Asymptote C

Percentage Accurate: 54.6% → 99.6%

Time: 9.3s

Alternatives: 9

Speedup: 1.0×

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: 54.6% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{x + 1}{x + -1}} + \left(-1 - x\right)}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 5e-8)
   (/ (+ -3.0 (/ -1.0 x)) x)
   (/ (+ (/ x (/ (+ x 1.0) (+ x -1.0))) (- -1.0 x)) (+ x -1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 5e-8) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = ((x / ((x + 1.0) / (x + -1.0))) + (-1.0 - x)) / (x + -1.0);
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 5e-8) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = ((x / ((x + 1.0) / (x + -1.0))) + (-1.0 - x)) / (x + -1.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 5e-8:
		tmp = (-3.0 + (-1.0 / x)) / x
	else:
		tmp = ((x / ((x + 1.0) / (x + -1.0))) + (-1.0 - x)) / (x + -1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 5e-8)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = Float64(Float64(Float64(x / Float64(Float64(x + 1.0) / Float64(x + -1.0))) + Float64(-1.0 - x)) / Float64(x + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 5e-8)
		tmp = (-3.0 + (-1.0 / x)) / x;
	else
		tmp = ((x / ((x + 1.0) / (x + -1.0))) + (-1.0 - x)) / (x + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(x / N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 4.9999999999999998e-8

   1. Initial program 7.7%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{3 + \frac{1}{x}}{x}\right) \]

      2. distribute-neg-frac N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)\right), \color{blue}{x}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(3\right)\right) + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-3 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{-1}{x}\right)\right), x\right) \]

      9. /-lowering-/.f64 99.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(-1, x\right)\right), x\right) \]

   5. Simplified 99.2%

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

   if 4.9999999999999998e-8 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.9%

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

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{1}{\color{blue}{\frac{x - 1}{x + 1}}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x - 1}{x + 1}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x - 1\right), \color{blue}{\left(x + 1\right)}\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{x} + 1\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{x} + 1\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + 1\right)\right)\right)\right) \]

      7. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

      1. frac-sub N/A

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

      2. metadata-eval N/A

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

      3. sub-neg N/A

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

      4. associate-*r/ N/A

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

      5. difference-of-sqr-1 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. flip-- N/A

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

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{x + -1}{x + 1} - \left(x + 1\right) \cdot 1\right), \color{blue}{\left(x - 1\right)}\right) \]

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{x + 1}{x + -1}} + \left(-1 - x\right)}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 5e-8) (/ (+ -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-8) {
		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-8) 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-8) {
		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-8:
		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-8)
		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-8)
		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-8], 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 #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 4.9999999999999998e-8

   1. Initial program 7.7%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{3 + \frac{1}{x}}{x}\right) \]

      2. distribute-neg-frac N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)\right), \color{blue}{x}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(3\right)\right) + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-3 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{-1}{x}\right)\right), x\right) \]

      9. /-lowering-/.f64 99.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(-1, x\right)\right), x\right) \]

   5. Simplified 99.2%

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

   if 4.9999999999999998e-8 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(1 + x \cdot 3\right) \cdot \left(1 + x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double t_0 = (-3.0 + (-1.0 / x)) / x;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (1.0 + (x * 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 = ((-3.0d0) + ((-1.0d0) / x)) / x
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = (1.0d0 + (x * 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 = (-3.0 + (-1.0 / x)) / x;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (1.0 + (x * 3.0)) * (1.0 + (x * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (-3.0 + (-1.0 / x)) / x
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 1.0:
		tmp = (1.0 + (x * 3.0)) * (1.0 + (x * x))
	else:
		tmp = t_0
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 7.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{3 + \frac{1}{x}}{x}\right) \]

      2. distribute-neg-frac N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)\right), \color{blue}{x}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(3\right)\right) + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-3 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{-1}{x}\right)\right), x\right) \]

      9. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(-1, x\right)\right), x\right) \]

   5. Simplified 100.0%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + 3 \cdot x\right), \color{blue}{\left({x}^{2} + 1\right)}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(3 \cdot x\right)\right), \left(\color{blue}{{x}^{2}} + 1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \left({x}^{\color{blue}{2}} + 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \left(1 + \color{blue}{{x}^{2}}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      14. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 98.2%

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

4. Final simplification 99.0%

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

Alternative 4: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double t_0 = (-3.0 + (-1.0 / x)) / x;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * (x + 3.0));
	} 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 = ((-3.0d0) + ((-1.0d0) / x)) / x
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 + (x * (x + 3.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	t_0 = (-3.0 + (-1.0 / x)) / x
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0 + (x * (x + 3.0))
	else:
		tmp = t_0
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 7.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{3 + \frac{1}{x}}{x}\right) \]

      2. distribute-neg-frac N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)\right), \color{blue}{x}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(3\right)\right) + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-3 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{-1}{x}\right)\right), x\right) \]

      9. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(-1, x\right)\right), x\right) \]

   5. Simplified 100.0%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(3 + x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(3 + x\right)}\right)\right) \]

      3. +-lowering-+.f64 97.6%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 97.6%

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

4. Final simplification 98.7%

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

Alternative 5: 98.6% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ 1.0 (+ 0.1111111111111111 (* x -0.3333333333333333)))
   (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 = 1.0 / (0.1111111111111111 + (x * -0.3333333333333333));
	} 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 = 1.0d0 / (0.1111111111111111d0 + (x * (-0.3333333333333333d0)))
    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 = 1.0 / (0.1111111111111111 + (x * -0.3333333333333333));
	} 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 = 1.0 / (0.1111111111111111 + (x * -0.3333333333333333))
	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(1.0 / Float64(0.1111111111111111 + Float64(x * -0.3333333333333333)));
	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 = 1.0 / (0.1111111111111111 + (x * -0.3333333333333333));
	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[(1.0 / N[(0.1111111111111111 + N[(x * -0.3333333333333333), $MachinePrecision]), $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 7.8%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{3 + \frac{1}{x}}{x}\right) \]

      2. distribute-neg-frac N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)\right), \color{blue}{x}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(3\right)\right) + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-3 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \left(\frac{-1}{x}\right)\right), x\right) \]

      9. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(-1, x\right)\right), x\right) \]

   5. Simplified 99.9%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{-3 + \frac{-1}{x}}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left(-3 + \frac{-1}{x}\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(-3, \color{blue}{\left(\frac{-1}{x}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 99.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right)\right)\right) \]

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{9} \cdot \frac{1}{x} - \frac{1}{3}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\frac{1}{9} \cdot \frac{1}{x} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\frac{1}{9} \cdot \frac{1}{x} + \frac{-1}{3}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{1}{9} \cdot \frac{1}{x}\right) \cdot x + \color{blue}{\frac{-1}{3} \cdot x}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{9} \cdot \left(\frac{1}{x} \cdot x\right) + \color{blue}{\frac{-1}{3}} \cdot x\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{9} \cdot 1 + \frac{-1}{3} \cdot x\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{9} + \color{blue}{\frac{-1}{3}} \cdot x\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{9} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot x\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{9} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot x\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{9} + \left(\mathsf{neg}\left(x \cdot \frac{1}{3}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{9}, \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{3}\right)\right)}\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{9}, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{9}, \left(x \cdot \frac{-1}{3}\right)\right)\right) \]

      13. *-lowering-*.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{3}}\right)\right)\right) \]

   10. Simplified 99.6%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(3 + x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(3 + x\right)}\right)\right) \]

      3. +-lowering-+.f64 97.6%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 97.6%

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

   if 1 < x

   1. Initial program 6.2%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 99.5%

         \[\leadsto \mathsf{/.f64}\left(-3, \color{blue}{x}\right) \]

   5. Simplified 99.5%

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

4. Final simplification 98.5%

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

Alternative 6: 98.5% accurate, 0.8× 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 7.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 99.1%

         \[\leadsto \mathsf{/.f64}\left(-3, \color{blue}{x}\right) \]

   5. Simplified 99.1%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(3 + x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(3 + x\right)}\right)\right) \]

      3. +-lowering-+.f64 97.6%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(3, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 97.6%

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

4. Final simplification 98.3%

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

Alternative 7: 98.4% accurate, 0.9× 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 7.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 99.1%

         \[\leadsto \mathsf{/.f64}\left(-3, \color{blue}{x}\right) \]

   5. Simplified 99.1%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(3 \cdot x\right)}\right) \]

      2. *-lowering-*.f64 97.3%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(3, \color{blue}{x}\right)\right) \]

   5. Simplified 97.3%

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

4. Final simplification 98.2%

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

Alternative 8: 97.7% accurate, 1.0× 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 7.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 99.1%

         \[\leadsto \mathsf{/.f64}\left(-3, \color{blue}{x}\right) \]

   5. Simplified 99.1%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 96.0%

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

Alternative 9: 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 55.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 52.0%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))