commutative 81.4%

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

6. Applied egg-rr 81.4%

   \[\leadsto \frac{\color{blue}{\left(x - \left(x + 1\right)\right) \cdot \frac{1}{x + 1}}}{x} \]
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: 99.5% 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 81.0%

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

   1. sub-neg 81.0%

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

   2. +-commutative 81.0%

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

   3. distribute-neg-frac 81.0%

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

   4. metadata-eval 81.0%

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

4. Applied egg-rr 81.0%

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

   1. *-rgt-identity 81.0%

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

   2. *-lft-identity 81.0%

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

   3. fma-undefine 81.0%

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

   4. *-rgt-identity 81.0%

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

   5. metadata-eval 81.0%

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

   6. distribute-neg-frac 81.0%

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

   7. *-inverses 81.0%

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

   8. fma-neg 81.0%

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

   9. associate-*l/ 81.0%

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

   10. associate-/l* 81.0%

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

   11. *-rgt-identity 81.0%

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

   12. *-lft-identity 81.0%

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

   13. associate-/l* 81.0%

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

   14. associate-*l/ 81.1%

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

   15. *-rgt-identity 81.1%

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

   16. distribute-lft-out-- 81.1%

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

   17. *-inverses 81.1%

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

   18. div-sub 81.4%

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

   19. associate-*l/ 81.4%

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

   20. associate-*r/ 81.4%

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

   21. *-lft-identity 81.4%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 51.8% 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 81.0%

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

3. Taylor expanded in x around 0 53.3%

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

4. Final simplification 53.3%

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

Alternative 4: 3.2% 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 81.0%

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

3. Taylor expanded in x around 0 52.5%

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

   1. neg-mul-1 52.5%

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

   2. unsub-neg 52.5%

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

5. Simplified 52.5%

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

6. Taylor expanded in x around inf 3.3%

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

   1. neg-mul-1 3.3%

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

8. Simplified 3.3%

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

9. Final simplification 3.3%

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

Reproduce

herbie shell --seed 2024046 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))