Asymptote C

Percentage Accurate: 54.5% → 99.8%

Time: 11.4s

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: 54.5% 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: 99.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0002:\\ \;\;\;\;\frac{-3}{{x}^{3}} + \left(\left(\frac{-1}{{x}^{4}} - \frac{1}{{x}^{2}}\right) - \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot \left(x + 1\right) - x \cdot \left(x + -1\right)}{\left(x + 1\right) \cdot \left(1 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.0002)
   (+
    (/ -3.0 (pow x 3.0))
    (- (- (/ -1.0 (pow x 4.0)) (/ 1.0 (pow x 2.0))) (/ 3.0 x)))
   (/ (- (* (+ x 1.0) (+ x 1.0)) (* x (+ x -1.0))) (* (+ x 1.0) (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.0002) {
		tmp = (-3.0 / pow(x, 3.0)) + (((-1.0 / pow(x, 4.0)) - (1.0 / pow(x, 2.0))) - (3.0 / x));
	} else {
		tmp = (((x + 1.0) * (x + 1.0)) - (x * (x + -1.0))) / ((x + 1.0) * (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)))) <= 0.0002d0) then
        tmp = ((-3.0d0) / (x ** 3.0d0)) + ((((-1.0d0) / (x ** 4.0d0)) - (1.0d0 / (x ** 2.0d0))) - (3.0d0 / x))
    else
        tmp = (((x + 1.0d0) * (x + 1.0d0)) - (x * (x + (-1.0d0)))) / ((x + 1.0d0) * (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))) <= 0.0002) {
		tmp = (-3.0 / Math.pow(x, 3.0)) + (((-1.0 / Math.pow(x, 4.0)) - (1.0 / Math.pow(x, 2.0))) - (3.0 / x));
	} else {
		tmp = (((x + 1.0) * (x + 1.0)) - (x * (x + -1.0))) / ((x + 1.0) * (1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.0002:
		tmp = (-3.0 / math.pow(x, 3.0)) + (((-1.0 / math.pow(x, 4.0)) - (1.0 / math.pow(x, 2.0))) - (3.0 / x))
	else:
		tmp = (((x + 1.0) * (x + 1.0)) - (x * (x + -1.0))) / ((x + 1.0) * (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))) <= 0.0002)
		tmp = Float64(Float64(-3.0 / (x ^ 3.0)) + Float64(Float64(Float64(-1.0 / (x ^ 4.0)) - Float64(1.0 / (x ^ 2.0))) - Float64(3.0 / x)));
	else
		tmp = Float64(Float64(Float64(Float64(x + 1.0) * Float64(x + 1.0)) - Float64(x * Float64(x + -1.0))) / Float64(Float64(x + 1.0) * Float64(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))) <= 0.0002)
		tmp = (-3.0 / (x ^ 3.0)) + (((-1.0 / (x ^ 4.0)) - (1.0 / (x ^ 2.0))) - (3.0 / x));
	else
		tmp = (((x + 1.0) * (x + 1.0)) - (x * (x + -1.0))) / ((x + 1.0) * (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], 0.0002], N[(N[(-3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(3.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(1.0 - 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))) < 2.0000000000000001e-4

   1. Initial program 9.2%

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

   2. Taylor expanded in x around inf 99.5%

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. distribute-neg-in 99.5%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{{x}^{3}}\right) + \left(-\left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right)\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{{x}^{3}}\right) - \left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right)} \]

      3. associate-*r/ 99.5%

         \[\leadsto \left(-\color{blue}{\frac{3 \cdot 1}{{x}^{3}}}\right) - \left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

      4. metadata-eval 99.5%

         \[\leadsto \left(-\frac{\color{blue}{3}}{{x}^{3}}\right) - \left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

      5. distribute-neg-frac 99.5%

         \[\leadsto \color{blue}{\frac{-3}{{x}^{3}}} - \left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

      6. metadata-eval 99.5%

         \[\leadsto \frac{\color{blue}{-3}}{{x}^{3}} - \left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

      7. associate-*r/ 100.0%

         \[\leadsto \frac{-3}{{x}^{3}} - \left(\color{blue}{\frac{3 \cdot 1}{x}} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \frac{-3}{{x}^{3}} - \left(\frac{\color{blue}{3}}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-3}{{x}^{3}} - \left(\frac{3}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right)} \]

   if 2.0000000000000001e-4 < (-.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%

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

      2. associate-/r/ 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. frac-2neg 99.9%

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

      2. associate-*l/ 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. frac-sub 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0002:\\ \;\;\;\;\frac{-3}{{x}^{3}} + \left(\left(\frac{-1}{{x}^{4}} - \frac{1}{{x}^{2}}\right) - \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot \left(x + 1\right) - x \cdot \left(x + -1\right)}{\left(x + 1\right) \cdot \left(1 - x\right)}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0002:\\ \;\;\;\;\frac{-3}{{x}^{3}} + \left(\frac{-3}{x} - {x}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot \left(x + 1\right) - x \cdot \left(x + -1\right)}{\left(x + 1\right) \cdot \left(1 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.0002)
   (+ (/ -3.0 (pow x 3.0)) (- (/ -3.0 x) (pow x -2.0)))
   (/ (- (* (+ x 1.0) (+ x 1.0)) (* x (+ x -1.0))) (* (+ x 1.0) (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.0002) {
		tmp = (-3.0 / pow(x, 3.0)) + ((-3.0 / x) - pow(x, -2.0));
	} else {
		tmp = (((x + 1.0) * (x + 1.0)) - (x * (x + -1.0))) / ((x + 1.0) * (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)))) <= 0.0002d0) then
        tmp = ((-3.0d0) / (x ** 3.0d0)) + (((-3.0d0) / x) - (x ** (-2.0d0)))
    else
        tmp = (((x + 1.0d0) * (x + 1.0d0)) - (x * (x + (-1.0d0)))) / ((x + 1.0d0) * (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))) <= 0.0002) {
		tmp = (-3.0 / Math.pow(x, 3.0)) + ((-3.0 / x) - Math.pow(x, -2.0));
	} else {
		tmp = (((x + 1.0) * (x + 1.0)) - (x * (x + -1.0))) / ((x + 1.0) * (1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.0002:
		tmp = (-3.0 / math.pow(x, 3.0)) + ((-3.0 / x) - math.pow(x, -2.0))
	else:
		tmp = (((x + 1.0) * (x + 1.0)) - (x * (x + -1.0))) / ((x + 1.0) * (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))) <= 0.0002)
		tmp = Float64(Float64(-3.0 / (x ^ 3.0)) + Float64(Float64(-3.0 / x) - (x ^ -2.0)));
	else
		tmp = Float64(Float64(Float64(Float64(x + 1.0) * Float64(x + 1.0)) - Float64(x * Float64(x + -1.0))) / Float64(Float64(x + 1.0) * Float64(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))) <= 0.0002)
		tmp = (-3.0 / (x ^ 3.0)) + ((-3.0 / x) - (x ^ -2.0));
	else
		tmp = (((x + 1.0) * (x + 1.0)) - (x * (x + -1.0))) / ((x + 1.0) * (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], 0.0002], N[(N[(-3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-3.0 / x), $MachinePrecision] - N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(1.0 - 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))) < 2.0000000000000001e-4

   1. Initial program 9.2%

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

   2. Taylor expanded in x around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-neg-in 99.4%

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

      3. distribute-neg-in 99.4%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

      8. distribute-neg-frac 99.9%

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

      9. metadata-eval 99.9%

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

      10. associate-*r/ 99.9%

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

      11. metadata-eval 99.9%

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

      12. distribute-neg-frac 99.9%

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

      13. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-def 99.9%

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

      3. div-inv 99.9%

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

      4. pow-flip 99.9%

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

      5. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. *-lft-identity 99.9%

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

      3. mul-1-neg 99.9%

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

      4. unsub-neg 99.9%

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

   8. Simplified 99.9%

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

   if 2.0000000000000001e-4 < (-.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%

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

      2. associate-/r/ 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. frac-2neg 99.9%

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

      2. associate-*l/ 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. frac-sub 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0002:\\ \;\;\;\;\frac{-3}{{x}^{3}} + \left(\frac{-3}{x} - {x}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot \left(x + 1\right) - x \cdot \left(x + -1\right)}{\left(x + 1\right) \cdot \left(1 - x\right)}\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0002:\\ \;\;\;\;\frac{-3}{x} + \left(-3 \cdot {x}^{-3} - {x}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot \left(x + 1\right) - x \cdot \left(x + -1\right)}{\left(x + 1\right) \cdot \left(1 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.0002)
   (+ (/ -3.0 x) (- (* -3.0 (pow x -3.0)) (pow x -2.0)))
   (/ (- (* (+ x 1.0) (+ x 1.0)) (* x (+ x -1.0))) (* (+ x 1.0) (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.0002) {
		tmp = (-3.0 / x) + ((-3.0 * pow(x, -3.0)) - pow(x, -2.0));
	} else {
		tmp = (((x + 1.0) * (x + 1.0)) - (x * (x + -1.0))) / ((x + 1.0) * (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)))) <= 0.0002d0) then
        tmp = ((-3.0d0) / x) + (((-3.0d0) * (x ** (-3.0d0))) - (x ** (-2.0d0)))
    else
        tmp = (((x + 1.0d0) * (x + 1.0d0)) - (x * (x + (-1.0d0)))) / ((x + 1.0d0) * (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))) <= 0.0002) {
		tmp = (-3.0 / x) + ((-3.0 * Math.pow(x, -3.0)) - Math.pow(x, -2.0));
	} else {
		tmp = (((x + 1.0) * (x + 1.0)) - (x * (x + -1.0))) / ((x + 1.0) * (1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.0002:
		tmp = (-3.0 / x) + ((-3.0 * math.pow(x, -3.0)) - math.pow(x, -2.0))
	else:
		tmp = (((x + 1.0) * (x + 1.0)) - (x * (x + -1.0))) / ((x + 1.0) * (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))) <= 0.0002)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-3.0 * (x ^ -3.0)) - (x ^ -2.0)));
	else
		tmp = Float64(Float64(Float64(Float64(x + 1.0) * Float64(x + 1.0)) - Float64(x * Float64(x + -1.0))) / Float64(Float64(x + 1.0) * Float64(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))) <= 0.0002)
		tmp = (-3.0 / x) + ((-3.0 * (x ^ -3.0)) - (x ^ -2.0));
	else
		tmp = (((x + 1.0) * (x + 1.0)) - (x * (x + -1.0))) / ((x + 1.0) * (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], 0.0002], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-3.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision] - N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(1.0 - 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))) < 2.0000000000000001e-4

   1. Initial program 9.2%

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

   2. Taylor expanded in x around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-neg-in 99.4%

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

      3. distribute-neg-in 99.4%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

      8. distribute-neg-frac 99.9%

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

      9. metadata-eval 99.9%

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

      10. associate-*r/ 99.9%

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

      11. metadata-eval 99.9%

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

      12. distribute-neg-frac 99.9%

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

      13. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-def 99.9%

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

      3. div-inv 99.9%

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

      4. pow-flip 99.9%

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

      5. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. *-lft-identity 99.9%

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

      3. mul-1-neg 99.9%

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

      4. unsub-neg 99.9%

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

   8. Simplified 99.9%

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

      1. metadata-eval 99.9%

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

      2. pow-flip 99.9%

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

      3. associate-+l- 99.9%

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

      4. pow-flip 99.9%

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

      5. metadata-eval 99.9%

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

      6. div-inv 99.9%

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

      7. pow-flip 99.9%

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

      8. metadata-eval 99.9%

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

   10. Applied egg-rr 99.9%

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

   if 2.0000000000000001e-4 < (-.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%

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

      2. associate-/r/ 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. frac-2neg 99.9%

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

      2. associate-*l/ 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. frac-sub 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0002:\\ \;\;\;\;\frac{-3}{x} + \left(-3 \cdot {x}^{-3} - {x}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot \left(x + 1\right) - x \cdot \left(x + -1\right)}{\left(x + 1\right) \cdot \left(1 - x\right)}\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_0 \leq 10^{-7}:\\ \;\;\;\;\frac{-3}{x} + \frac{1}{x} \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 1e-7) (+ (/ -3.0 x) (* (/ 1.0 x) (/ -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-7) {
		tmp = (-3.0 / x) + ((1.0 / x) * (-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-7) then
        tmp = ((-3.0d0) / x) + ((1.0d0 / x) * ((-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-7) {
		tmp = (-3.0 / x) + ((1.0 / x) * (-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-7:
		tmp = (-3.0 / x) + ((1.0 / x) * (-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-7)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(1.0 / x) * 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-7)
		tmp = (-3.0 / x) + ((1.0 / x) * (-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-7], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * 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))) < 9.9999999999999995e-8

   1. Initial program 8.6%

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

      1. clear-num 8.5%

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

      2. associate-/r/ 8.5%

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

      3. sub-neg 8.5%

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

      4. metadata-eval 8.5%

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

   3. Applied egg-rr 8.5%

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

      1. distribute-rgt-in 8.4%

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

      2. un-div-inv 8.5%

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

      3. *-un-lft-identity 8.5%

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

   5. Applied egg-rr 8.5%

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

   6. Taylor expanded in x around inf 99.3%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

      3. distribute-neg-in 99.8%

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

      4. unsub-neg 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. metadata-eval 99.8%

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

   8. Simplified 99.8%

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

      1. inv-pow 99.8%

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

      2. unpow2 99.8%

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

      3. pow-prod-down 99.8%

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

      4. inv-pow 99.8%

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

      5. inv-pow 99.8%

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

   10. Applied egg-rr 99.8%

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

   if 9.9999999999999995e-8 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 10^{-7}:\\ \;\;\;\;\frac{-3}{x} + \frac{1}{x} \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 5: 99.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 9.2%

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

      1. clear-num 9.1%

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

      2. associate-/r/ 9.1%

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

      3. sub-neg 9.1%

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

      4. metadata-eval 9.1%

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

   3. Applied egg-rr 9.1%

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

      1. distribute-rgt-in 9.0%

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

      2. un-div-inv 9.1%

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

      3. *-un-lft-identity 9.1%

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

   5. Applied egg-rr 9.1%

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

   6. Taylor expanded in x around inf 99.0%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

      3. distribute-neg-in 99.5%

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

      4. unsub-neg 99.5%

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

      5. distribute-neg-frac 99.5%

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

      6. metadata-eval 99.5%

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

   8. Simplified 99.5%

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

      1. inv-pow 99.5%

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

      2. unpow2 99.5%

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

      3. pow-prod-down 99.5%

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

      4. inv-pow 99.5%

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

      5. inv-pow 99.5%

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

   10. Applied egg-rr 99.5%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.4%

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

      1. unpow2 98.4%

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

      2. distribute-rgt-out 98.4%

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

   4. Simplified 98.4%

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

4. Final simplification 98.9%

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

Alternative 6: 98.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 9.2%

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

   2. Taylor expanded in x around inf 97.6%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.4%

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

      1. unpow2 98.4%

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

      2. distribute-rgt-out 98.4%

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

   4. Simplified 98.4%

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

4. Final simplification 98.0%

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

Alternative 7: 98.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = 1.0d0 / ((x * (-0.3333333333333333d0)) + 0.1111111111111111d0)
    else
        tmp = 1.0d0 + (x * (x + 3.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(1.0 / Float64(Float64(x * -0.3333333333333333) + 0.1111111111111111));
	else
		tmp = Float64(1.0 + Float64(x * Float64(x + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = 1.0 / ((x * -0.3333333333333333) + 0.1111111111111111);
	else
		tmp = 1.0 + (x * (x + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(1.0 / N[(N[(x * -0.3333333333333333), $MachinePrecision] + 0.1111111111111111), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 9.2%

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

      1. clear-num 9.2%

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

      2. clear-num 9.1%

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

      3. frac-sub 9.3%

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

      4. *-un-lft-identity 9.3%

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

      5. sub-neg 9.3%

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

      6. metadata-eval 9.3%

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

      7. sub-neg 9.3%

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

      8. metadata-eval 9.3%

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

   3. Applied egg-rr 9.3%

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

   4. Taylor expanded in x around 0 9.3%

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

      1. clear-num 9.3%

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

      2. inv-pow 9.3%

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

      3. *-rgt-identity 9.3%

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

   6. Applied egg-rr 9.3%

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

      1. unpow-1 9.3%

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

      2. sub-neg 9.3%

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

      3. distribute-neg-frac 9.3%

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

      4. metadata-eval 9.3%

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

      5. +-commutative 9.3%

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

      6. +-commutative 9.3%

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

   8. Simplified 9.3%

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

   9. Taylor expanded in x around inf 98.9%

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

       1. +-commutative 98.9%

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

       2. *-commutative 98.9%

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

   11. Simplified 98.9%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.4%

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

      1. unpow2 98.4%

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

      2. distribute-rgt-out 98.4%

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

   4. Simplified 98.4%

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

4. Final simplification 98.7%

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

Alternative 8: 98.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 9.2%

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

   2. Taylor expanded in x around inf 97.6%

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

   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 97.4%

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

4. Final simplification 97.5%

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

Alternative 9: 98.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 9.2%

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

   2. Taylor expanded in x around inf 97.6%

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

   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 95.5%

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

4. Final simplification 96.5%

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

Alternative 10: 51.2% 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 55.3%

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

2. Taylor expanded in x around 0 50.4%

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

3. Final simplification 50.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023332 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))