2frac (problem 3.3.1)

Percentage Accurate: 76.9% → 99.9%

Time: 3.9s

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: 76.9% 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 82.5%

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

   1. frac-sub 83.5%

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

   2. div-inv 83.5%

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

   3. *-un-lft-identity 83.5%

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

   4. *-rgt-identity 83.5%

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

   5. +-commutative 83.5%

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

   6. metadata-eval 83.5%

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

   7. frac-times 83.5%

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

   8. clear-num 83.5%

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

   9. associate-*l/ 83.5%

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

   10. *-un-lft-identity 83.5%

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

   11. div-inv 83.5%

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

   12. metadata-eval 83.5%

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

   13. *-rgt-identity 83.5%

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

   14. +-commutative 83.5%

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

4. Applied egg-rr 83.5%

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

5. Taylor expanded in x around 0 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 75.2% 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 63.4%

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

   3. Taylor expanded in x around inf 61.4%

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

   if -1 < x < 3.79999999999999995e61

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0 93.7%

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

      1. neg-mul-1 93.7%

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

      2. unsub-neg 93.7%

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

   5. Simplified 93.7%

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

4. Final simplification 80.2%

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

Alternative 3: 99.9% 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 82.5%

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

   1. frac-sub 83.5%

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

   2. *-rgt-identity 83.5%

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

   3. metadata-eval 83.5%

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

   4. div-inv 83.5%

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

   5. associate-/r* 83.5%

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

   6. *-un-lft-identity 83.5%

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

   7. *-rgt-identity 83.5%

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

   8. +-commutative 83.5%

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

   9. div-inv 83.5%

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

   10. metadata-eval 83.5%

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

   11. *-rgt-identity 83.5%

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

   12. +-commutative 83.5%

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

4. Applied egg-rr 83.5%

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

5. Taylor expanded in x around 0 99.9%

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

6. Final simplification 99.9%

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

Alternative 4: 51.2% 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 82.5%

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

3. Taylor expanded in x around 0 56.0%

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

4. Final simplification 56.0%

   \[\leadsto \frac{-1}{x} \]
5. 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 82.5%

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

3. Taylor expanded in x around 0 55.2%

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

4. Taylor expanded in x around inf 3.4%

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

5. Final simplification 3.4%

   \[\leadsto 1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024061 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))