2frac (problem 3.3.1)

Percentage Accurate: 77.3% → 99.9%

Time: 4.8s

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: 77.3% 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 75.3%

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

   1. sub-neg 75.3%

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

   2. +-commutative 75.3%

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

   3. distribute-neg-frac 75.3%

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

   4. metadata-eval 75.3%

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

4. Applied egg-rr 75.3%

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

   1. metadata-eval 75.3%

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

   2. distribute-neg-frac 75.3%

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

   3. sub-neg 75.3%

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

   4. *-inverses 75.3%

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

   5. associate-/r* 53.9%

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

   6. *-commutative 53.9%

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

   7. associate-/r* 75.3%

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

   8. div-sub 75.3%

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

   9. *-inverses 75.3%

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

   10. div-sub 77.2%

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

   11. associate-/l/ 77.2%

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

   12. +-commutative 77.2%

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

   13. associate--r+ 99.5%

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

   14. div-sub 99.5%

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

   15. +-inverses 99.5%

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

   16. div0 99.5%

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

   17. associate-/r* 99.9%

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

   18. neg-sub0 99.9%

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

   19. associate-/r* 99.5%

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

   20. distribute-neg-frac 99.5%

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

   21. metadata-eval 99.5%

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

   22. distribute-rgt-in 99.6%

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

6. Simplified 99.6%

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

   1. +-commutative 99.6%

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

   2. unpow2 99.6%

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

   3. fma-define 99.6%

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

8. Applied egg-rr 99.6%

   \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]
9. Step-by-step derivation

   1. fma-undefine 99.6%

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

   2. *-rgt-identity 99.6%

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

   3. distribute-lft-in 99.5%

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

   4. associate-/l/ 99.9%

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

   5. div-inv 99.8%

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

10. Applied egg-rr 99.8%

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

    1. associate-*l/ 99.9%

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

    2. associate-*r/ 99.9%

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

    3. metadata-eval 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 3.8 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{1}{x} + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 3.8e+61)))
   (+ (/ 1.0 x) (/ -1.0 x))
   (+ (- 1.0 x) (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 3.8e+61)) {
		tmp = (1.0 / x) + (-1.0 / x);
	} else {
		tmp = (1.0 - x) + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 3.8d+61))) then
        tmp = (1.0d0 / x) + ((-1.0d0) / x)
    else
        tmp = (1.0d0 - x) + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 3.8e+61)) {
		tmp = (1.0 / x) + (-1.0 / x);
	} else {
		tmp = (1.0 - x) + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 3.8e+61):
		tmp = (1.0 / x) + (-1.0 / x)
	else:
		tmp = (1.0 - x) + (-1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 3.8e+61))
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 / x));
	else
		tmp = Float64(Float64(1.0 - x) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 3.8e+61)))
		tmp = (1.0 / x) + (-1.0 / x);
	else
		tmp = (1.0 - x) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 3.8e+61]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 3.79999999999999995e61 < x

   1. Initial program 57.1%

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

   3. Taylor expanded in x around inf 55.2%

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

   if -1 < x < 3.79999999999999995e61

   1. Initial program 89.7%

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

   3. Taylor expanded in x around 0 87.3%

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

      1. neg-mul-1 87.3%

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

      2. unsub-neg 87.3%

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

   5. Simplified 87.3%

      \[\leadsto \color{blue}{\left(1 - x\right)} - \frac{1}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 3.8 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{1}{x} + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% 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 75.3%

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

   1. sub-neg 75.3%

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

   2. +-commutative 75.3%

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

   3. distribute-neg-frac 75.3%

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

   4. metadata-eval 75.3%

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

4. Applied egg-rr 75.3%

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

   1. metadata-eval 75.3%

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

   2. distribute-neg-frac 75.3%

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

   3. sub-neg 75.3%

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

   4. *-inverses 75.3%

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

   5. associate-/r* 53.9%

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

   6. *-commutative 53.9%

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

   7. associate-/r* 75.3%

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

   8. div-sub 75.3%

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

   9. *-inverses 75.3%

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

   10. div-sub 77.2%

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

   11. associate-/l/ 77.2%

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

   12. +-commutative 77.2%

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

   13. associate--r+ 99.5%

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

   14. div-sub 99.5%

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

   15. +-inverses 99.5%

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

   16. div0 99.5%

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

   17. associate-/r* 99.9%

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

   18. neg-sub0 99.9%

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

   19. associate-/r* 99.5%

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

   20. distribute-neg-frac 99.5%

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

   21. metadata-eval 99.5%

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

   22. distribute-rgt-in 99.6%

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

6. Simplified 99.6%

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

   1. unpow2 99.6%

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

   2. distribute-rgt1-in 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 4: 51.5% 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 75.3%

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

3. Taylor expanded in x around 0 51.1%

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

4. Final simplification 51.1%

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

Reproduce

herbie shell --seed 2024076 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))