Asymptote C

Percentage Accurate: 54.2% → 100.0%

Time: 5.9s

Alternatives: 8

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{-3 - \frac{1}{x}}{x - \frac{1}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- -3.0 (/ 1.0 x)) (- x (/ 1.0 x))))

C

double code(double x) {
	return (-3.0 - (1.0 / x)) / (x - (1.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-3.0d0) - (1.0d0 / x)) / (x - (1.0d0 / x))
end function

Java

public static double code(double x) {
	return (-3.0 - (1.0 / x)) / (x - (1.0 / x));
}

Python

def code(x):
	return (-3.0 - (1.0 / x)) / (x - (1.0 / x))

Julia

function code(x)
	return Float64(Float64(-3.0 - Float64(1.0 / x)) / Float64(x - Float64(1.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = (-3.0 - (1.0 / x)) / (x - (1.0 / x));
end

Wolfram

code[x_] := N[(N[(-3.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.3%

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

   1. clear-num 52.3%

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

   2. frac-sub 52.6%

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

   3. *-un-lft-identity 52.6%

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

   4. sub-neg 52.6%

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

   5. metadata-eval 52.6%

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

   6. sub-neg 52.6%

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

   7. metadata-eval 52.6%

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

3. Applied egg-rr 52.6%

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

4. Taylor expanded in x around 0 100.0%

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

   1. distribute-neg-in 100.0%

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

   2. metadata-eval 100.0%

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

   3. unsub-neg 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in x around 0 100.0%

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

8. Final simplification 100.0%

   \[\leadsto \frac{-3 - \frac{1}{x}}{x - \frac{1}{x}} \]

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.7\right):\\ \;\;\;\;\frac{-3 - \frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1 + x}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.7)))
   (/ (- -3.0 (/ 1.0 x)) x)
   (- x (/ (+ 1.0 x) (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.7)) {
		tmp = (-3.0 - (1.0 / x)) / x;
	} else {
		tmp = x - ((1.0 + x) / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.7d0))) then
        tmp = ((-3.0d0) - (1.0d0 / x)) / x
    else
        tmp = x - ((1.0d0 + x) / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.7)) {
		tmp = (-3.0 - (1.0 / x)) / x;
	} else {
		tmp = x - ((1.0 + x) / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.7):
		tmp = (-3.0 - (1.0 / x)) / x
	else:
		tmp = x - ((1.0 + x) / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.7))
		tmp = Float64(Float64(-3.0 - Float64(1.0 / x)) / x);
	else
		tmp = Float64(x - Float64(Float64(1.0 + x) / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.7)))
		tmp = (-3.0 - (1.0 / x)) / x;
	else
		tmp = x - ((1.0 + x) / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.7]], $MachinePrecision]], N[(N[(-3.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x - N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.69999999999999996 < x

   1. Initial program 7.6%

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

   2. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{-\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 98.7%

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

      2. distribute-neg-in 98.7%

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

      3. sub-neg 98.7%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

      6. distribute-neg-frac 99.1%

         \[\leadsto \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]

      7. metadata-eval 99.1%

         \[\leadsto \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]

      8. unpow2 99.1%

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

      9. associate-/r* 99.1%

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

   4. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{\frac{1}{x}}{x}} \]
   5. Step-by-step derivation

      1. sub-neg 99.1%

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

      2. inv-pow 99.1%

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

      3. pow1 99.1%

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

      4. pow-div 99.1%

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

      5. metadata-eval 99.1%

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

   6. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{-3}{x} + \left(-{x}^{-2}\right)} \]
   7. Step-by-step derivation

      1. sub-neg 99.1%

         \[\leadsto \color{blue}{\frac{-3}{x} - {x}^{-2}} \]

      2. metadata-eval 99.1%

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

      3. pow-sqr 99.1%

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

      4. unpow-1 99.1%

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

      5. unpow-1 99.1%

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

      6. cancel-sign-sub-inv 99.1%

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

      7. metadata-eval 99.1%

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

      8. distribute-neg-frac 99.1%

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

      9. metadata-eval 99.1%

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

      10. associate-*r/ 98.7%

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

      11. *-commutative 98.7%

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

      12. distribute-rgt-neg-in 98.7%

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

      13. metadata-eval 98.7%

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

      14. distribute-lft-neg-in 98.7%

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

      15. distribute-rgt-neg-in 98.7%

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

      16. distribute-neg-frac 98.7%

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

      17. metadata-eval 98.7%

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

      18. distribute-lft-in 98.8%

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

      19. *-commutative 98.8%

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

      20. associate-*r/ 99.1%

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

      21. *-rgt-identity 99.1%

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

      22. metadata-eval 99.1%

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

      23. distribute-neg-frac 99.1%

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

      24. unsub-neg 99.1%

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

   8. Simplified 99.1%

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

   if -1 < x < 1.69999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.7\right):\\ \;\;\;\;\frac{-3 - \frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1 + x}{x + -1}\\ \end{array} \]

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 1.7:\\ \;\;\;\;x - \frac{1 + x}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 - \frac{1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (+ (/ -3.0 x) (/ (/ -1.0 x) x))
   (if (<= x 1.7) (- x (/ (+ 1.0 x) (+ x -1.0))) (/ (- -3.0 (/ 1.0 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else if (x <= 1.7) {
		tmp = x - ((1.0 + x) / (x + -1.0));
	} else {
		tmp = (-3.0 - (1.0 / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    else if (x <= 1.7d0) then
        tmp = x - ((1.0d0 + x) / (x + (-1.0d0)))
    else
        tmp = ((-3.0d0) - (1.0d0 / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else if (x <= 1.7) {
		tmp = x - ((1.0 + x) / (x + -1.0));
	} else {
		tmp = (-3.0 - (1.0 / x)) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	elif x <= 1.7:
		tmp = x - ((1.0 + x) / (x + -1.0))
	else:
		tmp = (-3.0 - (1.0 / x)) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x));
	elseif (x <= 1.7)
		tmp = Float64(x - Float64(Float64(1.0 + x) / Float64(x + -1.0)));
	else
		tmp = Float64(Float64(-3.0 - Float64(1.0 / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	elseif (x <= 1.7)
		tmp = x - ((1.0 + x) / (x + -1.0));
	else
		tmp = (-3.0 - (1.0 / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.7], N[(x - N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-3.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 6.1%

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

   2. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{-\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 99.6%

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

      2. distribute-neg-in 99.6%

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

      3. sub-neg 99.6%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-frac 100.0%

         \[\leadsto \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]

      8. unpow2 100.0%

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

      9. associate-/r* 100.0%

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

   4. Simplified 100.0%

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

   if -1 < x < 1.69999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.7%

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

   if 1.69999999999999996 < x

   1. Initial program 8.8%

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

   2. Taylor expanded in x around inf 98.0%

      \[\leadsto \color{blue}{-\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 98.0%

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

      2. distribute-neg-in 98.0%

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

      3. sub-neg 98.0%

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

      4. associate-*r/ 98.4%

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

      5. metadata-eval 98.4%

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

      6. distribute-neg-frac 98.4%

         \[\leadsto \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]

      7. metadata-eval 98.4%

         \[\leadsto \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]

      8. unpow2 98.4%

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

      9. associate-/r* 98.4%

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

   4. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{\frac{1}{x}}{x}} \]
   5. Step-by-step derivation

      1. sub-neg 98.4%

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

      2. inv-pow 98.4%

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

      3. pow1 98.4%

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

      4. pow-div 98.4%

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

      5. metadata-eval 98.4%

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

   6. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\frac{-3}{x} + \left(-{x}^{-2}\right)} \]
   7. Step-by-step derivation

      1. sub-neg 98.4%

         \[\leadsto \color{blue}{\frac{-3}{x} - {x}^{-2}} \]

      2. metadata-eval 98.4%

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

      3. pow-sqr 98.4%

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

      4. unpow-1 98.4%

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

      5. unpow-1 98.4%

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

      6. cancel-sign-sub-inv 98.4%

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

      7. metadata-eval 98.4%

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

      8. distribute-neg-frac 98.4%

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

      9. metadata-eval 98.4%

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

      10. associate-*r/ 98.0%

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

      11. *-commutative 98.0%

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

      12. distribute-rgt-neg-in 98.0%

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

      13. metadata-eval 98.0%

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

      14. distribute-lft-neg-in 98.0%

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

      15. distribute-rgt-neg-in 98.0%

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

      16. distribute-neg-frac 98.0%

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

      17. metadata-eval 98.0%

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

      18. distribute-lft-in 98.0%

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

      19. *-commutative 98.0%

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

      20. associate-*r/ 98.4%

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

      21. *-rgt-identity 98.4%

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

      22. metadata-eval 98.4%

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

      23. distribute-neg-frac 98.4%

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

      24. unsub-neg 98.4%

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

   8. Simplified 98.4%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 1.7:\\ \;\;\;\;x - \frac{1 + x}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 - \frac{1}{x}}{x}\\ \end{array} \]

Alternative 4: 99.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3 - \frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (/ (- -3.0 (/ 1.0 x)) x)
   (+ 1.0 (* x 3.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-3.0 - (1.0 / x)) / x;
	} else {
		tmp = 1.0 + (x * 3.0);
	}
	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 = ((-3.0d0) - (1.0d0 / x)) / x
    else
        tmp = 1.0d0 + (x * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-3.0 - (1.0 / x)) / x;
	} else {
		tmp = 1.0 + (x * 3.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = (-3.0 - (1.0 / x)) / x
	else:
		tmp = 1.0 + (x * 3.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(-3.0 - Float64(1.0 / x)) / x);
	else
		tmp = Float64(1.0 + Float64(x * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = (-3.0 - (1.0 / x)) / x;
	else
		tmp = 1.0 + (x * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(-3.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 + N[(x * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 7.6%

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

   2. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{-\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 98.7%

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

      2. distribute-neg-in 98.7%

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

      3. sub-neg 98.7%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

      6. distribute-neg-frac 99.1%

         \[\leadsto \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]

      7. metadata-eval 99.1%

         \[\leadsto \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]

      8. unpow2 99.1%

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

      9. associate-/r* 99.1%

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

   4. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{\frac{1}{x}}{x}} \]
   5. Step-by-step derivation

      1. sub-neg 99.1%

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

      2. inv-pow 99.1%

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

      3. pow1 99.1%

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

      4. pow-div 99.1%

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

      5. metadata-eval 99.1%

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

   6. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{-3}{x} + \left(-{x}^{-2}\right)} \]
   7. Step-by-step derivation

      1. sub-neg 99.1%

         \[\leadsto \color{blue}{\frac{-3}{x} - {x}^{-2}} \]

      2. metadata-eval 99.1%

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

      3. pow-sqr 99.1%

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

      4. unpow-1 99.1%

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

      5. unpow-1 99.1%

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

      6. cancel-sign-sub-inv 99.1%

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

      7. metadata-eval 99.1%

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

      8. distribute-neg-frac 99.1%

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

      9. metadata-eval 99.1%

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

      10. associate-*r/ 98.7%

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

      11. *-commutative 98.7%

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

      12. distribute-rgt-neg-in 98.7%

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

      13. metadata-eval 98.7%

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

      14. distribute-lft-neg-in 98.7%

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

      15. distribute-rgt-neg-in 98.7%

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

      16. distribute-neg-frac 98.7%

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

      17. metadata-eval 98.7%

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

      18. distribute-lft-in 98.8%

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

      19. *-commutative 98.8%

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

      20. associate-*r/ 99.1%

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

      21. *-rgt-identity 99.1%

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

      22. metadata-eval 99.1%

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

      23. distribute-neg-frac 99.1%

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

      24. unsub-neg 99.1%

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

   8. Simplified 99.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3 - \frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 3\\ \end{array} \]

Alternative 5: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot -0.3333333333333333 + 0.1111111111111111}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ -3.0 x)
   (if (<= x 1.0)
     (+ 1.0 (* x 3.0))
     (/ 1.0 (+ (* x -0.3333333333333333) 0.1111111111111111)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * 3.0);
	} else {
		tmp = 1.0 / ((x * -0.3333333333333333) + 0.1111111111111111);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (-3.0d0) / x
    else if (x <= 1.0d0) then
        tmp = 1.0d0 + (x * 3.0d0)
    else
        tmp = 1.0d0 / ((x * (-0.3333333333333333d0)) + 0.1111111111111111d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * 3.0);
	} else {
		tmp = 1.0 / ((x * -0.3333333333333333) + 0.1111111111111111);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -3.0 / x
	elif x <= 1.0:
		tmp = 1.0 + (x * 3.0)
	else:
		tmp = 1.0 / ((x * -0.3333333333333333) + 0.1111111111111111)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-3.0 / x);
	elseif (x <= 1.0)
		tmp = Float64(1.0 + Float64(x * 3.0));
	else
		tmp = Float64(1.0 / Float64(Float64(x * -0.3333333333333333) + 0.1111111111111111));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -3.0 / x;
	elseif (x <= 1.0)
		tmp = 1.0 + (x * 3.0);
	else
		tmp = 1.0 / ((x * -0.3333333333333333) + 0.1111111111111111);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 + N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * -0.3333333333333333), $MachinePrecision] + 0.1111111111111111), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 6.1%

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

   2. Taylor expanded in x around inf 99.4%

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.7%

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

   if 1 < x

   1. Initial program 8.8%

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

   2. Taylor expanded in x around inf 98.0%

      \[\leadsto \color{blue}{-\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 98.0%

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

      2. distribute-neg-in 98.0%

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

      3. sub-neg 98.0%

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

      4. associate-*r/ 98.4%

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

      5. metadata-eval 98.4%

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

      6. distribute-neg-frac 98.4%

         \[\leadsto \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]

      7. metadata-eval 98.4%

         \[\leadsto \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]

      8. unpow2 98.4%

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

      9. associate-/r* 98.4%

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

   4. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{\frac{1}{x}}{x}} \]
   5. Step-by-step derivation

      1. sub-div 98.4%

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

      2. clear-num 98.1%

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

      3. sub-neg 98.1%

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

      4. distribute-neg-frac 98.1%

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

      5. metadata-eval 98.1%

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in x around inf 97.9%

      \[\leadsto \frac{1}{\color{blue}{0.1111111111111111 + -0.3333333333333333 \cdot x}} \]
   8. Step-by-step derivation

      1. +-commutative 97.9%

         \[\leadsto \frac{1}{\color{blue}{-0.3333333333333333 \cdot x + 0.1111111111111111}} \]

      2. *-commutative 97.9%

         \[\leadsto \frac{1}{\color{blue}{x \cdot -0.3333333333333333} + 0.1111111111111111} \]

   9. Simplified 97.9%

      \[\leadsto \frac{1}{\color{blue}{x \cdot -0.3333333333333333 + 0.1111111111111111}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot -0.3333333333333333 + 0.1111111111111111}\\ \end{array} \]

Alternative 6: 98.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) (/ -3.0 x) (if (<= x 1.0) (+ 1.0 (* x 3.0)) (/ -3.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * 3.0);
	} else {
		tmp = -3.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (-3.0d0) / x
    else if (x <= 1.0d0) then
        tmp = 1.0d0 + (x * 3.0d0)
    else
        tmp = (-3.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * 3.0);
	} else {
		tmp = -3.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -3.0 / x
	elif x <= 1.0:
		tmp = 1.0 + (x * 3.0)
	else:
		tmp = -3.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-3.0 / x);
	elseif (x <= 1.0)
		tmp = Float64(1.0 + Float64(x * 3.0));
	else
		tmp = Float64(-3.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -3.0 / x;
	elseif (x <= 1.0)
		tmp = 1.0 + (x * 3.0);
	else
		tmp = -3.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 + N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(-3.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 7.6%

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

   2. Taylor expanded in x around inf 98.5%

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 7: 98.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) (/ -3.0 x) (if (<= x 1.0) (- x -1.0) (/ -3.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = x - -1.0;
	} else {
		tmp = -3.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (-3.0d0) / x
    else if (x <= 1.0d0) then
        tmp = x - (-1.0d0)
    else
        tmp = (-3.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = x - -1.0;
	} else {
		tmp = -3.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -3.0 / x
	elif x <= 1.0:
		tmp = x - -1.0
	else:
		tmp = -3.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-3.0 / x);
	elseif (x <= 1.0)
		tmp = Float64(x - -1.0);
	else
		tmp = Float64(-3.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -3.0 / x;
	elseif (x <= 1.0)
		tmp = x - -1.0;
	else
		tmp = -3.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 1.0], N[(x - -1.0), $MachinePrecision], N[(-3.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 7.6%

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

   2. Taylor expanded in x around inf 98.5%

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.7%

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

   3. Taylor expanded in x around 0 98.1%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 8: 51.0% accurate, 13.0× 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 52.3%

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

2. Taylor expanded in x around 0 49.5%

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

3. Final simplification 49.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023178 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))