2frac (problem 3.3.1)

Percentage Accurate: 77.6% → 99.9%

Time: 4.6s

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: 77.6% 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 76.5%

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

   1. frac-sub 77.7%

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

   2. *-rgt-identity 77.7%

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

   3. metadata-eval 77.7%

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

   4. div-inv 77.7%

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

   5. associate-/r* 77.7%

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

   6. *-un-lft-identity 77.7%

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

   7. *-rgt-identity 77.7%

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

   8. +-commutative 77.7%

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

   9. div-inv 77.7%

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

   10. metadata-eval 77.7%

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

   11. *-rgt-identity 77.7%

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

   12. +-commutative 77.7%

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

3. Applied egg-rr 77.7%

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

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: 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 76.5%

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

   1. frac-sub 77.7%

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

   2. *-rgt-identity 77.7%

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

   3. metadata-eval 77.7%

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

   4. div-inv 77.7%

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

   5. associate-/r* 77.7%

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

   6. *-un-lft-identity 77.7%

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

   7. *-rgt-identity 77.7%

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

   8. +-commutative 77.7%

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

   9. div-inv 77.7%

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

   10. metadata-eval 77.7%

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

   11. *-rgt-identity 77.7%

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

   12. +-commutative 77.7%

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

3. Applied egg-rr 77.7%

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

   1. associate--r+ 77.7%

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

   2. div-sub 76.5%

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

   3. sub-neg 76.5%

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

   4. metadata-eval 76.5%

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

   5. +-commutative 76.5%

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

   6. +-commutative 76.5%

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

5. Applied egg-rr 76.5%

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

   1. frac-2neg 76.5%

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

   2. div-inv 76.5%

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

   3. sub-div 77.7%

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

   4. distribute-neg-frac 77.7%

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

   5. associate--l+ 77.7%

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

7. Applied egg-rr 77.7%

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

   1. associate-*r/ 77.7%

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

   2. *-rgt-identity 77.7%

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

   3. associate-/l/ 77.7%

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

   4. sub-neg 77.7%

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

   5. distribute-neg-in 77.7%

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

   6. distribute-neg-in 77.7%

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

   7. metadata-eval 77.7%

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

   8. remove-double-neg 77.7%

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

   9. associate-+l+ 77.7%

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

   10. +-commutative 77.7%

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

   11. associate-+l+ 99.8%

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

   12. +-commutative 99.8%

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

   13. neg-mul-1 99.8%

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

   14. distribute-rgt1-in 99.8%

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

   15. metadata-eval 99.8%

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

   16. distribute-lft-in 99.8%

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

   17. *-rgt-identity 99.8%

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

   18. neg-mul-1 99.8%

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

   19. distribute-rgt-in 99.8%

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

   20. +-commutative 99.8%

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

   21. sub-neg 99.8%

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

9. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{1 + 0 \cdot x}{x \cdot \left(-1 - x\right)}} \]
10. Step-by-step derivation

    1. add-log-exp 29.5%

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

    2. *-un-lft-identity 29.5%

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

    3. log-prod 29.5%

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

    4. 1-exp 29.5%

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

    5. mul0-lft 29.5%

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

    6. add-log-exp 29.5%

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

    7. mul0-lft 29.5%

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

    8. add-log-exp 99.8%

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

    9. associate-/r* 99.9%

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

    10. mul0-lft 99.9%

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

    11. metadata-eval 99.9%

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

11. Applied egg-rr 99.9%

    \[\leadsto \color{blue}{0 + \frac{\frac{1}{x}}{-1 - x}} \]
12. Step-by-step derivation

    1. +-lft-identity 99.9%

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

    2. associate-/r* 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 3: 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 76.5%

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

   1. sub-neg 76.5%

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

   2. +-commutative 76.5%

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

   3. distribute-neg-frac 76.5%

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

   4. metadata-eval 76.5%

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

3. Applied egg-rr 76.5%

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 4: 52.2% 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 76.5%

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

2. Taylor expanded in x around 0 51.1%

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

3. Final simplification 51.1%

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

Alternative 5: 2.9% 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 76.5%

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

   1. frac-sub 77.7%

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

   2. *-rgt-identity 77.7%

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

   3. metadata-eval 77.7%

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

   4. div-inv 77.7%

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

   5. associate-/r* 77.7%

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

   6. *-un-lft-identity 77.7%

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

   7. *-rgt-identity 77.7%

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

   8. +-commutative 77.7%

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

   9. div-inv 77.7%

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

   10. metadata-eval 77.7%

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

   11. *-rgt-identity 77.7%

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

   12. +-commutative 77.7%

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

3. Applied egg-rr 77.7%

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

4. Taylor expanded in x around 0 50.0%

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

5. Taylor expanded in x around inf 2.8%

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

6. Final simplification 2.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023318 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))