Asymptote C

Percentage Accurate: 53.7% → 99.8%

Time: 8.6s

Alternatives: 12

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.7% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{{x}^{4}} + \left(\frac{-3 + \frac{-1}{x}}{x} + \frac{-3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 5e-5)
     (+
      (/ -1.0 (pow x 4.0))
      (+ (/ (+ -3.0 (/ -1.0 x)) x) (/ -3.0 (pow x 3.0))))
     t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = (-1.0 / pow(x, 4.0)) + (((-3.0 + (-1.0 / x)) / x) + (-3.0 / pow(x, 3.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (x + 1.0d0)) + (((-1.0d0) - x) / (x + (-1.0d0)))
    if (t_0 <= 5d-5) then
        tmp = ((-1.0d0) / (x ** 4.0d0)) + ((((-3.0d0) + ((-1.0d0) / x)) / x) + ((-3.0d0) / (x ** 3.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = (-1.0 / Math.pow(x, 4.0)) + (((-3.0 + (-1.0 / x)) / x) + (-3.0 / Math.pow(x, 3.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))
	tmp = 0
	if t_0 <= 5e-5:
		tmp = (-1.0 / math.pow(x, 4.0)) + (((-3.0 + (-1.0 / x)) / x) + (-3.0 / math.pow(x, 3.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 5e-5)
		tmp = Float64(Float64(-1.0 / (x ^ 4.0)) + Float64(Float64(Float64(-3.0 + Float64(-1.0 / x)) / x) + Float64(-3.0 / (x ^ 3.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 5e-5)
		tmp = (-1.0 / (x ^ 4.0)) + (((-3.0 + (-1.0 / x)) / x) + (-3.0 / (x ^ 3.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-5], N[(N[(-1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-3.0 / N[Power[x, 3.0], $MachinePrecision]), $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))) < 5.00000000000000024e-5

   1. Initial program 9.1%

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

      1. clear-num 9.1%

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

      2. clear-num 9.1%

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

      3. frac-sub 9.1%

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

      4. *-un-lft-identity 9.1%

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

      5. sub-neg 9.1%

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

      6. metadata-eval 9.1%

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

      7. sub-neg 9.1%

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

      8. metadata-eval 9.1%

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

   3. Applied egg-rr 9.1%

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

      1. clear-num 9.1%

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

      2. metadata-eval 9.1%

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

      3. sub-neg 9.1%

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

      4. inv-pow 9.1%

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

      5. sub-neg 9.1%

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

      6. metadata-eval 9.1%

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

   5. Applied egg-rr 9.1%

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

      1. unpow-1 9.1%

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

      2. +-commutative 9.1%

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

   7. Simplified 9.1%

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

   8. Taylor expanded in x around inf 98.5%

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

      1. distribute-neg-in 98.5%

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

      2. distribute-neg-frac 98.5%

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

      3. metadata-eval 98.5%

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

      4. +-commutative 98.5%

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

      5. associate-+r+ 98.5%

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

      6. distribute-neg-in 98.5%

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

   10. Simplified 99.0%

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

   if 5.00000000000000024e-5 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{{x}^{4}} + \left(\frac{-3 + \frac{-1}{x}}{x} + \frac{-3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := t_0 + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{2}{{x}^{3}} + \left(\frac{2}{x} + \left(-3 - \frac{2}{x \cdot x}\right)\right)}{t_0 \cdot \left(x + 1\right)}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))) (t_1 (+ t_0 (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_1 5e-5)
     (/
      (/
       (+ (/ 2.0 (pow x 3.0)) (+ (/ 2.0 x) (- -3.0 (/ 2.0 (* x x)))))
       (* t_0 (+ x 1.0)))
      (* (/ (+ x 1.0) x) (/ (+ x -1.0) (+ x 1.0))))
     t_1)))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double t_1 = t_0 + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = (((2.0 / pow(x, 3.0)) + ((2.0 / x) + (-3.0 - (2.0 / (x * x))))) / (t_0 * (x + 1.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    t_1 = t_0 + (((-1.0d0) - x) / (x + (-1.0d0)))
    if (t_1 <= 5d-5) then
        tmp = (((2.0d0 / (x ** 3.0d0)) + ((2.0d0 / x) + ((-3.0d0) - (2.0d0 / (x * x))))) / (t_0 * (x + 1.0d0))) / (((x + 1.0d0) / x) * ((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x / (x + 1.0);
	double t_1 = t_0 + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = (((2.0 / Math.pow(x, 3.0)) + ((2.0 / x) + (-3.0 - (2.0 / (x * x))))) / (t_0 * (x + 1.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (x + 1.0)
	t_1 = t_0 + ((-1.0 - x) / (x + -1.0))
	tmp = 0
	if t_1 <= 5e-5:
		tmp = (((2.0 / math.pow(x, 3.0)) + ((2.0 / x) + (-3.0 - (2.0 / (x * x))))) / (t_0 * (x + 1.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	t_1 = Float64(t_0 + Float64(Float64(-1.0 - x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= 5e-5)
		tmp = Float64(Float64(Float64(Float64(2.0 / (x ^ 3.0)) + Float64(Float64(2.0 / x) + Float64(-3.0 - Float64(2.0 / Float64(x * x))))) / Float64(t_0 * Float64(x + 1.0))) / Float64(Float64(Float64(x + 1.0) / x) * Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	t_1 = t_0 + ((-1.0 - x) / (x + -1.0));
	tmp = 0.0;
	if (t_1 <= 5e-5)
		tmp = (((2.0 / (x ^ 3.0)) + ((2.0 / x) + (-3.0 - (2.0 / (x * x))))) / (t_0 * (x + 1.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-5], N[(N[(N[(N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / x), $MachinePrecision] + N[(-3.0 - N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.00000000000000024e-5

   1. Initial program 9.1%

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

      1. clear-num 9.1%

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

      2. clear-num 9.1%

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

      3. frac-sub 9.1%

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

      4. *-un-lft-identity 9.1%

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

      5. sub-neg 9.1%

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

      6. metadata-eval 9.1%

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

      7. sub-neg 9.1%

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

      8. metadata-eval 9.1%

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

   3. Applied egg-rr 9.1%

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

      1. clear-num 9.1%

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

      2. metadata-eval 9.1%

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

      3. sub-neg 9.1%

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

      4. inv-pow 9.1%

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

      5. sub-neg 9.1%

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

      6. metadata-eval 9.1%

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

   5. Applied egg-rr 9.1%

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

      1. unpow-1 9.1%

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

      2. +-commutative 9.1%

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

   7. Simplified 9.1%

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

      1. +-commutative 9.1%

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

      2. clear-num 9.1%

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

      3. *-rgt-identity 9.1%

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

      4. clear-num 9.1%

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

      5. frac-sub 9.2%

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

      6. +-commutative 9.2%

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

      7. +-commutative 9.2%

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

      8. +-commutative 9.2%

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

      9. +-commutative 9.2%

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

   9. Applied egg-rr 9.2%

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

       1. +-commutative 9.2%

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

       2. +-commutative 9.2%

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

       3. *-rgt-identity 9.2%

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

       4. +-commutative 9.2%

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

       5. +-commutative 9.2%

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

       6. +-commutative 9.2%

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

   11. Simplified 9.2%

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

   12. Taylor expanded in x around inf 99.0%

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

       1. +-commutative 99.0%

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

       2. associate--l+ 99.0%

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

       3. associate-*r/ 99.0%

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

       4. metadata-eval 99.0%

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

       5. associate--r+ 99.0%

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

       6. sub-neg 99.0%

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

       7. metadata-eval 99.0%

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

       8. associate--l+ 99.0%

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

       9. associate-*r/ 99.0%

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

       10. metadata-eval 99.0%

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

       11. associate-*r/ 99.0%

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

       12. metadata-eval 99.0%

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

       13. unpow2 99.0%

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

   14. Simplified 99.0%

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

   if 5.00000000000000024e-5 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{2}{{x}^{3}} + \left(\frac{2}{x} + \left(-3 - \frac{2}{x \cdot x}\right)\right)}{\frac{x}{x + 1} \cdot \left(x + 1\right)}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-5], N[(N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-3.0 / N[Power[x, 3.0], $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))) < 5.00000000000000024e-5

   1. Initial program 9.1%

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

      1. clear-num 9.1%

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

      2. clear-num 9.1%

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

      3. frac-sub 9.1%

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

      4. *-un-lft-identity 9.1%

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

      5. sub-neg 9.1%

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

      6. metadata-eval 9.1%

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

      7. sub-neg 9.1%

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

      8. metadata-eval 9.1%

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

   3. Applied egg-rr 9.1%

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

      1. clear-num 9.1%

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

      2. metadata-eval 9.1%

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

      3. sub-neg 9.1%

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

      4. inv-pow 9.1%

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

      5. sub-neg 9.1%

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

      6. metadata-eval 9.1%

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

   5. Applied egg-rr 9.1%

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

      1. unpow-1 9.1%

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

      2. +-commutative 9.1%

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

   7. Simplified 9.1%

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

   8. Taylor expanded in x around inf 98.3%

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

      1. +-commutative 98.3%

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

      2. distribute-neg-in 98.3%

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

      3. neg-sub0 98.3%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)} + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]

      4. +-commutative 98.3%

         \[\leadsto \left(0 - \color{blue}{\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)}\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]

      5. associate--r+ 98.3%

         \[\leadsto \color{blue}{\left(\left(0 - 3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}\right)} + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]

      6. neg-sub0 98.3%

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

      7. associate-*r/ 98.8%

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

      8. metadata-eval 98.8%

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

      9. distribute-neg-frac 98.8%

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

      10. metadata-eval 98.8%

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

      11. unpow2 98.8%

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

      12. associate-/r* 98.8%

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

      13. div-sub 98.8%

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

      14. associate-*r/ 98.8%

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

      15. metadata-eval 98.8%

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

      16. distribute-neg-frac 98.8%

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

      17. metadata-eval 98.8%

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

   10. Simplified 98.8%

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

   if 5.00000000000000024e-5 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 100.0%

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

4. Final simplification 99.4%

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

Alternative 4: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := t_0 + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\left(-3 + \frac{2}{x}\right) - \frac{2}{x \cdot x}}{t_0 \cdot \left(x + 1\right)}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))) (t_1 (+ t_0 (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_1 5e-5)
     (/
      (/ (- (+ -3.0 (/ 2.0 x)) (/ 2.0 (* x x))) (* t_0 (+ x 1.0)))
      (* (/ (+ x 1.0) x) (/ (+ x -1.0) (+ x 1.0))))
     t_1)))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double t_1 = t_0 + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = (((-3.0 + (2.0 / x)) - (2.0 / (x * x))) / (t_0 * (x + 1.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    t_1 = t_0 + (((-1.0d0) - x) / (x + (-1.0d0)))
    if (t_1 <= 5d-5) then
        tmp = ((((-3.0d0) + (2.0d0 / x)) - (2.0d0 / (x * x))) / (t_0 * (x + 1.0d0))) / (((x + 1.0d0) / x) * ((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x / (x + 1.0);
	double t_1 = t_0 + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = (((-3.0 + (2.0 / x)) - (2.0 / (x * x))) / (t_0 * (x + 1.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (x + 1.0)
	t_1 = t_0 + ((-1.0 - x) / (x + -1.0))
	tmp = 0
	if t_1 <= 5e-5:
		tmp = (((-3.0 + (2.0 / x)) - (2.0 / (x * x))) / (t_0 * (x + 1.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	t_1 = Float64(t_0 + Float64(Float64(-1.0 - x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= 5e-5)
		tmp = Float64(Float64(Float64(Float64(-3.0 + Float64(2.0 / x)) - Float64(2.0 / Float64(x * x))) / Float64(t_0 * Float64(x + 1.0))) / Float64(Float64(Float64(x + 1.0) / x) * Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	t_1 = t_0 + ((-1.0 - x) / (x + -1.0));
	tmp = 0.0;
	if (t_1 <= 5e-5)
		tmp = (((-3.0 + (2.0 / x)) - (2.0 / (x * x))) / (t_0 * (x + 1.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-5], N[(N[(N[(N[(-3.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.00000000000000024e-5

   1. Initial program 9.1%

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

      1. clear-num 9.1%

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

      2. clear-num 9.1%

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

      3. frac-sub 9.1%

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

      4. *-un-lft-identity 9.1%

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

      5. sub-neg 9.1%

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

      6. metadata-eval 9.1%

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

      7. sub-neg 9.1%

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

      8. metadata-eval 9.1%

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

   3. Applied egg-rr 9.1%

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

      1. clear-num 9.1%

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

      2. metadata-eval 9.1%

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

      3. sub-neg 9.1%

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

      4. inv-pow 9.1%

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

      5. sub-neg 9.1%

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

      6. metadata-eval 9.1%

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

   5. Applied egg-rr 9.1%

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

      1. unpow-1 9.1%

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

      2. +-commutative 9.1%

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

   7. Simplified 9.1%

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

      1. +-commutative 9.1%

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

      2. clear-num 9.1%

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

      3. *-rgt-identity 9.1%

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

      4. clear-num 9.1%

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

      5. frac-sub 9.2%

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

      6. +-commutative 9.2%

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

      7. +-commutative 9.2%

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

      8. +-commutative 9.2%

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

      9. +-commutative 9.2%

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

   9. Applied egg-rr 9.2%

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

       1. +-commutative 9.2%

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

       2. +-commutative 9.2%

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

       3. *-rgt-identity 9.2%

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

       4. +-commutative 9.2%

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

       5. +-commutative 9.2%

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

       6. +-commutative 9.2%

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

   11. Simplified 9.2%

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

   12. Taylor expanded in x around inf 98.8%

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

       1. associate--r+ 98.8%

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

       2. sub-neg 98.8%

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

       3. associate-*r/ 98.8%

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

       4. metadata-eval 98.8%

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

       5. metadata-eval 98.8%

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

       6. associate-*r/ 98.8%

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

       7. metadata-eval 98.8%

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

       8. unpow2 98.8%

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

   14. Simplified 98.8%

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

   if 5.00000000000000024e-5 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\left(-3 + \frac{2}{x}\right) - \frac{2}{x \cdot x}}{\frac{x}{x + 1} \cdot \left(x + 1\right)}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 5: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x + -1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot \left(x + 1\right)}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 2e-8)
   (/ (+ -3.0 (/ -1.0 x)) x)
   (/
    (/ (- (* x (+ x -1.0)) (* (+ x 1.0) (+ x 1.0))) (* x (+ x 1.0)))
    (* (/ (+ x 1.0) x) (/ (+ x -1.0) (+ x 1.0))))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 2e-8) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = (((x * (x + -1.0)) - ((x + 1.0) * (x + 1.0))) / (x * (x + 1.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x / (x + 1.0d0)) + (((-1.0d0) - x) / (x + (-1.0d0)))) <= 2d-8) then
        tmp = ((-3.0d0) + ((-1.0d0) / x)) / x
    else
        tmp = (((x * (x + (-1.0d0))) - ((x + 1.0d0) * (x + 1.0d0))) / (x * (x + 1.0d0))) / (((x + 1.0d0) / x) * ((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 2e-8) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = (((x * (x + -1.0)) - ((x + 1.0) * (x + 1.0))) / (x * (x + 1.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 2e-8:
		tmp = (-3.0 + (-1.0 / x)) / x
	else:
		tmp = (((x * (x + -1.0)) - ((x + 1.0) * (x + 1.0))) / (x * (x + 1.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 2e-8)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = Float64(Float64(Float64(Float64(x * Float64(x + -1.0)) - Float64(Float64(x + 1.0) * Float64(x + 1.0))) / Float64(x * Float64(x + 1.0))) / Float64(Float64(Float64(x + 1.0) / x) * Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 2e-8)
		tmp = (-3.0 + (-1.0 / x)) / x;
	else
		tmp = (((x * (x + -1.0)) - ((x + 1.0) * (x + 1.0))) / (x * (x + 1.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x + -1.0), $MachinePrecision] / 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))) < 2e-8

   1. Initial program 8.5%

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

      1. clear-num 8.5%

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

      2. clear-num 8.5%

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

      3. frac-sub 8.5%

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

      4. *-un-lft-identity 8.5%

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

      5. sub-neg 8.5%

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

      6. metadata-eval 8.5%

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

      7. sub-neg 8.5%

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

      8. metadata-eval 8.5%

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

   3. Applied egg-rr 8.5%

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

      1. clear-num 8.6%

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

      2. metadata-eval 8.6%

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

      3. sub-neg 8.6%

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

      4. inv-pow 8.6%

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

      5. sub-neg 8.6%

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

      6. metadata-eval 8.6%

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

   5. Applied egg-rr 8.6%

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

      1. unpow-1 8.6%

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

      2. +-commutative 8.6%

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

   7. Simplified 8.6%

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

   8. Taylor expanded in x around inf 98.0%

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

      1. neg-sub0 98.0%

         \[\leadsto \color{blue}{0 - \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)} \]

      2. +-commutative 98.0%

         \[\leadsto 0 - \color{blue}{\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]

      3. associate--r+ 98.0%

         \[\leadsto \color{blue}{\left(0 - 3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]

      4. neg-sub0 98.0%

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

      5. associate-*r/ 98.5%

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

      6. metadata-eval 98.5%

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

      7. distribute-neg-frac 98.5%

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

      8. metadata-eval 98.5%

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

      9. unpow2 98.5%

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

      10. associate-/r* 98.5%

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

      11. div-sub 98.5%

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

   10. Simplified 98.5%

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

   if 2e-8 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.8%

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

      1. clear-num 99.8%

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

      2. clear-num 99.8%

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

      3. frac-sub 99.8%

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

      4. *-un-lft-identity 99.8%

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

      5. sub-neg 99.8%

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

      6. metadata-eval 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. metadata-eval 99.8%

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

      3. sub-neg 99.8%

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

      4. inv-pow 99.8%

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

      5. sub-neg 99.8%

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

      6. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. +-commutative 99.8%

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

      2. clear-num 99.8%

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

      3. associate-*l/ 99.8%

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

      4. frac-sub 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x + -1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot \left(x + 1\right)}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 6: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;t_0 + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0 \cdot \left(x + -1\right) + \left(-1 - x\right)}{x}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (+ t_0 (/ (- -1.0 x) (+ x -1.0))) 2e-8)
     (/ (+ -3.0 (/ -1.0 x)) x)
     (/
      (/ (+ (* t_0 (+ x -1.0)) (- -1.0 x)) x)
      (* (/ (+ x 1.0) x) (/ (+ x -1.0) (+ x 1.0)))))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 + ((-1.0 - x) / (x + -1.0))) <= 2e-8) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = (((t_0 * (x + -1.0)) + (-1.0 - x)) / x) / (((x + 1.0) / x) * ((x + -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 + (((-1.0d0) - x) / (x + (-1.0d0)))) <= 2d-8) then
        tmp = ((-3.0d0) + ((-1.0d0) / x)) / x
    else
        tmp = (((t_0 * (x + (-1.0d0))) + ((-1.0d0) - x)) / x) / (((x + 1.0d0) / x) * ((x + (-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 + ((-1.0 - x) / (x + -1.0))) <= 2e-8) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = (((t_0 * (x + -1.0)) + (-1.0 - x)) / x) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (x + 1.0)
	tmp = 0
	if (t_0 + ((-1.0 - x) / (x + -1.0))) <= 2e-8:
		tmp = (-3.0 + (-1.0 / x)) / x
	else:
		tmp = (((t_0 * (x + -1.0)) + (-1.0 - x)) / x) / (((x + 1.0) / x) * ((x + -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(-1.0 - x) / Float64(x + -1.0))) <= 2e-8)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = Float64(Float64(Float64(Float64(t_0 * Float64(x + -1.0)) + Float64(-1.0 - x)) / x) / Float64(Float64(Float64(x + 1.0) / x) * Float64(Float64(x + -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 + ((-1.0 - x) / (x + -1.0))) <= 2e-8)
		tmp = (-3.0 + (-1.0 / x)) / x;
	else
		tmp = (((t_0 * (x + -1.0)) + (-1.0 - x)) / x) / (((x + 1.0) / x) * ((x + -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[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(t$95$0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x + -1.0), $MachinePrecision] / 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))) < 2e-8

   1. Initial program 8.5%

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

      1. clear-num 8.5%

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

      2. clear-num 8.5%

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

      3. frac-sub 8.5%

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

      4. *-un-lft-identity 8.5%

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

      5. sub-neg 8.5%

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

      6. metadata-eval 8.5%

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

      7. sub-neg 8.5%

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

      8. metadata-eval 8.5%

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

   3. Applied egg-rr 8.5%

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

      1. clear-num 8.6%

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

      2. metadata-eval 8.6%

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

      3. sub-neg 8.6%

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

      4. inv-pow 8.6%

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

      5. sub-neg 8.6%

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

      6. metadata-eval 8.6%

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

   5. Applied egg-rr 8.6%

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

      1. unpow-1 8.6%

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

      2. +-commutative 8.6%

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

   7. Simplified 8.6%

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

   8. Taylor expanded in x around inf 98.0%

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

      1. neg-sub0 98.0%

         \[\leadsto \color{blue}{0 - \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)} \]

      2. +-commutative 98.0%

         \[\leadsto 0 - \color{blue}{\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]

      3. associate--r+ 98.0%

         \[\leadsto \color{blue}{\left(0 - 3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]

      4. neg-sub0 98.0%

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

      5. associate-*r/ 98.5%

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

      6. metadata-eval 98.5%

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

      7. distribute-neg-frac 98.5%

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

      8. metadata-eval 98.5%

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

      9. unpow2 98.5%

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

      10. associate-/r* 98.5%

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

      11. div-sub 98.5%

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

   10. Simplified 98.5%

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

   if 2e-8 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.8%

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

      1. clear-num 99.8%

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

      2. clear-num 99.8%

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

      3. frac-sub 99.8%

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

      4. *-un-lft-identity 99.8%

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

      5. sub-neg 99.8%

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

      6. metadata-eval 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. metadata-eval 99.8%

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

      3. sub-neg 99.8%

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

      4. inv-pow 99.8%

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

      5. sub-neg 99.8%

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

      6. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. +-commutative 99.8%

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

      2. clear-num 99.8%

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

      3. *-rgt-identity 99.8%

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

      4. clear-num 99.8%

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

      5. frac-sub 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. +-commutative 99.9%

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

       2. +-commutative 99.9%

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

       3. *-rgt-identity 99.9%

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

       4. +-commutative 99.9%

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

       5. +-commutative 99.9%

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

       6. +-commutative 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in x around 0 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{x + 1} \cdot \left(x + -1\right) + \left(-1 - x\right)}{x}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 7: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 2e-8) (/ (+ -3.0 (/ -1.0 x)) x) t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 2e-8) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (x + 1.0d0)) + (((-1.0d0) - x) / (x + (-1.0d0)))
    if (t_0 <= 2d-8) then
        tmp = ((-3.0d0) + ((-1.0d0) / x)) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 2e-8) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))
	tmp = 0
	if t_0 <= 2e-8:
		tmp = (-3.0 + (-1.0 / x)) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 2e-8)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 2e-8)
		tmp = (-3.0 + (-1.0 / x)) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-8], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 2e-8

   1. Initial program 8.5%

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

      1. clear-num 8.5%

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

      2. clear-num 8.5%

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

      3. frac-sub 8.5%

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

      4. *-un-lft-identity 8.5%

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

      5. sub-neg 8.5%

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

      6. metadata-eval 8.5%

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

      7. sub-neg 8.5%

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

      8. metadata-eval 8.5%

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

   3. Applied egg-rr 8.5%

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

      1. clear-num 8.6%

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

      2. metadata-eval 8.6%

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

      3. sub-neg 8.6%

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

      4. inv-pow 8.6%

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

      5. sub-neg 8.6%

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

      6. metadata-eval 8.6%

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

   5. Applied egg-rr 8.6%

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

      1. unpow-1 8.6%

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

      2. +-commutative 8.6%

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

   7. Simplified 8.6%

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

   8. Taylor expanded in x around inf 98.0%

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

      1. neg-sub0 98.0%

         \[\leadsto \color{blue}{0 - \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)} \]

      2. +-commutative 98.0%

         \[\leadsto 0 - \color{blue}{\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]

      3. associate--r+ 98.0%

         \[\leadsto \color{blue}{\left(0 - 3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]

      4. neg-sub0 98.0%

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

      5. associate-*r/ 98.5%

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

      6. metadata-eval 98.5%

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

      7. distribute-neg-frac 98.5%

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

      8. metadata-eval 98.5%

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

      9. unpow2 98.5%

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

      10. associate-/r* 98.5%

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

      11. div-sub 98.5%

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

   10. Simplified 98.5%

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

   if 2e-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 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 8: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.7\right):\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1 - x}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.7)))
   (/ (+ -3.0 (/ -1.0 x)) x)
   (+ x (/ (- -1.0 x) (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.7)) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = x + ((-1.0 - x) / (x + -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 <= 1.7d0))) then
        tmp = ((-3.0d0) + ((-1.0d0) / x)) / x
    else
        tmp = x + (((-1.0d0) - x) / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.7)) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = x + ((-1.0 - x) / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.7):
		tmp = (-3.0 + (-1.0 / x)) / x
	else:
		tmp = x + ((-1.0 - x) / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.7))
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = Float64(x + Float64(Float64(-1.0 - x) / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.7)))
		tmp = (-3.0 + (-1.0 / x)) / x;
	else
		tmp = x + ((-1.0 - x) / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.7]], $MachinePrecision]], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.69999999999999996 < x

   1. Initial program 9.1%

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

      1. clear-num 9.1%

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

      2. clear-num 9.1%

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

      3. frac-sub 9.1%

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

      4. *-un-lft-identity 9.1%

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

      5. sub-neg 9.1%

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

      6. metadata-eval 9.1%

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

      7. sub-neg 9.1%

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

      8. metadata-eval 9.1%

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

   3. Applied egg-rr 9.1%

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

      1. clear-num 9.1%

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

      2. metadata-eval 9.1%

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

      3. sub-neg 9.1%

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

      4. inv-pow 9.1%

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

      5. sub-neg 9.1%

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

      6. metadata-eval 9.1%

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

   5. Applied egg-rr 9.1%

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

      1. unpow-1 9.1%

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

      2. +-commutative 9.1%

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

   7. Simplified 9.1%

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

   8. Taylor expanded in x around inf 97.7%

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

      1. neg-sub0 97.7%

         \[\leadsto \color{blue}{0 - \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)} \]

      2. +-commutative 97.7%

         \[\leadsto 0 - \color{blue}{\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]

      3. associate--r+ 97.7%

         \[\leadsto \color{blue}{\left(0 - 3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]

      4. neg-sub0 97.7%

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

      5. associate-*r/ 98.3%

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

      6. metadata-eval 98.3%

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

      7. distribute-neg-frac 98.3%

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

      8. metadata-eval 98.3%

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

      9. unpow2 98.3%

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

      10. associate-/r* 98.3%

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

      11. div-sub 98.3%

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

   10. Simplified 98.3%

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

   if -1 < x < 1.69999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.7\right):\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 9: 99.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 9.1%

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

      1. clear-num 9.1%

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

      2. clear-num 9.1%

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

      3. frac-sub 9.1%

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

      4. *-un-lft-identity 9.1%

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

      5. sub-neg 9.1%

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

      6. metadata-eval 9.1%

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

      7. sub-neg 9.1%

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

      8. metadata-eval 9.1%

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

   3. Applied egg-rr 9.1%

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

      1. clear-num 9.1%

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

      2. metadata-eval 9.1%

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

      3. sub-neg 9.1%

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

      4. inv-pow 9.1%

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

      5. sub-neg 9.1%

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

      6. metadata-eval 9.1%

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

   5. Applied egg-rr 9.1%

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

      1. unpow-1 9.1%

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

      2. +-commutative 9.1%

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

   7. Simplified 9.1%

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

   8. Taylor expanded in x around inf 97.7%

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

      1. neg-sub0 97.7%

         \[\leadsto \color{blue}{0 - \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)} \]

      2. +-commutative 97.7%

         \[\leadsto 0 - \color{blue}{\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]

      3. associate--r+ 97.7%

         \[\leadsto \color{blue}{\left(0 - 3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]

      4. neg-sub0 97.7%

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

      5. associate-*r/ 98.3%

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

      6. metadata-eval 98.3%

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

      7. distribute-neg-frac 98.3%

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

      8. metadata-eval 98.3%

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

      9. unpow2 98.3%

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

      10. associate-/r* 98.3%

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

      11. div-sub 98.3%

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

   10. Simplified 98.3%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.2%

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

4. Final simplification 98.2%

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

Alternative 10: 98.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 9.1%

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

   2. Taylor expanded in x around inf 97.3%

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

   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 98.2%

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

4. Final simplification 97.8%

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

Alternative 11: 98.0% 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 9.1%

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

   2. Taylor expanded in x around inf 97.3%

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

   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 98.3%

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

   3. Taylor expanded in x around 0 97.1%

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

4. Final simplification 97.2%

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

Alternative 12: 50.3% 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 57.0%

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

2. Taylor expanded in x around 0 53.0%

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

3. Final simplification 53.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023175 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))