Asymptote C

Percentage Accurate: 54.7% → 99.8%

Time: 8.0s

Alternatives: 11

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: 54.7% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x / (x + 1.0d0)) - ((x + 1.0d0) / (x + (-1.0d0)))) <= 0.0002d0) then
        tmp = (((2.0d0 / x) + (2.0d0 / (x ** 3.0d0))) - (3.0d0 + (2.0d0 / (x * x)))) / (x + (-1.0d0))
    else
        tmp = ((x * ((x + (-1.0d0)) / (x + 1.0d0))) + ((-1.0d0) - x)) / (x + (-1.0d0))
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 0.0002:
		tmp = (((2.0 / x) + (2.0 / math.pow(x, 3.0))) - (3.0 + (2.0 / (x * x)))) / (x + -1.0)
	else:
		tmp = ((x * ((x + -1.0) / (x + 1.0))) + (-1.0 - x)) / (x + -1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x + -1.0))) <= 0.0002)
		tmp = Float64(Float64(Float64(Float64(2.0 / x) + Float64(2.0 / (x ^ 3.0))) - Float64(3.0 + Float64(2.0 / Float64(x * x)))) / Float64(x + -1.0));
	else
		tmp = Float64(Float64(Float64(x * Float64(Float64(x + -1.0) / Float64(x + 1.0))) + Float64(-1.0 - x)) / Float64(x + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 0.0002)
		tmp = (((2.0 / x) + (2.0 / (x ^ 3.0))) - (3.0 + (2.0 / (x * x)))) / (x + -1.0);
	else
		tmp = ((x * ((x + -1.0) / (x + 1.0))) + (-1.0 - x)) / (x + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(3.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 2.0000000000000001e-4

   1. Initial program 8.2%

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

      1. clear-num 8.2%

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

      2. associate-/r/ 7.8%

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

      3. sub-neg 7.8%

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

      4. metadata-eval 7.8%

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

   3. Applied egg-rr 7.8%

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

      1. *-commutative 7.8%

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

      2. metadata-eval 7.8%

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

      3. sub-neg 7.8%

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

      4. div-inv 8.2%

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

      5. clear-num 8.2%

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

      6. frac-sub 10.3%

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

      7. sub-neg 10.3%

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

      8. metadata-eval 10.3%

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

      9. metadata-eval 10.3%

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

      10. div-inv 10.3%

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

      11. /-rgt-identity 10.3%

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

      12. sub-neg 10.3%

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

      13. metadata-eval 10.3%

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

   5. Applied egg-rr 10.3%

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

   6. Taylor expanded in x around 0 10.3%

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

   7. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. unpow2 100.0%

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

   9. Simplified 100.0%

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

   if 2.0000000000000001e-4 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. metadata-eval 100.0%

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

      3. sub-neg 100.0%

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

      4. div-inv 99.9%

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

      5. clear-num 99.9%

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

      6. frac-sub 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. div-inv 100.0%

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

      11. /-rgt-identity 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x / (x + 1.0d0)) - ((x + 1.0d0) / (x + (-1.0d0)))) <= 0.0002d0) then
        tmp = ((-3.0d0) / x) + (((-3.0d0) / (x ** 3.0d0)) + ((-1.0d0) / (x * x)))
    else
        tmp = ((x * ((x + (-1.0d0)) / (x + 1.0d0))) + ((-1.0d0) - x)) / (x + (-1.0d0))
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 0.0002:
		tmp = (-3.0 / x) + ((-3.0 / math.pow(x, 3.0)) + (-1.0 / (x * x)))
	else:
		tmp = ((x * ((x + -1.0) / (x + 1.0))) + (-1.0 - x)) / (x + -1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x + -1.0))) <= 0.0002)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-3.0 / (x ^ 3.0)) + Float64(-1.0 / Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(x * Float64(Float64(x + -1.0) / Float64(x + 1.0))) + Float64(-1.0 - x)) / Float64(x + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 0.0002)
		tmp = (-3.0 / x) + ((-3.0 / (x ^ 3.0)) + (-1.0 / (x * x)));
	else
		tmp = ((x * ((x + -1.0) / (x + 1.0))) + (-1.0 - x)) / (x + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 2.0000000000000001e-4

   1. Initial program 8.2%

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

      1. clear-num 8.2%

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

      2. associate-/r/ 7.8%

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

      3. sub-neg 7.8%

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

      4. metadata-eval 7.8%

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

   3. Applied egg-rr 7.8%

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

      1. *-commutative 7.8%

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

      2. metadata-eval 7.8%

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

      3. sub-neg 7.8%

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

      4. div-inv 8.2%

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

      5. clear-num 8.2%

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

      6. frac-sub 10.3%

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

      7. sub-neg 10.3%

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

      8. metadata-eval 10.3%

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

      9. metadata-eval 10.3%

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

      10. div-inv 10.3%

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

      11. /-rgt-identity 10.3%

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

      12. sub-neg 10.3%

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

      13. metadata-eval 10.3%

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

   5. Applied egg-rr 10.3%

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

   6. Taylor expanded in x around inf 99.2%

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)} \]
   7. 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. +-commutative 99.2%

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

      3. associate-+r+ 99.2%

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

      4. distribute-neg-in 99.2%

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

      5. distribute-lft-neg-in 99.2%

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

      6. metadata-eval 99.2%

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

      7. associate-*r/ 99.9%

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

      8. metadata-eval 99.9%

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

      9. +-commutative 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. associate-*r/ 99.9%

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

      12. metadata-eval 99.9%

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

      13. distribute-neg-frac 99.9%

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

      14. metadata-eval 99.9%

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

      15. distribute-neg-frac 99.9%

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

      16. metadata-eval 99.9%

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

      17. unpow2 99.9%

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

   8. Simplified 99.9%

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

   if 2.0000000000000001e-4 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. metadata-eval 100.0%

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

      3. sub-neg 100.0%

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

      4. div-inv 99.9%

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

      5. clear-num 99.9%

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

      6. frac-sub 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. div-inv 100.0%

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

      11. /-rgt-identity 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.9%

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

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 0.0002) {
		tmp = (((2.0 / x) + -3.0) - (2.0 / (x * x))) / (x + -1.0);
	} else {
		tmp = ((x * ((x + -1.0) / (x + 1.0))) + (-1.0 - x)) / (x + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x / (x + 1.0d0)) - ((x + 1.0d0) / (x + (-1.0d0)))) <= 0.0002d0) then
        tmp = (((2.0d0 / x) + (-3.0d0)) - (2.0d0 / (x * x))) / (x + (-1.0d0))
    else
        tmp = ((x * ((x + (-1.0d0)) / (x + 1.0d0))) + ((-1.0d0) - x)) / (x + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 0.0002) {
		tmp = (((2.0 / x) + -3.0) - (2.0 / (x * x))) / (x + -1.0);
	} else {
		tmp = ((x * ((x + -1.0) / (x + 1.0))) + (-1.0 - x)) / (x + -1.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 0.0002:
		tmp = (((2.0 / x) + -3.0) - (2.0 / (x * x))) / (x + -1.0)
	else:
		tmp = ((x * ((x + -1.0) / (x + 1.0))) + (-1.0 - x)) / (x + -1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x + -1.0))) <= 0.0002)
		tmp = Float64(Float64(Float64(Float64(2.0 / x) + -3.0) - Float64(2.0 / Float64(x * x))) / Float64(x + -1.0));
	else
		tmp = Float64(Float64(Float64(x * Float64(Float64(x + -1.0) / Float64(x + 1.0))) + Float64(-1.0 - x)) / Float64(x + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 0.0002)
		tmp = (((2.0 / x) + -3.0) - (2.0 / (x * x))) / (x + -1.0);
	else
		tmp = ((x * ((x + -1.0) / (x + 1.0))) + (-1.0 - x)) / (x + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(N[(2.0 / x), $MachinePrecision] + -3.0), $MachinePrecision] - N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 2.0000000000000001e-4

   1. Initial program 8.2%

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

      1. clear-num 8.2%

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

      2. associate-/r/ 7.8%

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

      3. sub-neg 7.8%

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

      4. metadata-eval 7.8%

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

   3. Applied egg-rr 7.8%

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

      1. *-commutative 7.8%

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

      2. metadata-eval 7.8%

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

      3. sub-neg 7.8%

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

      4. div-inv 8.2%

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

      5. clear-num 8.2%

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

      6. frac-sub 10.3%

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

      7. sub-neg 10.3%

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

      8. metadata-eval 10.3%

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

      9. metadata-eval 10.3%

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

      10. div-inv 10.3%

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

      11. /-rgt-identity 10.3%

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

      12. sub-neg 10.3%

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

      13. metadata-eval 10.3%

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

   5. Applied egg-rr 10.3%

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

   6. Taylor expanded in x around 0 10.3%

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

   7. Taylor expanded in x around inf 99.8%

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

      1. associate--r+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. metadata-eval 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-*r/ 99.8%

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

      7. metadata-eval 99.8%

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

      8. unpow2 99.8%

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

   9. Simplified 99.8%

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

   if 2.0000000000000001e-4 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. metadata-eval 100.0%

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

      3. sub-neg 100.0%

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

      4. div-inv 99.9%

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

      5. clear-num 99.9%

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

      6. frac-sub 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. div-inv 100.0%

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

      11. /-rgt-identity 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.9%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;t_0 - \frac{x + 1}{x + -1} \leq 10^{-8}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(x + 1\right) \cdot \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (- t_0 (/ (+ x 1.0) (+ x -1.0))) 1e-8)
     (+ (/ -3.0 x) (/ (/ -1.0 x) x))
     (- t_0 (* (+ x 1.0) (/ 1.0 (+ x -1.0)))))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 - ((x + 1.0) / (x + -1.0))) <= 1e-8) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = t_0 - ((x + 1.0) * (1.0 / (x + -1.0)));
	}
	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)
    if ((t_0 - ((x + 1.0d0) / (x + (-1.0d0)))) <= 1d-8) then
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    else
        tmp = t_0 - ((x + 1.0d0) * (1.0d0 / (x + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 - ((x + 1.0) / (x + -1.0))) <= 1e-8) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = t_0 - ((x + 1.0) * (1.0 / (x + -1.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (x + 1.0)
	tmp = 0
	if (t_0 - ((x + 1.0) / (x + -1.0))) <= 1e-8:
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	else:
		tmp = t_0 - ((x + 1.0) * (1.0 / (x + -1.0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 - Float64(Float64(x + 1.0) / Float64(x + -1.0))) <= 1e-8)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x));
	else
		tmp = Float64(t_0 - Float64(Float64(x + 1.0) * Float64(1.0 / Float64(x + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if ((t_0 - ((x + 1.0) / (x + -1.0))) <= 1e-8)
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	else
		tmp = t_0 - ((x + 1.0) * (1.0 / (x + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-8], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(x + 1.0), $MachinePrecision] * N[(1.0 / N[(x + -1.0), $MachinePrecision]), $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))) < 1e-8

   1. Initial program 7.0%

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

   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%

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

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

   1. Initial program 99.7%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

4. Final simplification 99.8%

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

Alternative 5: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;t_0 - \frac{x + 1}{x + -1} \leq 0.0002:\\ \;\;\;\;\frac{\left(\frac{2}{x} + -3\right) - \frac{2}{x \cdot x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(x + 1\right) \cdot \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (- t_0 (/ (+ x 1.0) (+ x -1.0))) 0.0002)
     (/ (- (+ (/ 2.0 x) -3.0) (/ 2.0 (* x x))) (+ x -1.0))
     (- t_0 (* (+ x 1.0) (/ 1.0 (+ x -1.0)))))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 - ((x + 1.0) / (x + -1.0))) <= 0.0002) {
		tmp = (((2.0 / x) + -3.0) - (2.0 / (x * x))) / (x + -1.0);
	} else {
		tmp = t_0 - ((x + 1.0) * (1.0 / (x + -1.0)));
	}
	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)
    if ((t_0 - ((x + 1.0d0) / (x + (-1.0d0)))) <= 0.0002d0) then
        tmp = (((2.0d0 / x) + (-3.0d0)) - (2.0d0 / (x * x))) / (x + (-1.0d0))
    else
        tmp = t_0 - ((x + 1.0d0) * (1.0d0 / (x + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 - ((x + 1.0) / (x + -1.0))) <= 0.0002) {
		tmp = (((2.0 / x) + -3.0) - (2.0 / (x * x))) / (x + -1.0);
	} else {
		tmp = t_0 - ((x + 1.0) * (1.0 / (x + -1.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (x + 1.0)
	tmp = 0
	if (t_0 - ((x + 1.0) / (x + -1.0))) <= 0.0002:
		tmp = (((2.0 / x) + -3.0) - (2.0 / (x * x))) / (x + -1.0)
	else:
		tmp = t_0 - ((x + 1.0) * (1.0 / (x + -1.0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 - Float64(Float64(x + 1.0) / Float64(x + -1.0))) <= 0.0002)
		tmp = Float64(Float64(Float64(Float64(2.0 / x) + -3.0) - Float64(2.0 / Float64(x * x))) / Float64(x + -1.0));
	else
		tmp = Float64(t_0 - Float64(Float64(x + 1.0) * Float64(1.0 / Float64(x + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if ((t_0 - ((x + 1.0) / (x + -1.0))) <= 0.0002)
		tmp = (((2.0 / x) + -3.0) - (2.0 / (x * x))) / (x + -1.0);
	else
		tmp = t_0 - ((x + 1.0) * (1.0 / (x + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(N[(2.0 / x), $MachinePrecision] + -3.0), $MachinePrecision] - N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(x + 1.0), $MachinePrecision] * N[(1.0 / N[(x + -1.0), $MachinePrecision]), $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 8.2%

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

      1. clear-num 8.2%

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

      2. associate-/r/ 7.8%

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

      3. sub-neg 7.8%

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

      4. metadata-eval 7.8%

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

   3. Applied egg-rr 7.8%

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

      1. *-commutative 7.8%

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

      2. metadata-eval 7.8%

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

      3. sub-neg 7.8%

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

      4. div-inv 8.2%

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

      5. clear-num 8.2%

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

      6. frac-sub 10.3%

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

      7. sub-neg 10.3%

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

      8. metadata-eval 10.3%

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

      9. metadata-eval 10.3%

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

      10. div-inv 10.3%

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

      11. /-rgt-identity 10.3%

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

      12. sub-neg 10.3%

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

      13. metadata-eval 10.3%

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

   5. Applied egg-rr 10.3%

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

   6. Taylor expanded in x around 0 10.3%

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

   7. Taylor expanded in x around inf 99.8%

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

      1. associate--r+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. metadata-eval 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-*r/ 99.8%

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

      7. metadata-eval 99.8%

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

      8. unpow2 99.8%

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

   9. Simplified 99.8%

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

   if 2.0000000000000001e-4 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

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

Alternative 6: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0)))))
   (if (<= t_0 1e-8) (+ (/ -3.0 x) (/ (/ -1.0 x) x)) t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	double tmp;
	if (t_0 <= 1e-8) {
		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)) - ((x + 1.0d0) / (x + (-1.0d0)))
    if (t_0 <= 1d-8) 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)) - ((x + 1.0) / (x + -1.0));
	double tmp;
	if (t_0 <= 1e-8) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))
	tmp = 0
	if t_0 <= 1e-8:
		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(x + 1.0) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 1e-8)
		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)) - ((x + 1.0) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 1e-8)
		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[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-8], 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))) < 1e-8

   1. Initial program 7.0%

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

   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%

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

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

   1. Initial program 99.7%

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

4. Final simplification 99.8%

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

Alternative 7: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.9%

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

   2. Taylor expanded in x around inf 98.3%

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

      1. +-commutative 98.3%

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

      2. distribute-neg-in 98.3%

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

      3. sub-neg 98.3%

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

      4. associate-*r/ 98.9%

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

      5. metadata-eval 98.9%

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

      6. distribute-neg-frac 98.9%

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

      7. metadata-eval 98.9%

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

      8. unpow2 98.9%

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

      9. associate-/r* 98.9%

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

   4. Simplified 98.9%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.1%

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

4. Final simplification 99.0%

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

Alternative 8: 98.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ (+ (/ 2.0 x) -3.0) (+ x -1.0))
   (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 = ((2.0 / x) + -3.0) / (x + -1.0);
	} 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 = ((2.0d0 / x) + (-3.0d0)) / (x + (-1.0d0))
    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 = ((2.0 / x) + -3.0) / (x + -1.0);
	} 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 = ((2.0 / x) + -3.0) / (x + -1.0)
	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(Float64(2.0 / x) + -3.0) / Float64(x + -1.0));
	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 = ((2.0 / x) + -3.0) / (x + -1.0);
	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[(N[(2.0 / x), $MachinePrecision] + -3.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $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 11.2%

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

      1. clear-num 11.2%

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

      2. associate-/r/ 11.2%

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

      3. sub-neg 11.2%

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

      4. metadata-eval 11.2%

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

   3. Applied egg-rr 11.2%

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

      1. *-commutative 11.2%

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

      2. metadata-eval 11.2%

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

      3. sub-neg 11.2%

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

      4. div-inv 11.2%

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

      5. clear-num 11.2%

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

      6. frac-sub 12.2%

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

      7. sub-neg 12.2%

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

      8. metadata-eval 12.2%

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

      9. metadata-eval 12.2%

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

      10. div-inv 12.2%

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

      11. /-rgt-identity 12.2%

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

      12. sub-neg 12.2%

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

      13. metadata-eval 12.2%

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

   5. Applied egg-rr 12.2%

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

   6. Taylor expanded in x around 0 12.2%

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

   7. Taylor expanded in x around inf 97.8%

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

      1. sub-neg 97.8%

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

      2. associate-*r/ 97.8%

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

      3. metadata-eval 97.8%

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

      4. metadata-eval 97.8%

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

   9. Simplified 97.8%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.1%

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

   if 1 < x

   1. Initial program 6.7%

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

   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%

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

4. Final simplification 99.0%

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

Alternative 9: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.9%

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

   2. Taylor expanded in x around inf 97.9%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.1%

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

4. Final simplification 98.5%

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

Alternative 10: 97.9% 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 8.9%

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

   2. Taylor expanded in x around inf 97.9%

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

   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 99.0%

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

   3. Taylor expanded in x around 0 98.0%

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

4. Final simplification 98.0%

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

Alternative 11: 51.3% 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 53.7%

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

2. Taylor expanded in x around 0 50.1%

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

3. Final simplification 50.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023257 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))