2frac (problem 3.3.1)

Percentage Accurate: 76.9% → 99.9%

Time: 3.3s

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

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

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 76.1%

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

   1. frac-sub 77.0%

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

   2. *-rgt-identity 77.0%

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

   3. metadata-eval 77.0%

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

   4. div-inv 77.0%

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

   5. associate-/r* 77.0%

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

   6. *-un-lft-identity 77.0%

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

   7. *-rgt-identity 77.0%

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

   8. +-commutative 77.0%

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

   9. div-inv 77.0%

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

   10. metadata-eval 77.0%

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

   11. *-rgt-identity 77.0%

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

   12. +-commutative 77.0%

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

3. Applied egg-rr 77.0%

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

4. Taylor expanded in x around 0 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.76))) (/ -1.0 (* x x)) (+ 1.0 (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.76)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = 1.0 + (-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 <= 0.76d0))) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.76)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.76):
		tmp = -1.0 / (x * x)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.76))
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.76)))
		tmp = -1.0 / (x * x);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.76]], $MachinePrecision]], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.76000000000000001 < x

   1. Initial program 57.1%

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

   2. Taylor expanded in x around inf 95.2%

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

      1. unpow2 95.2%

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

   4. Simplified 95.2%

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

   if -1 < x < 0.76000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.3%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 3: 99.4% 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 76.1%

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

   1. sub-neg 76.1%

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

   2. +-commutative 76.1%

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

   3. distribute-neg-frac 76.1%

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

   4. metadata-eval 76.1%

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

3. Applied egg-rr 76.1%

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

   1. *-rgt-identity 76.1%

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

   2. cancel-sign-sub 76.1%

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

   3. distribute-neg-frac 76.1%

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

   4. metadata-eval 76.1%

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

   5. *-rgt-identity 76.1%

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

   6. *-inverses 76.1%

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

   7. associate-*r/ 76.1%

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

   8. associate-*l/ 49.5%

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

   9. associate-/r* 49.4%

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

   10. associate-/l/ 49.5%

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

   11. associate-*l/ 76.1%

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

   12. associate-*r/ 76.1%

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

   13. distribute-lft-out-- 76.1%

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

   14. *-inverses 76.1%

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

   15. div-sub 77.0%

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

   16. associate-*l/ 77.0%

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

   17. associate-*r/ 77.0%

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

   18. *-lft-identity 77.0%

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

   19. associate-/l/ 77.0%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 4: 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 76.1%

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

   1. sub-neg 76.1%

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

   2. +-commutative 76.1%

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

   3. distribute-neg-frac 76.1%

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

   4. metadata-eval 76.1%

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

3. Applied egg-rr 76.1%

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

   1. *-rgt-identity 76.1%

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

   2. cancel-sign-sub 76.1%

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

   3. distribute-neg-frac 76.1%

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

   4. metadata-eval 76.1%

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

   5. *-rgt-identity 76.1%

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

   6. *-inverses 76.1%

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

   7. associate-*r/ 76.1%

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

   8. associate-*l/ 49.5%

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

   9. associate-/r* 49.4%

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

   10. associate-/l/ 49.5%

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

   11. associate-*l/ 76.1%

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

   12. associate-*r/ 76.1%

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

   13. distribute-lft-out-- 76.1%

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

   14. *-inverses 76.1%

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

   15. div-sub 77.0%

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

   16. associate-*l/ 77.0%

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

   17. associate-*r/ 77.0%

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

   18. *-lft-identity 77.0%

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

   19. associate-/l/ 77.0%

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

5. Simplified 98.8%

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

   1. associate-/r* 99.9%

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

   2. +-commutative 99.9%

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

   3. flip-+ 98.8%

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

   4. associate-/r/ 94.1%

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

   5. metadata-eval 94.1%

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

7. Applied egg-rr 94.1%

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

   1. associate-*l/ 98.8%

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

   2. associate-/l* 98.8%

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

   3. metadata-eval 98.8%

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

   4. flip-+ 99.9%

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

   5. +-commutative 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 5: 52.5% 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 76.1%

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

2. Taylor expanded in x around 0 46.6%

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

3. Final simplification 46.6%

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

Reproduce

herbie shell --seed 2023178 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))