Asymptote C

Percentage Accurate: 52.9% → 99.6%

Time: 5.0s

Alternatives: 9

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: 52.9% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 5e-9) (/ (+ -3.0 (/ -1.0 x)) x) t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 5e-9) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (x + 1.0d0)) + (((-1.0d0) - x) / (x + (-1.0d0)))
    if (t_0 <= 5d-9) then
        tmp = ((-3.0d0) + ((-1.0d0) / x)) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 5e-9) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))
	tmp = 0
	if t_0 <= 5e-9:
		tmp = (-3.0 + (-1.0 / x)) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 5e-9)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 5e-9)
		tmp = (-3.0 + (-1.0 / x)) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-9], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.0000000000000001e-9

   1. Initial program 6.2%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. distribute-neg-in 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. metadata-eval 100.0%

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

      7. unpow2 100.0%

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

      8. associate-/r* 100.0%

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

   4. Simplified 100.0%

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

      1. sub-div 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 5.0000000000000001e-9 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.3%

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

   1. frac-sub 50.5%

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

   2. associate-/r* 50.5%

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

   3. sub-neg 50.5%

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

   4. distribute-rgt-in 50.5%

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

   5. metadata-eval 50.5%

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

   6. neg-mul-1 50.5%

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

   7. fma-def 50.5%

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

   8. fma-neg 50.5%

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

   9. pow2 50.5%

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

   10. sub-neg 50.5%

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

   11. metadata-eval 50.5%

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

3. Applied egg-rr 50.5%

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

4. Taylor expanded in x around 0 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 98.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / 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
    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;
	} 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
	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(-3.0 / 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;
	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[(-3.0 / 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 7.5%

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

   2. Taylor expanded in x around inf 98.7%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.4%

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

      1. unpow2 99.4%

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

      2. distribute-rgt-out 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 99.0%

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

Alternative 4: 99.0% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ (+ -3.0 (/ -1.0 x)) 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 + (-1.0 / x)) / 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) + ((-1.0d0) / x)) / 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 + (-1.0 / x)) / 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 + (-1.0 / x)) / 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(Float64(-3.0 + Float64(-1.0 / x)) / 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 + (-1.0 / x)) / 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[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / 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 3 regimes

2. if x < -1

   1. Initial program 9.6%

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

   2. Taylor expanded in x around inf 98.2%

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

      1. distribute-neg-in 98.2%

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

      2. unsub-neg 98.2%

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

      3. associate-*r/ 98.7%

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

      4. metadata-eval 98.7%

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

      5. distribute-neg-frac 98.7%

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

      6. metadata-eval 98.7%

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

      7. unpow2 98.7%

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

      8. associate-/r* 98.7%

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

   4. Simplified 98.7%

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

      1. sub-div 98.7%

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

   6. Applied egg-rr 98.7%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.4%

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

      1. unpow2 99.4%

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

      2. distribute-rgt-out 99.4%

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

   4. Simplified 99.4%

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

   if 1 < x

   1. Initial program 5.4%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.4%

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

Alternative 5: 99.0% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (+ (/ -3.0 x) (/ (/ -1.0 x) 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) + ((-1.0 / x) / 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) + (((-1.0d0) / x) / 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) + ((-1.0 / x) / 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) + ((-1.0 / x) / 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(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / 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) + ((-1.0 / x) / 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[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $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 3 regimes

2. if x < -1

   1. Initial program 9.6%

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

   2. Taylor expanded in x around inf 98.2%

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

      1. distribute-neg-in 98.2%

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

      2. unsub-neg 98.2%

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

      3. associate-*r/ 98.7%

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

      4. metadata-eval 98.7%

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

      5. distribute-neg-frac 98.7%

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

      6. metadata-eval 98.7%

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

      7. unpow2 98.7%

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

      8. associate-/r* 98.7%

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

   4. Simplified 98.7%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.4%

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

      1. unpow2 99.4%

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

      2. distribute-rgt-out 99.4%

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

   4. Simplified 99.4%

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

   if 1 < x

   1. Initial program 5.4%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.4%

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

Alternative 6: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ (+ -3.0 (/ 2.0 x)) (+ x -1.0))
   (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 + (2.0 / x)) / (x + -1.0);
	} 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) + (2.0d0 / x)) / (x + (-1.0d0))
    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 + (2.0 / x)) / (x + -1.0);
	} 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 + (2.0 / x)) / (x + -1.0)
	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(Float64(-3.0 + Float64(2.0 / x)) / Float64(x + -1.0));
	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 + (2.0 / x)) / (x + -1.0);
	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[(N[(-3.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $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 3 regimes

2. if x < -1

   1. Initial program 9.6%

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

      1. frac-sub 6.0%

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

      2. associate-/r* 6.0%

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

      3. sub-neg 6.0%

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

      4. distribute-rgt-in 6.0%

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

      5. metadata-eval 6.0%

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

      6. neg-mul-1 6.0%

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

      7. fma-def 6.0%

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

      8. fma-neg 6.0%

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

      9. pow2 6.0%

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

      10. sub-neg 6.0%

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

      11. metadata-eval 6.0%

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

   3. Applied egg-rr 6.0%

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

   4. Taylor expanded in x around inf 98.7%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{x} - 3}}{x + -1} \]
   5. Step-by-step derivation

      1. sub-neg 98.7%

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

      2. associate-*r/ 98.7%

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

      3. metadata-eval 98.7%

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

      4. metadata-eval 98.7%

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

   6. Simplified 98.7%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.4%

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

      1. unpow2 99.4%

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

      2. distribute-rgt-out 99.4%

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

   4. Simplified 99.4%

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

   if 1 < x

   1. Initial program 5.4%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 7: 98.6% 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 7.5%

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

   2. Taylor expanded in x around inf 98.7%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 98.7%

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

4. Final simplification 98.7%

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

Alternative 8: 98.1% 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 7.5%

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

   2. Taylor expanded in x around inf 98.7%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 96.6%

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

4. Final simplification 97.7%

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

Alternative 9: 49.6% 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 52.3%

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

2. Taylor expanded in x around 0 48.7%

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

3. Final simplification 48.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023278 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))