2frac (problem 3.3.1)

Percentage Accurate: 77.7% → 99.9%

Time: 6.0s

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: 77.7% 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

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

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(Float64(-1.0 / x) / Float64(x + 1.0))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

   1. frac-2neg 76.6%

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

   2. frac-2neg 76.6%

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

   3. frac-sub 77.0%

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

   4. distribute-rgt-neg-in 77.0%

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

   5. distribute-lft-neg-in 77.0%

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

   6. metadata-eval 77.0%

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

   7. metadata-eval 77.0%

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

   8. *-un-lft-identity 77.0%

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

   9. cancel-sign-sub 77.0%

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

   10. distribute-rgt-neg-in 77.0%

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

   11. *-rgt-identity 77.0%

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

   12. neg-sub0 77.0%

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

   13. +-commutative 77.0%

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

   14. associate--r+ 77.0%

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

   15. metadata-eval 77.0%

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

   16. *-commutative 77.0%

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

   17. neg-sub0 77.0%

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

   18. +-commutative 77.0%

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

   19. associate--r+ 77.0%

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

   20. metadata-eval 77.0%

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

3. Applied egg-rr 77.0%

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

   1. associate-+r- 77.0%

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

   2. +-commutative 77.0%

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

   3. associate--l+ 99.4%

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

   4. distribute-lft-neg-out 99.4%

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

   5. distribute-rgt-neg-in 99.4%

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

5. Simplified 99.4%

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

   1. associate-/r* 99.9%

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

   2. +-inverses 99.9%

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

   3. metadata-eval 99.9%

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

   4. metadata-eval 99.9%

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

   5. distribute-neg-frac 99.9%

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

   6. clear-num 99.3%

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

   7. associate-/r/ 99.8%

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

   8. neg-sub0 99.8%

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

   9. +-inverses 99.8%

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

   10. metadata-eval 99.8%

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

   11. +-inverses 99.8%

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

   12. associate--r- 99.8%

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

   13. +-inverses 99.8%

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

   14. +-inverses 99.8%

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

   15. metadata-eval 99.8%

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

   16. metadata-eval 99.8%

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

   17. +-commutative 99.8%

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

   18. distribute-neg-frac 99.8%

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

   19. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.9%

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

   2. *-commutative 99.9%

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

9. Applied egg-rr 47.7%

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

    1. div-inv 47.7%

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

    2. add-sqr-sqrt 21.7%

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

    3. sqrt-unprod 48.6%

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

    4. frac-times 48.4%

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

    5. metadata-eval 48.4%

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

    6. metadata-eval 48.4%

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

    7. frac-times 48.6%

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

    8. sqrt-unprod 38.6%

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

    9. add-sqr-sqrt 74.6%

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

    10. div-inv 74.6%

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

    11. mul-1-neg 74.6%

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

    12. distribute-rgt-neg-in 74.6%

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

    13. add-sqr-sqrt 35.6%

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

    14. associate-*r* 35.6%

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

    15. distribute-rgt-neg-in 35.6%

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

11. Applied egg-rr 46.6%

    \[\leadsto \color{blue}{\left(\frac{1}{1 + x} \cdot {x}^{-0.5}\right) \cdot \left(-{x}^{-0.5}\right)} \]
12. Step-by-step derivation

    1. associate-*l/ 46.6%

       \[\leadsto \color{blue}{\frac{1 \cdot {x}^{-0.5}}{1 + x}} \cdot \left(-{x}^{-0.5}\right) \]

    2. associate-*l/ 46.6%

       \[\leadsto \color{blue}{\frac{\left(1 \cdot {x}^{-0.5}\right) \cdot \left(-{x}^{-0.5}\right)}{1 + x}} \]

    3. *-lft-identity 46.6%

       \[\leadsto \frac{\color{blue}{{x}^{-0.5}} \cdot \left(-{x}^{-0.5}\right)}{1 + x} \]

    4. distribute-rgt-neg-out 46.6%

       \[\leadsto \frac{\color{blue}{-{x}^{-0.5} \cdot {x}^{-0.5}}}{1 + x} \]

    5. pow-sqr 99.9%

       \[\leadsto \frac{-\color{blue}{{x}^{\left(2 \cdot -0.5\right)}}}{1 + x} \]

    6. metadata-eval 99.9%

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

    7. unpow-1 99.9%

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

    8. distribute-neg-frac 99.9%

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

    9. metadata-eval 99.9%

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

    10. +-commutative 99.9%

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

13. Simplified 99.9%

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

14. Final simplification 99.9%

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

Alternative 2: 98.5% accurate, 0.8× 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 51.4%

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

      1. frac-2neg 51.4%

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

      2. frac-2neg 51.4%

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

      3. frac-sub 52.1%

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

      4. distribute-rgt-neg-in 52.1%

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

      5. distribute-lft-neg-in 52.1%

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

      6. metadata-eval 52.1%

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

      7. metadata-eval 52.1%

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

      8. *-un-lft-identity 52.1%

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

      9. cancel-sign-sub 52.1%

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

      10. distribute-rgt-neg-in 52.1%

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

      11. *-rgt-identity 52.1%

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

      12. neg-sub0 52.1%

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

      13. +-commutative 52.1%

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

      14. associate--r+ 52.1%

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

      15. metadata-eval 52.1%

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

      16. *-commutative 52.1%

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

      17. neg-sub0 52.1%

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

      18. +-commutative 52.1%

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

      19. associate--r+ 52.1%

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

      20. metadata-eval 52.1%

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

   3. Applied egg-rr 52.1%

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

      1. associate-+r- 52.1%

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

      2. +-commutative 52.1%

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

      3. associate--l+ 98.8%

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

      4. distribute-lft-neg-out 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

   5. Simplified 98.8%

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

      1. associate-/r* 99.8%

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

      2. +-inverses 99.8%

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

      3. metadata-eval 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. clear-num 98.6%

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

      7. associate-/r/ 99.6%

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

      8. neg-sub0 99.6%

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

      9. +-inverses 99.6%

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

      10. metadata-eval 99.6%

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

      11. +-inverses 99.6%

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

      12. associate--r- 99.6%

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

      13. +-inverses 99.6%

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

      14. +-inverses 99.6%

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

      15. metadata-eval 99.6%

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

      16. metadata-eval 99.6%

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

      17. +-commutative 99.6%

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

      18. distribute-neg-frac 99.6%

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

      19. metadata-eval 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.8%

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

      2. *-commutative 99.8%

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

   9. Applied egg-rr 98.6%

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

   10. Taylor expanded in x around inf 98.7%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. sub-neg 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 99.1%

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

Alternative 3: 98.4% accurate, 1.0× 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 51.4%

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

      1. frac-2neg 51.4%

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

      2. frac-2neg 51.4%

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

      3. frac-sub 52.1%

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

      4. distribute-rgt-neg-in 52.1%

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

      5. distribute-lft-neg-in 52.1%

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

      6. metadata-eval 52.1%

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

      7. metadata-eval 52.1%

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

      8. *-un-lft-identity 52.1%

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

      9. cancel-sign-sub 52.1%

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

      10. distribute-rgt-neg-in 52.1%

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

      11. *-rgt-identity 52.1%

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

      12. neg-sub0 52.1%

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

      13. +-commutative 52.1%

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

      14. associate--r+ 52.1%

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

      15. metadata-eval 52.1%

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

      16. *-commutative 52.1%

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

      17. neg-sub0 52.1%

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

      18. +-commutative 52.1%

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

      19. associate--r+ 52.1%

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

      20. metadata-eval 52.1%

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

   3. Applied egg-rr 52.1%

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

      1. associate-+r- 52.1%

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

      2. +-commutative 52.1%

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

      3. associate--l+ 98.8%

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

      4. distribute-lft-neg-out 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

   5. Simplified 98.8%

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

      1. associate-/r* 99.8%

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

      2. +-inverses 99.8%

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

      3. metadata-eval 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. clear-num 98.6%

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

      7. associate-/r/ 99.6%

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

      8. neg-sub0 99.6%

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

      9. +-inverses 99.6%

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

      10. metadata-eval 99.6%

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

      11. +-inverses 99.6%

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

      12. associate--r- 99.6%

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

      13. +-inverses 99.6%

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

      14. +-inverses 99.6%

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

      15. metadata-eval 99.6%

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

      16. metadata-eval 99.6%

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

      17. +-commutative 99.6%

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

      18. distribute-neg-frac 99.6%

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

      19. metadata-eval 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.8%

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

      2. *-commutative 99.8%

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

   9. Applied egg-rr 98.6%

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

   10. Taylor expanded in x around inf 98.7%

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

   if -1 < x < 0.75

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.5%

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

4. Final simplification 99.1%

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

Alternative 4: 51.3% 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 76.6%

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

2. Taylor expanded in x around 0 53.8%

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

3. Final simplification 53.8%

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

Alternative 5: 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 76.6%

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

2. Taylor expanded in x around 0 52.9%

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

   1. mul-1-neg 52.9%

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

   2. sub-neg 52.9%

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

4. Simplified 52.9%

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

5. Taylor expanded in x around inf 3.1%

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

   1. mul-1-neg 3.1%

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

7. Simplified 3.1%

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

8. Final simplification 3.1%

   \[\leadsto -x \]

Alternative 6: 2.0% accurate, 9.0× 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 x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 76.6%

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

2. Taylor expanded in x around 0 52.9%

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

   1. mul-1-neg 52.9%

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

   2. sub-neg 52.9%

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

4. Simplified 52.9%

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

   1. sub-neg 52.9%

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

   2. +-commutative 52.9%

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

   3. flip-- 52.8%

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

   4. metadata-eval 52.8%

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

   5. metadata-eval 52.8%

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

   6. metadata-eval 52.8%

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

   7. +-inverses 52.8%

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

   8. metadata-eval 52.8%

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

   9. +-inverses 52.8%

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

   10. associate--r- 52.8%

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

   11. +-inverses 52.8%

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

   12. +-inverses 52.8%

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

   13. metadata-eval 52.8%

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

   14. neg-sub0 52.8%

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

   15. metadata-eval 52.8%

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

   16. +-inverses 52.8%

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

   17. metadata-eval 52.8%

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

   18. +-inverses 52.8%

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

   19. add-sqr-sqrt 52.4%

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

   20. sqrt-unprod 52.8%

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

   21. sqr-neg 52.8%

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

   22. sqrt-unprod 0.8%

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

   23. add-sqr-sqrt 52.3%

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

6. Applied egg-rr 52.3%

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

7. Taylor expanded in x around inf 2.0%

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

8. Final simplification 2.0%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023305 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))