2frac (problem 3.3.1)

Percentage Accurate: 78.0% → 99.9%

Time: 4.9s

Alternatives: 8

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: 78.0% 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 / 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 74.7%

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

   1. sub-neg 74.7%

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

   2. +-commutative 74.7%

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

   3. distribute-neg-frac 74.7%

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

   4. metadata-eval 74.7%

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

4. Applied egg-rr 74.7%

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

   1. *-rgt-identity 74.7%

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

   2. cancel-sign-sub 74.7%

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

   3. distribute-neg-frac 74.7%

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

   4. metadata-eval 74.7%

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

   5. *-rgt-identity 74.7%

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

   6. *-inverses 74.7%

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

   7. associate-/r* 53.4%

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

   8. *-commutative 53.4%

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

   9. associate-/r* 74.6%

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

   10. div-sub 74.7%

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

   11. *-inverses 74.7%

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

   12. div-sub 76.0%

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

   13. associate-/l/ 76.0%

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

   14. +-commutative 76.0%

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

   15. associate--r+ 99.0%

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

   16. div-sub 99.0%

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

   17. +-inverses 99.0%

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

   18. div0 99.0%

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

   19. associate-/r* 99.9%

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

   20. neg-sub0 99.9%

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

   21. associate-/r* 99.0%

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

6. Simplified 99.0%

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

   1. unpow2 99.0%

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

8. Applied egg-rr 99.0%

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

   1. frac-2neg 99.0%

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

   2. metadata-eval 99.0%

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

   3. inv-pow 99.0%

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

   4. neg-mul-1 99.0%

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

   5. distribute-rgt1-in 99.0%

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

   6. add-sqr-sqrt 51.0%

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

   7. sqrt-prod 52.4%

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

   8. sqr-neg 52.4%

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

   9. sqrt-unprod 12.4%

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

   10. add-sqr-sqrt 24.5%

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

   11. distribute-rgt-neg-in 24.5%

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

   12. distribute-rgt1-in 24.5%

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

   13. distribute-neg-in 24.5%

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

   14. add-sqr-sqrt 12.4%

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

   15. sqrt-unprod 13.5%

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

   16. sqr-neg 13.5%

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

   17. sqrt-prod 35.2%

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

   18. add-sqr-sqrt 70.8%

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

   19. sub-neg 70.8%

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

   20. unpow-prod-down 70.8%

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

   21. metadata-eval 70.8%

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

10. Applied egg-rr 51.7%

    \[\leadsto \color{blue}{-1 \cdot {\left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-2}} \]
11. Step-by-step derivation

    1. neg-mul-1 51.7%

       \[\leadsto \color{blue}{-{\left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-2}} \]

    2. exp-to-pow 48.2%

       \[\leadsto -\color{blue}{e^{\log \left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right) \cdot -2}} \]

    3. metadata-eval 48.2%

       \[\leadsto -e^{\log \left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right) \cdot \color{blue}{\left(-2\right)}} \]

    4. distribute-rgt-neg-in 48.2%

       \[\leadsto -e^{\color{blue}{-\log \left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right) \cdot 2}} \]

    5. exp-neg 47.7%

       \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right) \cdot 2}}} \]

    6. exp-to-pow 51.1%

       \[\leadsto -\frac{1}{\color{blue}{{\left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{2}}} \]

    7. unpow2 51.1%

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

    8. hypot-undefine 51.1%

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

    9. rem-square-sqrt 51.1%

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

    10. fma-undefine 51.1%

        \[\leadsto -\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \cdot \mathsf{hypot}\left(x, \sqrt{x}\right)} \]

    11. hypot-undefine 51.1%

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

    12. rem-square-sqrt 75.0%

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

    13. fma-undefine 75.0%

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

    14. rem-square-sqrt 99.0%

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

    15. fma-undefine 99.0%

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

    16. *-lft-identity 99.0%

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

    17. distribute-rgt-in 99.0%

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

    18. associate-/r* 99.9%

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

    19. unpow-1 99.9%

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

12. Simplified 99.9%

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

Alternative 2: 98.3% 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 50.9%

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

      1. sub-neg 50.9%

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

      2. +-commutative 50.9%

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

      3. distribute-neg-frac 50.9%

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

      4. metadata-eval 50.9%

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

   4. Applied egg-rr 50.9%

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

      1. *-rgt-identity 50.9%

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

      2. cancel-sign-sub 50.9%

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

      3. distribute-neg-frac 50.9%

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

      4. metadata-eval 50.9%

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

      5. *-rgt-identity 50.9%

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

      6. *-inverses 50.9%

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

      7. associate-/r* 9.7%

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

      8. *-commutative 9.7%

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

      9. associate-/r* 50.8%

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

      10. div-sub 50.9%

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

      11. *-inverses 50.9%

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

      12. div-sub 53.5%

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

      13. associate-/l/ 53.5%

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

      14. +-commutative 53.5%

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

      15. associate--r+ 98.1%

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

      16. div-sub 98.1%

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

      17. +-inverses 98.1%

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

      18. div0 98.1%

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

      19. associate-/r* 99.8%

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

      20. neg-sub0 99.8%

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

      21. associate-/r* 98.1%

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

   6. Simplified 98.1%

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

      1. unpow2 98.1%

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

   8. Applied egg-rr 98.1%

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

      1. frac-2neg 98.1%

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

      2. metadata-eval 98.1%

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

      3. inv-pow 98.1%

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

      4. neg-mul-1 98.1%

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

      5. distribute-rgt1-in 98.1%

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

      6. add-sqr-sqrt 51.7%

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

      7. sqrt-prod 75.7%

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

      8. sqr-neg 75.7%

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

      9. sqrt-unprod 23.8%

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

      10. add-sqr-sqrt 46.8%

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

      11. distribute-rgt-neg-in 46.8%

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

      12. distribute-rgt1-in 46.8%

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

      13. distribute-neg-in 46.8%

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

      14. add-sqr-sqrt 23.8%

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

      15. sqrt-unprod 2.3%

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

      16. sqr-neg 2.3%

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

      17. sqrt-prod 23.0%

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

      18. add-sqr-sqrt 46.8%

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

      19. sub-neg 46.8%

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

      20. unpow-prod-down 46.8%

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

      21. metadata-eval 46.8%

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

   10. Applied egg-rr 53.0%

       \[\leadsto \color{blue}{-1 \cdot {\left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-2}} \]
   11. Step-by-step derivation

       1. neg-mul-1 53.0%

          \[\leadsto \color{blue}{-{\left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-2}} \]

       2. exp-to-pow 50.2%

          \[\leadsto -\color{blue}{e^{\log \left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right) \cdot -2}} \]

       3. metadata-eval 50.2%

          \[\leadsto -e^{\log \left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right) \cdot \color{blue}{\left(-2\right)}} \]

       4. distribute-rgt-neg-in 50.2%

          \[\leadsto -e^{\color{blue}{-\log \left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right) \cdot 2}} \]

       5. exp-neg 49.2%

          \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right) \cdot 2}}} \]

       6. exp-to-pow 51.8%

          \[\leadsto -\frac{1}{\color{blue}{{\left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{2}}} \]

       7. unpow2 51.8%

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

       8. hypot-undefine 51.8%

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

       9. rem-square-sqrt 51.8%

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

       10. fma-undefine 51.8%

           \[\leadsto -\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \cdot \mathsf{hypot}\left(x, \sqrt{x}\right)} \]

       11. hypot-undefine 51.8%

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

       12. rem-square-sqrt 98.1%

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

       13. fma-undefine 98.1%

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

       14. rem-square-sqrt 98.1%

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

       15. fma-undefine 98.1%

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

       16. *-lft-identity 98.1%

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

       17. distribute-rgt-in 98.1%

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

       18. associate-/r* 99.8%

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

       19. unpow-1 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around inf 97.7%

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

       1. neg-mul-1 97.7%

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

   15. Simplified 97.7%

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

   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 98.1%

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

      1. neg-mul-1 98.1%

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

      2. sub-neg 98.1%

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

   5. Simplified 98.1%

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

4. Final simplification 97.9%

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

Alternative 3: 98.2% 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 50.9%

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

      1. sub-neg 50.9%

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

      2. +-commutative 50.9%

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

      3. distribute-neg-frac 50.9%

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

      4. metadata-eval 50.9%

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

   4. Applied egg-rr 50.9%

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

      1. *-rgt-identity 50.9%

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

      2. cancel-sign-sub 50.9%

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

      3. distribute-neg-frac 50.9%

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

      4. metadata-eval 50.9%

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

      5. *-rgt-identity 50.9%

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

      6. *-inverses 50.9%

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

      7. associate-/r* 9.7%

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

      8. *-commutative 9.7%

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

      9. associate-/r* 50.8%

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

      10. div-sub 50.9%

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

      11. *-inverses 50.9%

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

      12. div-sub 53.5%

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

      13. associate-/l/ 53.5%

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

      14. +-commutative 53.5%

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

      15. associate--r+ 98.1%

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

      16. div-sub 98.1%

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

      17. +-inverses 98.1%

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

      18. div0 98.1%

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

      19. associate-/r* 99.8%

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

      20. neg-sub0 99.8%

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

      21. associate-/r* 98.1%

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

   6. Simplified 98.1%

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

      1. unpow2 98.1%

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

   8. Applied egg-rr 98.1%

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

      1. frac-2neg 98.1%

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

      2. metadata-eval 98.1%

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

      3. inv-pow 98.1%

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

      4. neg-mul-1 98.1%

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

      5. distribute-rgt1-in 98.1%

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

      6. add-sqr-sqrt 51.7%

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

      7. sqrt-prod 75.7%

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

      8. sqr-neg 75.7%

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

      9. sqrt-unprod 23.8%

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

      10. add-sqr-sqrt 46.8%

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

      11. distribute-rgt-neg-in 46.8%

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

      12. distribute-rgt1-in 46.8%

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

      13. distribute-neg-in 46.8%

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

      14. add-sqr-sqrt 23.8%

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

      15. sqrt-unprod 2.3%

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

      16. sqr-neg 2.3%

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

      17. sqrt-prod 23.0%

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

      18. add-sqr-sqrt 46.8%

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

      19. sub-neg 46.8%

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

      20. unpow-prod-down 46.8%

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

      21. metadata-eval 46.8%

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

   10. Applied egg-rr 53.0%

       \[\leadsto \color{blue}{-1 \cdot {\left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-2}} \]
   11. Step-by-step derivation

       1. neg-mul-1 53.0%

          \[\leadsto \color{blue}{-{\left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-2}} \]

       2. exp-to-pow 50.2%

          \[\leadsto -\color{blue}{e^{\log \left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right) \cdot -2}} \]

       3. metadata-eval 50.2%

          \[\leadsto -e^{\log \left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right) \cdot \color{blue}{\left(-2\right)}} \]

       4. distribute-rgt-neg-in 50.2%

          \[\leadsto -e^{\color{blue}{-\log \left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right) \cdot 2}} \]

       5. exp-neg 49.2%

          \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right) \cdot 2}}} \]

       6. exp-to-pow 51.8%

          \[\leadsto -\frac{1}{\color{blue}{{\left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{2}}} \]

       7. unpow2 51.8%

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

       8. hypot-undefine 51.8%

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

       9. rem-square-sqrt 51.8%

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

       10. fma-undefine 51.8%

           \[\leadsto -\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \cdot \mathsf{hypot}\left(x, \sqrt{x}\right)} \]

       11. hypot-undefine 51.8%

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

       12. rem-square-sqrt 98.1%

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

       13. fma-undefine 98.1%

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

       14. rem-square-sqrt 98.1%

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

       15. fma-undefine 98.1%

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

       16. *-lft-identity 98.1%

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

       17. distribute-rgt-in 98.1%

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

       18. associate-/r* 99.8%

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

       19. unpow-1 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around inf 97.7%

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

       1. neg-mul-1 97.7%

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

   15. Simplified 97.7%

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

   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 97.6%

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

4. Final simplification 97.7%

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

Alternative 4: 75.8% 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 50.9%

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

   3. Taylor expanded in x around inf 47.6%

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

   4. Taylor expanded in x around 0 47.6%

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

   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 71.8%

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

Alternative 5: 75.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.45 \cdot 10^{+102}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 4.45e+102) (/ -1.0 x) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 4.45e+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 <= 4.45d+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 <= 4.45e+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 <= 4.45e+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 <= 4.45e+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 <= 4.45e+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, 4.45e+102], N[(-1.0 / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 4.4499999999999999e102 < x

   1. Initial program 60.2%

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

   3. Taylor expanded in x around inf 57.3%

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

   4. Taylor expanded in x around 0 57.3%

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

   if -1 < x < 4.4499999999999999e102

   1. Initial program 85.2%

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

   3. Taylor expanded in x around 0 82.0%

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

Alternative 6: 99.3% 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 74.7%

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

   1. sub-neg 74.7%

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

   2. +-commutative 74.7%

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

   3. distribute-neg-frac 74.7%

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

   4. metadata-eval 74.7%

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

4. Applied egg-rr 74.7%

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

   1. *-rgt-identity 74.7%

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

   2. cancel-sign-sub 74.7%

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

   3. distribute-neg-frac 74.7%

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

   4. metadata-eval 74.7%

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

   5. *-rgt-identity 74.7%

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

   6. *-inverses 74.7%

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

   7. associate-/r* 53.4%

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

   8. *-commutative 53.4%

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

   9. associate-/r* 74.6%

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

   10. div-sub 74.7%

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

   11. *-inverses 74.7%

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

   12. div-sub 76.0%

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

   13. associate-/l/ 76.0%

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

   14. +-commutative 76.0%

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

   15. associate--r+ 99.0%

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

   16. div-sub 99.0%

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

   17. +-inverses 99.0%

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

   18. div0 99.0%

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

   19. associate-/r* 99.9%

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

   20. neg-sub0 99.9%

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

   21. associate-/r* 99.0%

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

6. Simplified 99.0%

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

   1. unpow2 99.0%

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

8. Applied egg-rr 99.0%

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

Alternative 7: 99.3% 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 74.7%

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

   1. sub-neg 74.7%

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

   2. +-commutative 74.7%

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

   3. distribute-neg-frac 74.7%

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

   4. metadata-eval 74.7%

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

4. Applied egg-rr 74.7%

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

   1. *-rgt-identity 74.7%

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

   2. cancel-sign-sub 74.7%

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

   3. distribute-neg-frac 74.7%

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

   4. metadata-eval 74.7%

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

   5. *-rgt-identity 74.7%

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

   6. *-inverses 74.7%

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

   7. associate-/r* 53.4%

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

   8. *-commutative 53.4%

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

   9. associate-/r* 74.6%

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

   10. div-sub 74.7%

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

   11. *-inverses 74.7%

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

   12. div-sub 76.0%

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

   13. associate-/l/ 76.0%

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

   14. +-commutative 76.0%

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

   15. associate--r+ 99.0%

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

   16. div-sub 99.0%

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

   17. +-inverses 99.0%

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

   18. div0 99.0%

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

   19. associate-/r* 99.9%

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

   20. neg-sub0 99.9%

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

   21. associate-/r* 99.0%

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

6. Simplified 99.0%

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

   1. unpow2 99.0%

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

   2. distribute-rgt1-in 99.0%

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 8: 27.1% 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 74.7%

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

3. Taylor expanded in x around inf 25.6%

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

4. Taylor expanded in x around 0 25.6%

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

Developer Target 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]

Developer Target 2: 99.3% 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]

Reproduce

herbie shell --seed 2024180 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ -1 x) (+ x 1)))

  :alt
  (! :herbie-platform default (/ 1 (* x (- -1 x))))

  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))