2frac (problem 3.3.1)

Percentage Accurate: 77.0% → 99.9%

Time: 8.3s

Alternatives: 7

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

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

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 72.2%

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

   1. frac-sub 72.6%

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

   2. *-rgt-identity 72.6%

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

   3. metadata-eval 72.6%

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

   4. div-inv 72.6%

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

   5. associate-/r* 72.6%

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

   6. *-un-lft-identity 72.6%

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

   7. *-rgt-identity 72.6%

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

   8. +-commutative 72.6%

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

   9. div-inv 72.6%

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

   10. metadata-eval 72.6%

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

   11. *-rgt-identity 72.6%

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

   12. +-commutative 72.6%

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

4. Applied egg-rr 72.6%

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

   1. div-inv 72.6%

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

   2. +-commutative 72.6%

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

   3. +-commutative 72.6%

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

6. Applied egg-rr 72.6%

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

   1. associate-*r/ 72.6%

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

   2. *-rgt-identity 72.6%

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

   3. associate--r+ 99.9%

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

   4. +-inverses 99.9%

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

   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: 98.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(-1.0 / x) / 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 <= 1.0)))
		tmp = (-1.0 / x) / 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, 1.0]], $MachinePrecision]], N[(N[(-1.0 / x), $MachinePrecision] / x), $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 1 < x

   1. Initial program 47.2%

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

      1. frac-sub 48.0%

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

      2. *-rgt-identity 48.0%

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

      3. metadata-eval 48.0%

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

      4. div-inv 48.0%

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

      5. associate-/r* 48.0%

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

      6. *-un-lft-identity 48.0%

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

      7. *-rgt-identity 48.0%

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

      8. +-commutative 48.0%

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

      9. div-inv 48.0%

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

      10. metadata-eval 48.0%

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

      11. *-rgt-identity 48.0%

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

      12. +-commutative 48.0%

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

   4. Applied egg-rr 48.0%

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

      1. div-inv 48.0%

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

      2. +-commutative 48.0%

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

      3. +-commutative 48.0%

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

   6. Applied egg-rr 48.0%

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

      1. associate-*r/ 48.0%

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

      2. *-rgt-identity 48.0%

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

      3. associate--r+ 99.8%

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

      4. +-inverses 99.8%

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

      5. metadata-eval 99.8%

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

   8. Simplified 99.8%

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

      1. add-cube-cbrt 98.9%

         \[\leadsto \frac{\frac{-1}{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}}{x} \]

      2. unpow2 98.9%

         \[\leadsto \frac{\frac{-1}{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{2}} \cdot \sqrt[3]{x + 1}}}{x} \]

      3. associate-/l/ 98.9%

         \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{\sqrt[3]{x + 1}}}{{\left(\sqrt[3]{x + 1}\right)}^{2}}}}{x} \]

      4. div-inv 98.9%

         \[\leadsto \frac{\color{blue}{\frac{-1}{\sqrt[3]{x + 1}} \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2}}}}{x} \]

      5. metadata-eval 98.9%

         \[\leadsto \frac{\frac{-1}{\sqrt[3]{x + 1}} \cdot \frac{\color{blue}{-1 \cdot -1}}{{\left(\sqrt[3]{x + 1}\right)}^{2}}}{x} \]

      6. unpow2 98.9%

         \[\leadsto \frac{\frac{-1}{\sqrt[3]{x + 1}} \cdot \frac{-1 \cdot -1}{\color{blue}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}}{x} \]

      7. frac-times 98.9%

         \[\leadsto \frac{\frac{-1}{\sqrt[3]{x + 1}} \cdot \color{blue}{\left(\frac{-1}{\sqrt[3]{x + 1}} \cdot \frac{-1}{\sqrt[3]{x + 1}}\right)}}{x} \]

      8. associate-*l* 98.9%

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{\sqrt[3]{x + 1}} \cdot \frac{-1}{\sqrt[3]{x + 1}}\right) \cdot \frac{-1}{\sqrt[3]{x + 1}}}}{x} \]

      9. pow3 98.8%

         \[\leadsto \frac{\color{blue}{{\left(\frac{-1}{\sqrt[3]{x + 1}}\right)}^{3}}}{x} \]

   10. Applied egg-rr 98.8%

       \[\leadsto \frac{\color{blue}{{\left(\frac{-1}{\sqrt[3]{x + 1}}\right)}^{3}}}{x} \]

   11. Taylor expanded in x around inf 97.7%

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

   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 98.5%

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

      1. neg-mul-1 98.5%

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

      2. unsub-neg 98.5%

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

   5. Simplified 98.5%

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

4. Final simplification 98.1%

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

Alternative 3: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.75))) (/ (/ -1.0 x) x) (+ 1.0 (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.75)) {
		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 <= 0.75d0))) 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 <= 0.75)) {
		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 <= 0.75):
		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 <= 0.75))
		tmp = Float64(Float64(-1.0 / 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 <= 0.75)))
		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, 0.75]], $MachinePrecision]], N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.75 < x

   1. Initial program 47.6%

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

      1. frac-sub 48.4%

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

      2. *-rgt-identity 48.4%

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

      3. metadata-eval 48.4%

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

      4. div-inv 48.4%

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

      5. associate-/r* 48.4%

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

      6. *-un-lft-identity 48.4%

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

      7. *-rgt-identity 48.4%

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

      8. +-commutative 48.4%

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

      9. div-inv 48.4%

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

      10. metadata-eval 48.4%

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

      11. *-rgt-identity 48.4%

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

      12. +-commutative 48.4%

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

   4. Applied egg-rr 48.4%

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

      1. div-inv 48.4%

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

      2. +-commutative 48.4%

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

      3. +-commutative 48.4%

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

   6. Applied egg-rr 48.4%

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

      1. associate-*r/ 48.4%

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

      2. *-rgt-identity 48.4%

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

      3. associate--r+ 99.8%

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

      4. +-inverses 99.8%

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

      5. metadata-eval 99.8%

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

   8. Simplified 99.8%

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

      1. add-cube-cbrt 98.9%

         \[\leadsto \frac{\frac{-1}{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}}{x} \]

      2. unpow2 98.9%

         \[\leadsto \frac{\frac{-1}{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{2}} \cdot \sqrt[3]{x + 1}}}{x} \]

      3. associate-/l/ 98.9%

         \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{\sqrt[3]{x + 1}}}{{\left(\sqrt[3]{x + 1}\right)}^{2}}}}{x} \]

      4. div-inv 98.9%

         \[\leadsto \frac{\color{blue}{\frac{-1}{\sqrt[3]{x + 1}} \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2}}}}{x} \]

      5. metadata-eval 98.9%

         \[\leadsto \frac{\frac{-1}{\sqrt[3]{x + 1}} \cdot \frac{\color{blue}{-1 \cdot -1}}{{\left(\sqrt[3]{x + 1}\right)}^{2}}}{x} \]

      6. unpow2 98.9%

         \[\leadsto \frac{\frac{-1}{\sqrt[3]{x + 1}} \cdot \frac{-1 \cdot -1}{\color{blue}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}}{x} \]

      7. frac-times 98.9%

         \[\leadsto \frac{\frac{-1}{\sqrt[3]{x + 1}} \cdot \color{blue}{\left(\frac{-1}{\sqrt[3]{x + 1}} \cdot \frac{-1}{\sqrt[3]{x + 1}}\right)}}{x} \]

      8. associate-*l* 98.9%

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{\sqrt[3]{x + 1}} \cdot \frac{-1}{\sqrt[3]{x + 1}}\right) \cdot \frac{-1}{\sqrt[3]{x + 1}}}}{x} \]

      9. pow3 98.9%

         \[\leadsto \frac{\color{blue}{{\left(\frac{-1}{\sqrt[3]{x + 1}}\right)}^{3}}}{x} \]

   10. Applied egg-rr 98.9%

       \[\leadsto \frac{\color{blue}{{\left(\frac{-1}{\sqrt[3]{x + 1}}\right)}^{3}}}{x} \]

   11. Taylor expanded in x around inf 97.1%

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

   if -1 < x < 0.75

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.0%

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

      1. div-sub 99.0%

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

      2. *-inverses 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 72.2%

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

   1. clear-num 72.2%

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

   2. frac-sub 72.6%

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

   3. *-un-lft-identity 72.6%

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

   4. div-inv 72.6%

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

   5. metadata-eval 72.6%

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

   6. *-rgt-identity 72.6%

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

   7. *-rgt-identity 72.6%

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

   8. +-commutative 72.6%

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

   9. *-commutative 72.6%

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

   10. div-inv 72.6%

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

   11. metadata-eval 72.6%

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

   12. *-rgt-identity 72.6%

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

   13. +-commutative 72.6%

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

4. Applied egg-rr 72.6%

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

5. Taylor expanded in x around 0 99.1%

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

6. Final simplification 99.1%

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

Alternative 5: 51.7% 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 72.2%

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

3. Taylor expanded in x around 0 49.0%

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

4. Final simplification 49.0%

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

Alternative 6: 3.1% 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 72.2%

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

3. Taylor expanded in x around 0 47.9%

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

   1. neg-mul-1 47.9%

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

   2. unsub-neg 47.9%

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

5. Simplified 47.9%

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

6. Taylor expanded in x around inf 3.0%

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

   1. neg-mul-1 3.0%

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

8. Simplified 3.0%

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

9. Final simplification 3.0%

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

Alternative 7: 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 72.2%

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

3. Taylor expanded in x around 0 47.8%

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

   1. div-sub 47.8%

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

   2. *-inverses 47.8%

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

5. Simplified 47.8%

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

6. Taylor expanded in x around inf 2.7%

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

7. Final simplification 2.7%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))