2frac (problem 3.3.1)

Percentage Accurate: 77.0% → 99.9%

Time: 4.5s

Alternatives: 6

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: 77.0% 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 70.1%

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

   1. sub-neg 70.1%

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

   2. +-commutative 70.1%

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

   3. distribute-neg-frac 70.1%

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

   4. metadata-eval 70.1%

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

4. Applied egg-rr 70.1%

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

   1. metadata-eval 70.1%

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

   2. distribute-neg-frac 70.1%

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

   3. unsub-neg 70.1%

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

   4. *-inverses 70.1%

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

   5. associate-/r* 49.3%

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

   6. *-commutative 49.3%

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

   7. associate-/r* 70.1%

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

   8. div-sub 70.1%

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

   9. *-inverses 70.1%

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

   10. div-sub 70.6%

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

   11. associate-/r* 70.6%

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

   12. +-commutative 70.6%

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

   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. distribute-lft-in 99.3%

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

   17. unpow2 99.3%

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

   18. *-rgt-identity 99.3%

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

6. Simplified 99.3%

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

   1. *-un-lft-identity 99.3%

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

   2. unpow2 99.3%

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

   3. distribute-rgt-in 99.2%

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

   4. associate-/r* 99.8%

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

   5. expm1-log1p-u 76.0%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{x}\right)\right)}}{1 + x} \]

   6. expm1-undefine 46.3%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{-1}{x}\right)} - 1}}{1 + x} \]

   7. log1p-undefine 46.3%

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

   8. div-inv 46.3%

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

   9. neg-mul-1 46.3%

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

   10. sub-neg 46.3%

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

   11. add-exp-log 70.1%

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

   12. *-inverses 70.1%

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

   13. div-sub 70.1%

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

   14. *-inverses 70.1%

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

   15. div-sub 70.6%

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

   16. associate--r+ 70.6%

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

   17. associate-/l/ 70.6%

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

   18. *-un-lft-identity 70.6%

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

   19. times-frac 70.6%

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

   20. +-commutative 70.6%

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

   21. associate--r+ 70.6%

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

   22. div-sub 70.1%

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

8. Applied egg-rr 99.7%

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

   1. frac-times 99.2%

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

   2. metadata-eval 99.2%

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

   3. associate-/r* 99.8%

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

   4. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

    \[\leadsto \color{blue}{\frac{\frac{-1}{1 + x}}{x}} \]
11. Add Preprocessing

Alternative 2: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+61}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 3.9e+61) (+ (- 1.0 x) (/ -1.0 x)) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 3.9e+61) {
		tmp = (1.0 - x) + (-1.0 / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 0.0d0
    else if (x <= 3.9d+61) then
        tmp = (1.0d0 - x) + ((-1.0d0) / x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 3.9e+61) {
		tmp = (1.0 - x) + (-1.0 / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0
	elif x <= 3.9e+61:
		tmp = (1.0 - x) + (-1.0 / x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 3.9e+61)
		tmp = Float64(Float64(1.0 - x) + Float64(-1.0 / x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 3.9e+61)
		tmp = (1.0 - x) + (-1.0 / x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 3.9e+61], N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 3.89999999999999987e61 < x

   1. Initial program 48.0%

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

   3. Taylor expanded in x around inf 46.6%

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

   4. Taylor expanded in x around 0 46.6%

      \[\leadsto \color{blue}{0} \]

   if -1 < x < 3.89999999999999987e61

   1. Initial program 93.7%

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

   3. Taylor expanded in x around 0 91.6%

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

      1. neg-mul-1 91.6%

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

      2. sub-neg 91.6%

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

   5. Simplified 91.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+61}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 1.0) (+ 1.0 (/ -1.0 x)) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 0.0d0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0
	elif x <= 1.0:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 1.0)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 1.0)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 1.0], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 45.8%

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

   3. Taylor expanded in x around inf 43.9%

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

   4. Taylor expanded in x around 0 43.9%

      \[\leadsto \color{blue}{0} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.6%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 4.5e+102) (/ -1.0 x) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 4.5e+102) {
		tmp = -1.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 0.0d0
    else if (x <= 4.5d+102) then
        tmp = (-1.0d0) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 4.5e+102) {
		tmp = -1.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0
	elif x <= 4.5e+102:
		tmp = -1.0 / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 4.5e+102)
		tmp = Float64(-1.0 / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 4.5e+102)
		tmp = -1.0 / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 4.5e+102], N[(-1.0 / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 4.50000000000000021e102 < x

   1. Initial program 54.0%

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

   3. Taylor expanded in x around inf 52.4%

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

   4. Taylor expanded in x around 0 52.4%

      \[\leadsto \color{blue}{0} \]

   if -1 < x < 4.50000000000000021e102

   1. Initial program 83.5%

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

   3. Taylor expanded in x around 0 80.3%

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

Alternative 5: 99.4% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ -1.0 (+ x (* x x))))

C

double code(double x) {
	return -1.0 / (x + (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / (x + (x * x))
end function

Java

public static double code(double x) {
	return -1.0 / (x + (x * x));
}

Python

def code(x):
	return -1.0 / (x + (x * x))

Julia

function code(x)
	return Float64(-1.0 / Float64(x + Float64(x * x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. sub-neg 70.1%

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

   2. +-commutative 70.1%

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

   3. distribute-neg-frac 70.1%

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

   4. metadata-eval 70.1%

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

4. Applied egg-rr 70.1%

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

   1. metadata-eval 70.1%

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

   2. distribute-neg-frac 70.1%

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

   3. unsub-neg 70.1%

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

   4. *-inverses 70.1%

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

   5. associate-/r* 49.3%

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

   6. *-commutative 49.3%

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

   7. associate-/r* 70.1%

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

   8. div-sub 70.1%

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

   9. *-inverses 70.1%

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

   10. div-sub 70.6%

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

   11. associate-/r* 70.6%

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

   12. +-commutative 70.6%

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

   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. distribute-lft-in 99.3%

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

   17. unpow2 99.3%

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

   18. *-rgt-identity 99.3%

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

6. Simplified 99.3%

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

   1. unpow2 99.3%

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

8. Applied egg-rr 99.3%

   \[\leadsto \frac{-1}{x + \color{blue}{x \cdot x}} \]
9. Add Preprocessing

Alternative 6: 26.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 70.1%

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

3. Taylor expanded in x around inf 25.2%

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

4. Taylor expanded in x around 0 25.2%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))