Asymptote C

Percentage Accurate: 54.4% → 100.0%

Time: 7.2s

Alternatives: 7

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: 54.4% 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 51.2%

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

   1. clear-num 51.1%

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

   2. frac-sub 52.0%

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

   3. *-un-lft-identity 52.0%

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

   4. sub-neg 52.0%

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

   5. metadata-eval 52.0%

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

   6. sub-neg 52.0%

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

   7. metadata-eval 52.0%

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

3. Applied egg-rr 52.0%

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

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. distribute-neg-frac 100.0%

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

   4. metadata-eval 100.0%

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

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: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{x \cdot -0.3333333333333333 + 0.1111111111111111}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (/ 1.0 (+ (* x -0.3333333333333333) 0.1111111111111111))
   (+ 1.0 (* x (+ x 3.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = 1.0 / ((x * -0.3333333333333333) + 0.1111111111111111);
	} else {
		tmp = 1.0 + (x * (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 = 1.0d0 / ((x * (-0.3333333333333333d0)) + 0.1111111111111111d0)
    else
        tmp = 1.0d0 + (x * (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 = 1.0 / ((x * -0.3333333333333333) + 0.1111111111111111);
	} else {
		tmp = 1.0 + (x * (x + 3.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = 1.0 / ((x * -0.3333333333333333) + 0.1111111111111111)
	else:
		tmp = 1.0 + (x * (x + 3.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(1.0 / Float64(Float64(x * -0.3333333333333333) + 0.1111111111111111));
	else
		tmp = Float64(1.0 + Float64(x * 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 = 1.0 / ((x * -0.3333333333333333) + 0.1111111111111111);
	else
		tmp = 1.0 + (x * (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[(1.0 / N[(N[(x * -0.3333333333333333), $MachinePrecision] + 0.1111111111111111), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.1%

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

   2. Taylor expanded in x around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-neg-in 99.4%

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

      3. sub-neg 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. unsub-neg 99.4%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. distribute-neg-frac 99.9%

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

      9. metadata-eval 99.9%

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

      10. unpow2 99.9%

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

      11. associate-/r* 99.9%

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

      12. associate-*r/ 99.9%

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

      13. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.2%

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

      1. unpow2 99.2%

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

      2. distribute-neg-in 99.2%

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

      3. unsub-neg 99.2%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

      6. distribute-neg-frac 99.6%

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

      7. metadata-eval 99.6%

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

      8. associate-/r* 99.6%

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

      9. div-sub 99.7%

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

      10. sub-neg 99.7%

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

      11. metadata-eval 99.7%

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

      12. distribute-neg-in 99.7%

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

      13. distribute-neg-in 99.7%

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

      14. metadata-eval 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

   7. Simplified 99.7%

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

      1. clear-num 99.4%

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

      2. inv-pow 99.4%

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

   9. Applied egg-rr 99.4%

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

       1. unpow-1 99.4%

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

       2. metadata-eval 99.4%

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

       3. distribute-neg-frac 99.4%

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

       4. unsub-neg 99.4%

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

   11. Simplified 99.4%

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

   12. Taylor expanded in x around inf 99.1%

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

       1. +-commutative 99.1%

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

       2. *-commutative 99.1%

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

   14. Simplified 99.1%

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

   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 99.9%

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

      1. unpow2 99.9%

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

      2. distribute-rgt-out 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{x \cdot -0.3333333333333333 + 0.1111111111111111}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array} \]

Alternative 3: 99.1% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (/ (- -3.0 (/ 1.0 x)) x)
   (+ 1.0 (* x (+ 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 * (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 * (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 * (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 * (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 * 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 * (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 * N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.1%

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

   2. Taylor expanded in x around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-neg-in 99.4%

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

      3. sub-neg 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. unsub-neg 99.4%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. distribute-neg-frac 99.9%

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

      9. metadata-eval 99.9%

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

      10. unpow2 99.9%

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

      11. associate-/r* 99.9%

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

      12. associate-*r/ 99.9%

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

      13. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.2%

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

      1. unpow2 99.2%

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

      2. distribute-neg-in 99.2%

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

      3. unsub-neg 99.2%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

      6. distribute-neg-frac 99.6%

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

      7. metadata-eval 99.6%

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

      8. associate-/r* 99.6%

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

      9. div-sub 99.7%

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

      10. sub-neg 99.7%

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

      11. metadata-eval 99.7%

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

      12. distribute-neg-in 99.7%

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

      13. distribute-neg-in 99.7%

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

      14. metadata-eval 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

   7. Simplified 99.7%

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

   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 99.9%

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

      1. unpow2 99.9%

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

      2. distribute-rgt-out 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.8%

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

Alternative 4: 98.7% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ -3.0 x)
   (if (<= x 1.0) (+ 1.0 (* x (+ 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 * (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 * (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 * (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 * (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 * 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 * (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 * N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-3.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.1%

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

   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 99.9%

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

      1. unpow2 99.9%

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

      2. distribute-rgt-out 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.2%

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

Alternative 5: 98.5% 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 8.1%

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

   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 99.5%

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

4. Final simplification 99.0%

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

Alternative 6: 97.9% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) (/ -3.0 x) (if (<= x 1.0) 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 = 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 = 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 = 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 = 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 = 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 = 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], 1.0, N[(-3.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.1%

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

   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 98.5%

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

4. Final simplification 98.5%

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

Alternative 7: 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 51.2%

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

2. Taylor expanded in x around 0 48.3%

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

3. Final simplification 48.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023290 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))