Asymptote C

Percentage Accurate: 54.2% → 100.0%

Time: 5.3s

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: 54.2% 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 51.8%

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

   1. clear-num 51.8%

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

   2. frac-sub 52.5%

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

   3. *-un-lft-identity 52.5%

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

   4. sub-neg 52.5%

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

   5. metadata-eval 52.5%

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

   6. sub-neg 52.5%

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

   7. metadata-eval 52.5%

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

3. Applied egg-rr 52.5%

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

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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (+ (/ -3.0 x) (/ (/ -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 / x) + ((-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) / x) + (((-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 / x) + ((-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 / x) + ((-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 / x) + Float64(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 / x) + ((-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 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $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 7.9%

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

   2. Taylor expanded in x around inf 98.7%

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

      1. distribute-neg-in 98.7%

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

      2. associate-*r/ 99.2%

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

      3. metadata-eval 99.2%

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

      4. distribute-neg-frac 99.2%

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

      5. metadata-eval 99.2%

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

      6. unpow2 99.2%

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

      7. associate-/r* 99.2%

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

      8. distribute-neg-frac 99.2%

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

      9. distribute-neg-frac 99.2%

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

      10. metadata-eval 99.2%

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

   4. Simplified 99.2%

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

   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 99.1%

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

      1. +-commutative 99.1%

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

      2. unpow2 99.1%

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

      3. distribute-rgt-out 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 99.2%

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

Alternative 3: 98.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = 1.0 / (0.1111111111111111 + (x * -0.3333333333333333));
	} 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 / (0.1111111111111111d0 + (x * (-0.3333333333333333d0)))
    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 / (0.1111111111111111 + (x * -0.3333333333333333));
	} 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 / (0.1111111111111111 + (x * -0.3333333333333333))
	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(0.1111111111111111 + Float64(x * -0.3333333333333333)));
	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 / (0.1111111111111111 + (x * -0.3333333333333333));
	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[(0.1111111111111111 + N[(x * -0.3333333333333333), $MachinePrecision]), $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 7.9%

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

      1. clear-num 8.0%

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

      2. frac-sub 9.2%

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

      3. *-un-lft-identity 9.2%

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

      4. sub-neg 9.2%

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

      5. metadata-eval 9.2%

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

      6. sub-neg 9.2%

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

      7. metadata-eval 9.2%

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

   3. Applied egg-rr 9.2%

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

      1. clear-num 9.2%

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

      2. inv-pow 9.2%

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

   5. Applied egg-rr 7.3%

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

      1. unpow-1 7.3%

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

      2. *-commutative 7.3%

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

      3. *-lft-identity 7.3%

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

      4. times-frac 7.4%

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

      5. associate-/r* 7.4%

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

      6. *-rgt-identity 7.4%

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

      7. div-sub 7.4%

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

      8. sub-neg 7.4%

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

      9. *-inverses 7.4%

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

      10. distribute-neg-frac 7.4%

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

      11. metadata-eval 7.4%

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

      12. sub-neg 7.4%

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

      13. metadata-eval 7.4%

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

   7. Simplified 7.4%

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

   8. Taylor expanded in x around inf 98.7%

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

      1. *-commutative 98.7%

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

   10. Simplified 98.7%

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

   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 99.1%

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

      1. +-commutative 99.1%

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

      2. unpow2 99.1%

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

      3. distribute-rgt-out 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 98.9%

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

Alternative 4: 98.6% 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 7.9%

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

   2. Taylor expanded in x around inf 98.2%

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

   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 99.1%

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

      1. +-commutative 99.1%

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

      2. unpow2 99.1%

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

      3. distribute-rgt-out 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 98.7%

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

Alternative 5: 98.4% 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 7.9%

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

   2. Taylor expanded in x around inf 98.2%

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

   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 98.9%

      \[\leadsto \color{blue}{1 + 3 \cdot x} \]
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 6: 97.8% 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 7.9%

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

   2. Taylor expanded in x around inf 98.2%

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

   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 98.1%

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

4. Final simplification 98.2%

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

Alternative 7: 50.6% 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 51.8%

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

2. Taylor expanded in x around 0 48.7%

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

3. Final simplification 48.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023268 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))