Asymptote C

Percentage Accurate: 53.8% → 99.7%

Time: 6.7s

Alternatives: 10

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: 53.8% 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: 99.7% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 1e-10)
   (- (+ (/ -1.0 (* x x)) (/ -3.0 x)) (/ 3.0 (pow x 3.0)))
   (/
    (+ (+ x -1.0) (* (+ x 1.0) (/ (- -1.0 x) x)))
    (* (+ x -1.0) (/ (+ x 1.0) x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 1e-10) {
		tmp = ((-1.0 / (x * x)) + (-3.0 / x)) - (3.0 / pow(x, 3.0));
	} else {
		tmp = ((x + -1.0) + ((x + 1.0) * ((-1.0 - x) / x))) / ((x + -1.0) * ((x + 1.0) / x));
	}
	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)))) <= 1d-10) then
        tmp = (((-1.0d0) / (x * x)) + ((-3.0d0) / x)) - (3.0d0 / (x ** 3.0d0))
    else
        tmp = ((x + (-1.0d0)) + ((x + 1.0d0) * (((-1.0d0) - x) / x))) / ((x + (-1.0d0)) * ((x + 1.0d0) / x))
    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))) <= 1e-10) {
		tmp = ((-1.0 / (x * x)) + (-3.0 / x)) - (3.0 / Math.pow(x, 3.0));
	} else {
		tmp = ((x + -1.0) + ((x + 1.0) * ((-1.0 - x) / x))) / ((x + -1.0) * ((x + 1.0) / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 1e-10:
		tmp = ((-1.0 / (x * x)) + (-3.0 / x)) - (3.0 / math.pow(x, 3.0))
	else:
		tmp = ((x + -1.0) + ((x + 1.0) * ((-1.0 - x) / x))) / ((x + -1.0) * ((x + 1.0) / x))
	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))) <= 1e-10)
		tmp = Float64(Float64(Float64(-1.0 / Float64(x * x)) + Float64(-3.0 / x)) - Float64(3.0 / (x ^ 3.0)));
	else
		tmp = Float64(Float64(Float64(x + -1.0) + Float64(Float64(x + 1.0) * Float64(Float64(-1.0 - x) / x))) / Float64(Float64(x + -1.0) * Float64(Float64(x + 1.0) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 1e-10)
		tmp = ((-1.0 / (x * x)) + (-3.0 / x)) - (3.0 / (x ^ 3.0));
	else
		tmp = ((x + -1.0) + ((x + 1.0) * ((-1.0 - x) / x))) / ((x + -1.0) * ((x + 1.0) / x));
	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], 1e-10], N[(N[(N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-3.0 / x), $MachinePrecision]), $MachinePrecision] - N[(3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + -1.0), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[(N[(-1.0 - x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 1.00000000000000004e-10

   1. Initial program 8.1%

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

   2. Taylor expanded in x around inf 99.2%

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

      1. +-commutative 99.2%

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

      2. distribute-neg-in 99.2%

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

      3. sub-neg 99.2%

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

      4. distribute-neg-in 99.2%

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

      5. distribute-neg-frac 99.2%

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

      6. metadata-eval 99.2%

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

      7. unpow2 99.2%

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

      8. associate-*r/ 99.7%

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

      9. metadata-eval 99.7%

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

      10. distribute-neg-frac 99.7%

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

      11. metadata-eval 99.7%

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

      12. associate-*r/ 99.7%

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

      13. metadata-eval 99.7%

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

   4. Simplified 99.7%

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

   if 1.00000000000000004e-10 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. frac-sub 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 1e-10)
   (+ (/ (- -3.0 (/ 1.0 x)) x) (/ -3.0 (pow x 3.0)))
   (/
    (+ (+ x -1.0) (* (+ x 1.0) (/ (- -1.0 x) x)))
    (* (+ x -1.0) (/ (+ x 1.0) x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 1e-10) {
		tmp = ((-3.0 - (1.0 / x)) / x) + (-3.0 / pow(x, 3.0));
	} else {
		tmp = ((x + -1.0) + ((x + 1.0) * ((-1.0 - x) / x))) / ((x + -1.0) * ((x + 1.0) / x));
	}
	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)))) <= 1d-10) then
        tmp = (((-3.0d0) - (1.0d0 / x)) / x) + ((-3.0d0) / (x ** 3.0d0))
    else
        tmp = ((x + (-1.0d0)) + ((x + 1.0d0) * (((-1.0d0) - x) / x))) / ((x + (-1.0d0)) * ((x + 1.0d0) / x))
    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))) <= 1e-10) {
		tmp = ((-3.0 - (1.0 / x)) / x) + (-3.0 / Math.pow(x, 3.0));
	} else {
		tmp = ((x + -1.0) + ((x + 1.0) * ((-1.0 - x) / x))) / ((x + -1.0) * ((x + 1.0) / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 1e-10:
		tmp = ((-3.0 - (1.0 / x)) / x) + (-3.0 / math.pow(x, 3.0))
	else:
		tmp = ((x + -1.0) + ((x + 1.0) * ((-1.0 - x) / x))) / ((x + -1.0) * ((x + 1.0) / x))
	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))) <= 1e-10)
		tmp = Float64(Float64(Float64(-3.0 - Float64(1.0 / x)) / x) + Float64(-3.0 / (x ^ 3.0)));
	else
		tmp = Float64(Float64(Float64(x + -1.0) + Float64(Float64(x + 1.0) * Float64(Float64(-1.0 - x) / x))) / Float64(Float64(x + -1.0) * Float64(Float64(x + 1.0) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 1e-10)
		tmp = ((-3.0 - (1.0 / x)) / x) + (-3.0 / (x ^ 3.0));
	else
		tmp = ((x + -1.0) + ((x + 1.0) * ((-1.0 - x) / x))) / ((x + -1.0) * ((x + 1.0) / x));
	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], 1e-10], N[(N[(N[(-3.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + -1.0), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[(N[(-1.0 - x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 1.00000000000000004e-10

   1. Initial program 8.1%

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

      1. clear-num 8.2%

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

      2. clear-num 8.1%

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

      3. frac-sub 8.2%

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

      4. *-un-lft-identity 8.2%

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

      5. sub-neg 8.2%

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

      6. metadata-eval 8.2%

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

      7. sub-neg 8.2%

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

      8. metadata-eval 8.2%

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

   3. Applied egg-rr 8.2%

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

      1. div-sub 8.1%

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

      2. sub-neg 8.1%

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

      3. *-un-lft-identity 8.1%

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

      4. *-rgt-identity 8.1%

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

      5. times-frac 8.1%

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

      6. *-rgt-identity 8.1%

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

      7. clear-num 8.1%

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

      8. *-rgt-identity 8.1%

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

      9. *-rgt-identity 8.1%

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

   5. Applied egg-rr 8.1%

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

      1. sub-neg 8.1%

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

      2. *-inverses 8.1%

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

      3. *-commutative 8.1%

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

      4. associate-/r/ 8.1%

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

      5. distribute-rgt-out-- 8.1%

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

      6. +-commutative 8.1%

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

      7. *-lft-identity 8.1%

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

      8. associate-*l/ 7.8%

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

      9. distribute-lft-in 7.8%

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

      10. lft-mult-inverse 8.1%

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

      11. *-rgt-identity 8.1%

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

      12. +-commutative 8.1%

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

   7. Simplified 8.1%

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

   8. Taylor expanded in x around inf 99.2%

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

      1. distribute-neg-in 99.2%

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

      2. associate-*r/ 99.2%

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

      3. metadata-eval 99.2%

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

      4. neg-mul-1 99.2%

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

      5. distribute-lft-in 99.2%

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

      6. associate-*r* 99.2%

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

      7. metadata-eval 99.2%

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

      8. associate-*r/ 99.7%

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

      9. metadata-eval 99.7%

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

      10. associate-*r/ 99.7%

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

      11. metadata-eval 99.7%

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

      12. unpow2 99.7%

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

      13. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   if 1.00000000000000004e-10 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. frac-sub 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

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

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x} - \frac{3}{x}}{1 + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + -1\right) + \left(x + 1\right) \cdot \frac{-1 - x}{x}}{\left(x + -1\right) \cdot \frac{x + 1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 1e-10)
   (/ (- (/ 2.0 (* x x)) (/ 3.0 x)) (+ 1.0 (/ -1.0 x)))
   (/
    (+ (+ x -1.0) (* (+ x 1.0) (/ (- -1.0 x) x)))
    (* (+ x -1.0) (/ (+ x 1.0) x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 1e-10) {
		tmp = ((2.0 / (x * x)) - (3.0 / x)) / (1.0 + (-1.0 / x));
	} else {
		tmp = ((x + -1.0) + ((x + 1.0) * ((-1.0 - x) / x))) / ((x + -1.0) * ((x + 1.0) / x));
	}
	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)))) <= 1d-10) then
        tmp = ((2.0d0 / (x * x)) - (3.0d0 / x)) / (1.0d0 + ((-1.0d0) / x))
    else
        tmp = ((x + (-1.0d0)) + ((x + 1.0d0) * (((-1.0d0) - x) / x))) / ((x + (-1.0d0)) * ((x + 1.0d0) / x))
    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))) <= 1e-10) {
		tmp = ((2.0 / (x * x)) - (3.0 / x)) / (1.0 + (-1.0 / x));
	} else {
		tmp = ((x + -1.0) + ((x + 1.0) * ((-1.0 - x) / x))) / ((x + -1.0) * ((x + 1.0) / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 1e-10:
		tmp = ((2.0 / (x * x)) - (3.0 / x)) / (1.0 + (-1.0 / x))
	else:
		tmp = ((x + -1.0) + ((x + 1.0) * ((-1.0 - x) / x))) / ((x + -1.0) * ((x + 1.0) / x))
	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))) <= 1e-10)
		tmp = Float64(Float64(Float64(2.0 / Float64(x * x)) - Float64(3.0 / x)) / Float64(1.0 + Float64(-1.0 / x)));
	else
		tmp = Float64(Float64(Float64(x + -1.0) + Float64(Float64(x + 1.0) * Float64(Float64(-1.0 - x) / x))) / Float64(Float64(x + -1.0) * Float64(Float64(x + 1.0) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 1e-10)
		tmp = ((2.0 / (x * x)) - (3.0 / x)) / (1.0 + (-1.0 / x));
	else
		tmp = ((x + -1.0) + ((x + 1.0) * ((-1.0 - x) / x))) / ((x + -1.0) * ((x + 1.0) / x));
	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], 1e-10], N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(3.0 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + -1.0), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[(N[(-1.0 - x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 1.00000000000000004e-10

   1. Initial program 8.1%

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

      1. clear-num 8.2%

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

      2. clear-num 8.1%

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

      3. frac-sub 8.2%

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

      4. *-un-lft-identity 8.2%

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

      5. sub-neg 8.2%

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

      6. metadata-eval 8.2%

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

      7. sub-neg 8.2%

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

      8. metadata-eval 8.2%

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

   3. Applied egg-rr 8.2%

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

   4. Taylor expanded in x around inf 99.0%

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

      1. unpow2 99.0%

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

      2. associate-*r/ 99.0%

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

      3. metadata-eval 99.0%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 99.6%

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

   if 1.00000000000000004e-10 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. frac-sub 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x} - \frac{3}{x}}{1 + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + -1\right) + \left(x + 1\right) \cdot \frac{-1 - x}{x}}{\left(x + -1\right) \cdot \frac{x + 1}{x}}\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_0 \leq 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x} - \frac{3}{x}}{1 + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 1e-10)
     (/ (- (/ 2.0 (* x x)) (/ 3.0 x)) (+ 1.0 (/ -1.0 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 <= 1e-10) {
		tmp = ((2.0 / (x * x)) - (3.0 / x)) / (1.0 + (-1.0 / 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 <= 1d-10) then
        tmp = ((2.0d0 / (x * x)) - (3.0d0 / x)) / (1.0d0 + ((-1.0d0) / 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 <= 1e-10) {
		tmp = ((2.0 / (x * x)) - (3.0 / x)) / (1.0 + (-1.0 / 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 <= 1e-10:
		tmp = ((2.0 / (x * x)) - (3.0 / x)) / (1.0 + (-1.0 / 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 <= 1e-10)
		tmp = Float64(Float64(Float64(2.0 / Float64(x * x)) - Float64(3.0 / x)) / Float64(1.0 + Float64(-1.0 / 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 <= 1e-10)
		tmp = ((2.0 / (x * x)) - (3.0 / x)) / (1.0 + (-1.0 / 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, 1e-10], N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(3.0 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $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))) < 1.00000000000000004e-10

   1. Initial program 8.1%

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

      1. clear-num 8.2%

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

      2. clear-num 8.1%

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

      3. frac-sub 8.2%

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

      4. *-un-lft-identity 8.2%

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

      5. sub-neg 8.2%

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

      6. metadata-eval 8.2%

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

      7. sub-neg 8.2%

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

      8. metadata-eval 8.2%

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

   3. Applied egg-rr 8.2%

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

   4. Taylor expanded in x around inf 99.0%

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

      1. unpow2 99.0%

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

      2. associate-*r/ 99.0%

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

      3. metadata-eval 99.0%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 99.6%

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

   if 1.00000000000000004e-10 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x} - \frac{3}{x}}{1 + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 5: 99.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 1e-10) (- (/ -3.0 x) (/ (/ 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 <= 1e-10) {
		tmp = (-3.0 / x) - ((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 <= 1d-10) then
        tmp = ((-3.0d0) / x) - ((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 <= 1e-10) {
		tmp = (-3.0 / x) - ((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 <= 1e-10:
		tmp = (-3.0 / x) - ((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 <= 1e-10)
		tmp = Float64(Float64(-3.0 / x) - Float64(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 <= 1e-10)
		tmp = (-3.0 / x) - ((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, 1e-10], N[(N[(-3.0 / x), $MachinePrecision] - N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $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))) < 1.00000000000000004e-10

   1. Initial program 8.1%

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

   2. Taylor expanded in x around inf 99.0%

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

      1. +-commutative 99.0%

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

      2. distribute-neg-in 99.0%

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

      3. sub-neg 99.0%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

      6. distribute-neg-frac 99.6%

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

      7. metadata-eval 99.6%

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

      8. unpow2 99.6%

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

      9. associate-/r* 99.6%

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

   4. Simplified 99.6%

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

   if 1.00000000000000004e-10 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 100.0%

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

4. Final simplification 99.8%

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

Alternative 6: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ (- -3.0 (/ 1.0 x)) x)
   (if (<= x 1.0) (+ 1.0 (* x 3.0)) (- (/ -3.0 x) (/ (/ 1.0 x) 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 * 3.0);
	} else {
		tmp = (-3.0 / x) - ((1.0 / x) / x);
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-3.0 - (1.0 / x)) / x;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * 3.0);
	} else {
		tmp = (-3.0 / x) - ((1.0 / x) / 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 * 3.0)
	else:
		tmp = (-3.0 / x) - ((1.0 / x) / 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 * 3.0));
	else
		tmp = Float64(Float64(-3.0 / x) - Float64(Float64(1.0 / x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (-3.0 - (1.0 / x)) / x;
	elseif (x <= 1.0)
		tmp = 1.0 + (x * 3.0);
	else
		tmp = (-3.0 / x) - ((1.0 / x) / 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 * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-3.0 / x), $MachinePrecision] - N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 6.1%

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

   2. Taylor expanded in x around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-neg-in 99.4%

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

      3. sub-neg 99.4%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. unpow2 100.0%

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

      9. associate-/r* 100.0%

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

   4. Simplified 100.0%

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.7%

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

   if 1 < x

   1. Initial program 10.2%

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

   2. Taylor expanded in x around inf 98.6%

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

      1. +-commutative 98.6%

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

      2. distribute-neg-in 98.6%

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

      3. sub-neg 98.6%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

      6. distribute-neg-frac 99.1%

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

      7. metadata-eval 99.1%

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

      8. unpow2 99.1%

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

      9. associate-/r* 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 99.6%

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

Alternative 7: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.1%

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

   2. Taylor expanded in x around inf 99.0%

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

      1. +-commutative 99.0%

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

      2. distribute-neg-in 99.0%

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

      3. sub-neg 99.0%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

      6. distribute-neg-frac 99.6%

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

      7. metadata-eval 99.6%

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

      8. unpow2 99.6%

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

      9. associate-/r* 99.6%

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

   4. Simplified 99.6%

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

      1. sub-div 99.6%

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

   6. Applied egg-rr 99.6%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.7%

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

4. Final simplification 99.6%

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

Alternative 8: 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 8.1%

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

   2. Taylor expanded in x around inf 98.5%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.7%

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

4. Final simplification 99.1%

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

Alternative 9: 98.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.1%

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

   2. Taylor expanded in x around inf 98.5%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in x around 0 99.1%

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

4. Final simplification 98.8%

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

Alternative 10: 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 50.8%

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

2. Taylor expanded in x around 0 48.1%

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

3. Final simplification 48.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023215 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))