Asymptote C

Percentage Accurate: 54.0% → 99.8%

Time: 9.1s

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: 54.0% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;t_0 + \frac{-1 - x}{x + -1} \leq 0:\\ \;\;\;\;\frac{\left(\frac{2}{{x}^{4}} + \left(\frac{\frac{2}{x}}{x} + \frac{-3}{x}\right)\right) - \frac{2}{{x}^{3}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x + -1} \cdot \left(x + \left(-2 - x\right)\right) - \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (+ t_0 (/ (- -1.0 x) (+ x -1.0))) 0.0)
     (/
      (-
       (+ (/ 2.0 (pow x 4.0)) (+ (/ (/ 2.0 x) x) (/ -3.0 x)))
       (/ 2.0 (pow x 3.0)))
      (* (/ (+ x 1.0) x) (/ (+ x -1.0) (+ x 1.0))))
     (- (* (/ t_0 (+ x -1.0)) (+ x (- -2.0 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))) <= 0.0) {
		tmp = (((2.0 / pow(x, 4.0)) + (((2.0 / x) / x) + (-3.0 / x))) - (2.0 / pow(x, 3.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)));
	} else {
		tmp = ((t_0 / (x + -1.0)) * (x + (-2.0 - 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)))) <= 0.0d0) then
        tmp = (((2.0d0 / (x ** 4.0d0)) + (((2.0d0 / x) / x) + ((-3.0d0) / x))) - (2.0d0 / (x ** 3.0d0))) / (((x + 1.0d0) / x) * ((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = ((t_0 / (x + (-1.0d0))) * (x + ((-2.0d0) - 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))) <= 0.0) {
		tmp = (((2.0 / Math.pow(x, 4.0)) + (((2.0 / x) / x) + (-3.0 / x))) - (2.0 / Math.pow(x, 3.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)));
	} else {
		tmp = ((t_0 / (x + -1.0)) * (x + (-2.0 - 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))) <= 0.0:
		tmp = (((2.0 / math.pow(x, 4.0)) + (((2.0 / x) / x) + (-3.0 / x))) - (2.0 / math.pow(x, 3.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)))
	else:
		tmp = ((t_0 / (x + -1.0)) * (x + (-2.0 - 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))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(2.0 / (x ^ 4.0)) + Float64(Float64(Float64(2.0 / x) / x) + Float64(-3.0 / x))) - Float64(2.0 / (x ^ 3.0))) / Float64(Float64(Float64(x + 1.0) / x) * Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	else
		tmp = Float64(Float64(Float64(t_0 / Float64(x + -1.0)) * Float64(x + Float64(-2.0 - x))) - Float64(1.0 / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if ((t_0 + ((-1.0 - x) / (x + -1.0))) <= 0.0)
		tmp = (((2.0 / (x ^ 4.0)) + (((2.0 / x) / x) + (-3.0 / x))) - (2.0 / (x ^ 3.0))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)));
	else
		tmp = ((t_0 / (x + -1.0)) * (x + (-2.0 - 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], 0.0], N[(N[(N[(N[(2.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 / x), $MachinePrecision] / x), $MachinePrecision] + N[(-3.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[Power[x, 3.0], $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], N[(N[(N[(t$95$0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x + N[(-2.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(x + -1.0), $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))) < 0.0

   1. Initial program 6.9%

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

      1. clear-num 6.9%

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

      2. clear-num 6.8%

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

      3. frac-sub 6.9%

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

      4. *-un-lft-identity 6.9%

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

      5. sub-neg 6.9%

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

      6. metadata-eval 6.9%

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

      7. sub-neg 6.9%

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

      8. metadata-eval 6.9%

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

   3. Applied egg-rr 6.9%

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

   4. Taylor expanded in x around inf 98.8%

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

      1. associate--r+ 98.8%

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

      2. associate--l+ 98.8%

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

      3. associate-*r/ 98.8%

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

      4. metadata-eval 98.8%

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

      5. sub-neg 98.8%

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

      6. associate-*r/ 98.8%

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

      7. metadata-eval 98.8%

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

      8. unpow2 98.8%

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

      9. associate-/r* 98.8%

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

      10. associate-*r/ 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. associate-*r/ 99.4%

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

      15. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

   if 0.0 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 98.8%

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

      1. clear-num 98.8%

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

      2. clear-num 98.8%

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

      3. frac-sub 98.8%

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

      4. *-un-lft-identity 98.8%

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

      5. sub-neg 98.8%

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

      6. metadata-eval 98.8%

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

      7. sub-neg 98.8%

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

      8. metadata-eval 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. *-rgt-identity 98.8%

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

      2. *-rgt-identity 98.8%

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

      3. *-un-lft-identity 98.8%

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

      4. frac-sub 98.8%

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

      5. *-rgt-identity 98.8%

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

      6. clear-num 98.8%

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

      7. clear-num 98.8%

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

      8. *-un-lft-identity 98.8%

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

      9. associate-*l/ 98.8%

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

      10. distribute-rgt-in 98.8%

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

      11. *-un-lft-identity 98.8%

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

      12. associate--r+ 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. frac-sub 98.9%

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

      2. fma-neg 98.9%

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

   7. Applied egg-rr 98.9%

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

      1. fma-udef 98.9%

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

      2. *-commutative 98.9%

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

      3. distribute-lft-neg-in 98.9%

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

      4. distribute-rgt-out 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. associate--r+ 100.0%

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

      8. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

       1. expm1-log1p-u 100.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\left(x + 1\right) \cdot \left(x + -1\right)}\right)\right)} - \frac{1}{x + -1} \]

       2. expm1-udef 98.8%

          \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\left(x + 1\right) \cdot \left(x + -1\right)}\right)} - 1\right)} - \frac{1}{x + -1} \]

       3. *-commutative 98.8%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\color{blue}{\left(x + -1\right) \cdot \left(x + 1\right)}}\right)} - 1\right) - \frac{1}{x + -1} \]

   11. Applied egg-rr 98.8%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\left(x + -1\right) \cdot \left(x + 1\right)}\right)} - 1\right)} - \frac{1}{x + -1} \]
   12. Step-by-step derivation

       1. expm1-def 100.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\left(x + -1\right) \cdot \left(x + 1\right)}\right)\right)} - \frac{1}{x + -1} \]

       2. expm1-log1p 100.0%

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

       3. *-lft-identity 100.0%

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

       4. associate-*l/ 100.0%

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

       5. associate-*r* 100.0%

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

       6. associate-/r/ 100.0%

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

       7. associate-*r/ 100.0%

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

       8. *-commutative 100.0%

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

       9. associate-/r* 100.0%

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

       10. associate-/l* 100.0%

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

       11. *-lft-identity 100.0%

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

       12. associate-+l+ 100.0%

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

       13. associate-+r- 100.0%

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

       14. metadata-eval 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0:\\ \;\;\;\;\frac{\left(\frac{2}{{x}^{4}} + \left(\frac{\frac{2}{x}}{x} + \frac{-3}{x}\right)\right) - \frac{2}{{x}^{3}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1}}{x + -1} \cdot \left(x + \left(-2 - x\right)\right) - \frac{1}{x + -1}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;t_0 + \frac{-1 - x}{x + -1} \leq 0:\\ \;\;\;\;\frac{-1}{{x}^{4}} + \left(\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) - \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x + -1} \cdot \left(x + \left(-2 - x\right)\right) - \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (+ t_0 (/ (- -1.0 x) (+ x -1.0))) 0.0)
     (+
      (/ -1.0 (pow x 4.0))
      (- (+ (/ -3.0 x) (/ -1.0 (* x x))) (/ 3.0 (pow x 3.0))))
     (- (* (/ t_0 (+ x -1.0)) (+ x (- -2.0 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))) <= 0.0) {
		tmp = (-1.0 / pow(x, 4.0)) + (((-3.0 / x) + (-1.0 / (x * x))) - (3.0 / pow(x, 3.0)));
	} else {
		tmp = ((t_0 / (x + -1.0)) * (x + (-2.0 - 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)))) <= 0.0d0) then
        tmp = ((-1.0d0) / (x ** 4.0d0)) + ((((-3.0d0) / x) + ((-1.0d0) / (x * x))) - (3.0d0 / (x ** 3.0d0)))
    else
        tmp = ((t_0 / (x + (-1.0d0))) * (x + ((-2.0d0) - 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))) <= 0.0) {
		tmp = (-1.0 / Math.pow(x, 4.0)) + (((-3.0 / x) + (-1.0 / (x * x))) - (3.0 / Math.pow(x, 3.0)));
	} else {
		tmp = ((t_0 / (x + -1.0)) * (x + (-2.0 - 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))) <= 0.0:
		tmp = (-1.0 / math.pow(x, 4.0)) + (((-3.0 / x) + (-1.0 / (x * x))) - (3.0 / math.pow(x, 3.0)))
	else:
		tmp = ((t_0 / (x + -1.0)) * (x + (-2.0 - 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))) <= 0.0)
		tmp = Float64(Float64(-1.0 / (x ^ 4.0)) + Float64(Float64(Float64(-3.0 / x) + Float64(-1.0 / Float64(x * x))) - Float64(3.0 / (x ^ 3.0))));
	else
		tmp = Float64(Float64(Float64(t_0 / Float64(x + -1.0)) * Float64(x + Float64(-2.0 - x))) - Float64(1.0 / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if ((t_0 + ((-1.0 - x) / (x + -1.0))) <= 0.0)
		tmp = (-1.0 / (x ^ 4.0)) + (((-3.0 / x) + (-1.0 / (x * x))) - (3.0 / (x ^ 3.0)));
	else
		tmp = ((t_0 / (x + -1.0)) * (x + (-2.0 - 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], 0.0], N[(N[(-1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-3.0 / x), $MachinePrecision] + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x + N[(-2.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(x + -1.0), $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))) < 0.0

   1. Initial program 6.9%

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

   2. Taylor expanded in x around inf 98.8%

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

      1. distribute-neg-in 98.8%

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

      2. distribute-neg-frac 98.8%

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

      3. metadata-eval 98.8%

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

      4. +-commutative 98.8%

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

      5. associate-+r+ 98.8%

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

      6. distribute-neg-in 98.8%

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

      7. sub-neg 98.8%

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

   4. Simplified 99.4%

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

   if 0.0 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 98.8%

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

      1. clear-num 98.8%

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

      2. clear-num 98.8%

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

      3. frac-sub 98.8%

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

      4. *-un-lft-identity 98.8%

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

      5. sub-neg 98.8%

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

      6. metadata-eval 98.8%

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

      7. sub-neg 98.8%

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

      8. metadata-eval 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. *-rgt-identity 98.8%

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

      2. *-rgt-identity 98.8%

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

      3. *-un-lft-identity 98.8%

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

      4. frac-sub 98.8%

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

      5. *-rgt-identity 98.8%

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

      6. clear-num 98.8%

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

      7. clear-num 98.8%

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

      8. *-un-lft-identity 98.8%

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

      9. associate-*l/ 98.8%

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

      10. distribute-rgt-in 98.8%

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

      11. *-un-lft-identity 98.8%

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

      12. associate--r+ 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. frac-sub 98.9%

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

      2. fma-neg 98.9%

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

   7. Applied egg-rr 98.9%

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

      1. fma-udef 98.9%

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

      2. *-commutative 98.9%

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

      3. distribute-lft-neg-in 98.9%

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

      4. distribute-rgt-out 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. associate--r+ 100.0%

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

      8. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

       1. expm1-log1p-u 100.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\left(x + 1\right) \cdot \left(x + -1\right)}\right)\right)} - \frac{1}{x + -1} \]

       2. expm1-udef 98.8%

          \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\left(x + 1\right) \cdot \left(x + -1\right)}\right)} - 1\right)} - \frac{1}{x + -1} \]

       3. *-commutative 98.8%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\color{blue}{\left(x + -1\right) \cdot \left(x + 1\right)}}\right)} - 1\right) - \frac{1}{x + -1} \]

   11. Applied egg-rr 98.8%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\left(x + -1\right) \cdot \left(x + 1\right)}\right)} - 1\right)} - \frac{1}{x + -1} \]
   12. Step-by-step derivation

       1. expm1-def 100.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\left(x + -1\right) \cdot \left(x + 1\right)}\right)\right)} - \frac{1}{x + -1} \]

       2. expm1-log1p 100.0%

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

       3. *-lft-identity 100.0%

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

       4. associate-*l/ 100.0%

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

       5. associate-*r* 100.0%

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

       6. associate-/r/ 100.0%

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

       7. associate-*r/ 100.0%

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

       8. *-commutative 100.0%

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

       9. associate-/r* 100.0%

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

       10. associate-/l* 100.0%

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

       11. *-lft-identity 100.0%

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

       12. associate-+l+ 100.0%

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

       13. associate-+r- 100.0%

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

       14. metadata-eval 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0:\\ \;\;\;\;\frac{-1}{{x}^{4}} + \left(\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) - \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1}}{x + -1} \cdot \left(x + \left(-2 - x\right)\right) - \frac{1}{x + -1}\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;t_0 + \frac{-1 - x}{x + -1} \leq 0:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x + -1} \cdot \left(x + \left(-2 - x\right)\right) - \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (+ t_0 (/ (- -1.0 x) (+ x -1.0))) 0.0)
     (+ (/ -3.0 x) (/ (/ -1.0 x) x))
     (- (* (/ t_0 (+ x -1.0)) (+ x (- -2.0 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))) <= 0.0) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = ((t_0 / (x + -1.0)) * (x + (-2.0 - 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)))) <= 0.0d0) then
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    else
        tmp = ((t_0 / (x + (-1.0d0))) * (x + ((-2.0d0) - 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))) <= 0.0) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = ((t_0 / (x + -1.0)) * (x + (-2.0 - 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))) <= 0.0:
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	else:
		tmp = ((t_0 / (x + -1.0)) * (x + (-2.0 - 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))) <= 0.0)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x));
	else
		tmp = Float64(Float64(Float64(t_0 / Float64(x + -1.0)) * Float64(x + Float64(-2.0 - x))) - Float64(1.0 / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if ((t_0 + ((-1.0 - x) / (x + -1.0))) <= 0.0)
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	else
		tmp = ((t_0 / (x + -1.0)) * (x + (-2.0 - 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], 0.0], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x + N[(-2.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(x + -1.0), $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))) < 0.0

   1. Initial program 6.9%

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

   2. Taylor expanded in x around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. distribute-neg-in 98.5%

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

      3. sub-neg 98.5%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

      6. distribute-neg-frac 99.1%

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

      7. metadata-eval 99.1%

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

      8. unpow2 99.1%

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

      9. associate-/r* 99.1%

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

   4. Simplified 99.1%

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

   if 0.0 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 98.8%

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

      1. clear-num 98.8%

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

      2. clear-num 98.8%

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

      3. frac-sub 98.8%

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

      4. *-un-lft-identity 98.8%

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

      5. sub-neg 98.8%

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

      6. metadata-eval 98.8%

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

      7. sub-neg 98.8%

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

      8. metadata-eval 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. *-rgt-identity 98.8%

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

      2. *-rgt-identity 98.8%

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

      3. *-un-lft-identity 98.8%

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

      4. frac-sub 98.8%

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

      5. *-rgt-identity 98.8%

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

      6. clear-num 98.8%

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

      7. clear-num 98.8%

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

      8. *-un-lft-identity 98.8%

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

      9. associate-*l/ 98.8%

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

      10. distribute-rgt-in 98.8%

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

      11. *-un-lft-identity 98.8%

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

      12. associate--r+ 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. frac-sub 98.9%

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

      2. fma-neg 98.9%

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

   7. Applied egg-rr 98.9%

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

      1. fma-udef 98.9%

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

      2. *-commutative 98.9%

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

      3. distribute-lft-neg-in 98.9%

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

      4. distribute-rgt-out 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. associate--r+ 100.0%

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

      8. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

       1. expm1-log1p-u 100.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\left(x + 1\right) \cdot \left(x + -1\right)}\right)\right)} - \frac{1}{x + -1} \]

       2. expm1-udef 98.8%

          \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\left(x + 1\right) \cdot \left(x + -1\right)}\right)} - 1\right)} - \frac{1}{x + -1} \]

       3. *-commutative 98.8%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\color{blue}{\left(x + -1\right) \cdot \left(x + 1\right)}}\right)} - 1\right) - \frac{1}{x + -1} \]

   11. Applied egg-rr 98.8%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\left(x + -1\right) \cdot \left(x + 1\right)}\right)} - 1\right)} - \frac{1}{x + -1} \]
   12. Step-by-step derivation

       1. expm1-def 100.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\left(x + -1\right) \cdot \left(x + 1\right)}\right)\right)} - \frac{1}{x + -1} \]

       2. expm1-log1p 100.0%

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

       3. *-lft-identity 100.0%

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

       4. associate-*l/ 100.0%

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

       5. associate-*r* 100.0%

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

       6. associate-/r/ 100.0%

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

       7. associate-*r/ 100.0%

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

       8. *-commutative 100.0%

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

       9. associate-/r* 100.0%

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

       10. associate-/l* 100.0%

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

       11. *-lft-identity 100.0%

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

       12. associate-+l+ 100.0%

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

       13. associate-+r- 100.0%

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

       14. metadata-eval 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1}}{x + -1} \cdot \left(x + \left(-2 - x\right)\right) - \frac{1}{x + -1}\\ \end{array} \]

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;t_0 + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 - \frac{x}{x + -1}\right) + \frac{-1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (+ t_0 (/ (- -1.0 x) (+ x -1.0))) 2e-13)
     (+ (/ -3.0 x) (/ (/ -1.0 x) x))
     (+ (- t_0 (/ x (+ x -1.0))) (/ -1.0 (+ x -1.0))))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 + ((-1.0 - x) / (x + -1.0))) <= 2e-13) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = (t_0 - (x / (x + -1.0))) + (-1.0 / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    if ((t_0 + (((-1.0d0) - x) / (x + (-1.0d0)))) <= 2d-13) then
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    else
        tmp = (t_0 - (x / (x + (-1.0d0)))) + ((-1.0d0) / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 + ((-1.0 - x) / (x + -1.0))) <= 2e-13) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = (t_0 - (x / (x + -1.0))) + (-1.0 / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (x + 1.0)
	tmp = 0
	if (t_0 + ((-1.0 - x) / (x + -1.0))) <= 2e-13:
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	else:
		tmp = (t_0 - (x / (x + -1.0))) + (-1.0 / (x + -1.0))
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 2e-13)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x));
	else
		tmp = Float64(Float64(t_0 - Float64(x / Float64(x + -1.0))) + Float64(-1.0 / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if ((t_0 + ((-1.0 - x) / (x + -1.0))) <= 2e-13)
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	else
		tmp = (t_0 - (x / (x + -1.0))) + (-1.0 / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-13], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 2.0000000000000001e-13

   1. Initial program 7.3%

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

   2. Taylor expanded in x around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. distribute-neg-in 98.5%

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

      3. sub-neg 98.5%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

      6. distribute-neg-frac 99.1%

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

      7. metadata-eval 99.1%

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

      8. unpow2 99.1%

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

      9. associate-/r* 99.1%

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

   4. Simplified 99.1%

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

   if 2.0000000000000001e-13 < (-.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 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. sub-neg 99.9%

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

      6. metadata-eval 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. *-rgt-identity 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. frac-sub 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. clear-num 99.8%

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

      7. clear-num 99.8%

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

      8. *-un-lft-identity 99.8%

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

      9. associate-*l/ 99.8%

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

      10. distribute-rgt-in 99.8%

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

      11. *-un-lft-identity 99.8%

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

      12. associate--r+ 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{x + 1} - \frac{x}{x + -1}\right) + \frac{-1}{x + -1}\\ \end{array} \]

Alternative 5: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{\left(x + 1\right) \cdot \left(x + -1\right)} + \frac{-1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.0)
   (+ (/ -3.0 x) (/ (/ -1.0 x) x))
   (+ (/ (* x -2.0) (* (+ x 1.0) (+ x -1.0))) (/ -1.0 (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.0) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = ((x * -2.0) / ((x + 1.0) * (x + -1.0))) + (-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)))) <= 0.0d0) then
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    else
        tmp = ((x * (-2.0d0)) / ((x + 1.0d0) * (x + (-1.0d0)))) + ((-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))) <= 0.0) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = ((x * -2.0) / ((x + 1.0) * (x + -1.0))) + (-1.0 / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.0:
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	else:
		tmp = ((x * -2.0) / ((x + 1.0) * (x + -1.0))) + (-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))) <= 0.0)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x));
	else
		tmp = Float64(Float64(Float64(x * -2.0) / Float64(Float64(x + 1.0) * Float64(x + -1.0))) + Float64(-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))) <= 0.0)
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	else
		tmp = ((x * -2.0) / ((x + 1.0) * (x + -1.0))) + (-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], 0.0], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * -2.0), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(x + -1.0), $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))) < 0.0

   1. Initial program 6.9%

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

   2. Taylor expanded in x around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. distribute-neg-in 98.5%

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

      3. sub-neg 98.5%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

      6. distribute-neg-frac 99.1%

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

      7. metadata-eval 99.1%

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

      8. unpow2 99.1%

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

      9. associate-/r* 99.1%

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

   4. Simplified 99.1%

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

   if 0.0 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 98.8%

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

      1. clear-num 98.8%

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

      2. clear-num 98.8%

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

      3. frac-sub 98.8%

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

      4. *-un-lft-identity 98.8%

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

      5. sub-neg 98.8%

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

      6. metadata-eval 98.8%

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

      7. sub-neg 98.8%

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

      8. metadata-eval 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. *-rgt-identity 98.8%

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

      2. *-rgt-identity 98.8%

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

      3. *-un-lft-identity 98.8%

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

      4. frac-sub 98.8%

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

      5. *-rgt-identity 98.8%

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

      6. clear-num 98.8%

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

      7. clear-num 98.8%

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

      8. *-un-lft-identity 98.8%

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

      9. associate-*l/ 98.8%

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

      10. distribute-rgt-in 98.8%

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

      11. *-un-lft-identity 98.8%

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

      12. associate--r+ 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. frac-sub 98.9%

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

      2. fma-neg 98.9%

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

   7. Applied egg-rr 98.9%

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

      1. fma-udef 98.9%

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

      2. *-commutative 98.9%

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

      3. distribute-lft-neg-in 98.9%

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

      4. distribute-rgt-out 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. associate--r+ 100.0%

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

      8. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{\left(x + 1\right) \cdot \left(x + -1\right)} + \frac{-1}{x + -1}\\ \end{array} \]

Alternative 6: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;t_0 + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{-1}{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (+ t_0 (/ (- -1.0 x) (+ x -1.0))) 2e-13)
     (+ (/ -3.0 x) (/ (/ -1.0 x) x))
     (+ t_0 (/ -1.0 (/ (+ 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-13) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = t_0 + (-1.0 / ((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-13) then
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    else
        tmp = t_0 + ((-1.0d0) / ((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-13) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = t_0 + (-1.0 / ((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-13:
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	else:
		tmp = t_0 + (-1.0 / ((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-13)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x));
	else
		tmp = Float64(t_0 + Float64(-1.0 / 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-13)
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	else
		tmp = t_0 + (-1.0 / ((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-13], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.0 / 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))) < 2.0000000000000001e-13

   1. Initial program 7.3%

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

   2. Taylor expanded in x around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. distribute-neg-in 98.5%

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

      3. sub-neg 98.5%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

      6. distribute-neg-frac 99.1%

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

      7. metadata-eval 99.1%

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

      8. unpow2 99.1%

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

      9. associate-/r* 99.1%

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

   4. Simplified 99.1%

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

   if 2.0000000000000001e-13 < (-.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%

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

      2. associate-/r/ 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1}{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 7: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 2e-13) (+ (/ -3.0 x) (/ (/ -1.0 x) x)) t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 2e-13) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (x + 1.0d0)) + (((-1.0d0) - x) / (x + (-1.0d0)))
    if (t_0 <= 2d-13) then
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 2e-13) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))
	tmp = 0
	if t_0 <= 2e-13:
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 2e-13)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 2e-13)
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-13], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 2.0000000000000001e-13

   1. Initial program 7.3%

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

   2. Taylor expanded in x around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. distribute-neg-in 98.5%

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

      3. sub-neg 98.5%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

      6. distribute-neg-frac 99.1%

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

      7. metadata-eval 99.1%

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

      8. unpow2 99.1%

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

      9. associate-/r* 99.1%

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

   4. Simplified 99.1%

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

   if 2.0000000000000001e-13 < (-.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 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 8: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \frac{-1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.15)))
   (+ (/ -3.0 x) (/ (/ -1.0 x) x))
   (+ (* x 2.0) (/ -1.0 (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.15)) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = (x * 2.0) + (-1.0 / (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 <= 1.15d0))) then
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    else
        tmp = (x * 2.0d0) + ((-1.0d0) / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.15)) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = (x * 2.0) + (-1.0 / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.15):
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	else:
		tmp = (x * 2.0) + (-1.0 / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.15))
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x));
	else
		tmp = Float64(Float64(x * 2.0) + Float64(-1.0 / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.15)))
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	else
		tmp = (x * 2.0) + (-1.0 / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] + N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.1499999999999999 < x

   1. Initial program 8.5%

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

   2. Taylor expanded in x around inf 97.6%

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

      1. +-commutative 97.6%

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

      2. distribute-neg-in 97.6%

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

      3. sub-neg 97.6%

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

      4. associate-*r/ 98.2%

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

      5. metadata-eval 98.2%

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

      6. distribute-neg-frac 98.2%

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

      7. metadata-eval 98.2%

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

      8. unpow2 98.2%

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

      9. associate-/r* 98.2%

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

   4. Simplified 98.2%

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

   if -1 < x < 1.1499999999999999

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. clear-num 99.9%

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

      3. frac-sub 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. frac-sub 99.9%

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

      5. *-rgt-identity 99.9%

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

      6. clear-num 99.9%

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

      7. clear-num 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. associate-*l/ 99.9%

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

      10. distribute-rgt-in 99.9%

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

      11. *-un-lft-identity 99.9%

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

      12. associate--r+ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 98.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \frac{-1}{x + -1}\\ \end{array} \]

Alternative 9: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;x \cdot 2 + \frac{-1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ -3.0 x)
   (if (<= x 1.2) (+ (* x 2.0) (/ -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 <= 1.2) {
		tmp = (x * 2.0) + (-1.0 / (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 <= 1.2d0) then
        tmp = (x * 2.0d0) + ((-1.0d0) / (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 <= 1.2) {
		tmp = (x * 2.0) + (-1.0 / (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 <= 1.2:
		tmp = (x * 2.0) + (-1.0 / (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 <= 1.2)
		tmp = Float64(Float64(x * 2.0) + Float64(-1.0 / 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 <= 1.2)
		tmp = (x * 2.0) + (-1.0 / (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, 1.2], N[(N[(x * 2.0), $MachinePrecision] + N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-3.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.19999999999999996 < x

   1. Initial program 8.5%

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

   2. Taylor expanded in x around inf 97.8%

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

   if -1 < x < 1.19999999999999996

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. clear-num 99.9%

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

      3. frac-sub 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. frac-sub 99.9%

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

      5. *-rgt-identity 99.9%

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

      6. clear-num 99.9%

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

      7. clear-num 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. associate-*l/ 99.9%

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

      10. distribute-rgt-in 99.9%

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

      11. *-un-lft-identity 99.9%

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

      12. associate--r+ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 98.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;x \cdot 2 + \frac{-1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 10: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.5%

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

   2. Taylor expanded in x around inf 97.8%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 98.5%

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

4. Final simplification 98.2%

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

Alternative 11: 97.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.5%

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

   2. Taylor expanded in x around inf 97.8%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 98.5%

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

   3. Taylor expanded in x around 0 97.2%

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

4. Final simplification 97.5%

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

Alternative 12: 50.6% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 52.1%

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

2. Taylor expanded in x around 0 48.4%

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

3. Final simplification 48.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023213 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))