Asymptote C

Percentage Accurate: 54.8% → 99.5%

Time: 9.0s

Alternatives: 8

Speedup: 1.0×

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: 54.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{x + 1}{1 - x} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(3 + \frac{2}{{x}^{2}}\right) - \frac{2}{x}}{1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + -1}{x + 1} - \frac{x + 1}{x}}{1 + \frac{-1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (+ x 1.0) (- 1.0 x))) 4e-12)
   (/ (- (+ 3.0 (/ 2.0 (pow x 2.0))) (/ 2.0 x)) (- 1.0 x))
   (/ (- (/ (+ x -1.0) (+ x 1.0)) (/ (+ x 1.0) x)) (+ 1.0 (/ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((x + 1.0) / (1.0 - x))) <= 4e-12) {
		tmp = ((3.0 + (2.0 / pow(x, 2.0))) - (2.0 / x)) / (1.0 - x);
	} else {
		tmp = (((x + -1.0) / (x + 1.0)) - ((x + 1.0) / x)) / (1.0 + (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x / (x + 1.0d0)) + ((x + 1.0d0) / (1.0d0 - x))) <= 4d-12) then
        tmp = ((3.0d0 + (2.0d0 / (x ** 2.0d0))) - (2.0d0 / x)) / (1.0d0 - x)
    else
        tmp = (((x + (-1.0d0)) / (x + 1.0d0)) - ((x + 1.0d0) / x)) / (1.0d0 + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((x + 1.0) / (1.0 - x))) <= 4e-12) {
		tmp = ((3.0 + (2.0 / Math.pow(x, 2.0))) - (2.0 / x)) / (1.0 - x);
	} else {
		tmp = (((x + -1.0) / (x + 1.0)) - ((x + 1.0) / x)) / (1.0 + (-1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) + ((x + 1.0) / (1.0 - x))) <= 4e-12:
		tmp = ((3.0 + (2.0 / math.pow(x, 2.0))) - (2.0 / x)) / (1.0 - x)
	else:
		tmp = (((x + -1.0) / (x + 1.0)) - ((x + 1.0) / x)) / (1.0 + (-1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(x + 1.0) / Float64(1.0 - x))) <= 4e-12)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / (x ^ 2.0))) - Float64(2.0 / x)) / Float64(1.0 - x));
	else
		tmp = Float64(Float64(Float64(Float64(x + -1.0) / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / x)) / Float64(1.0 + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) + ((x + 1.0) / (1.0 - x))) <= 4e-12)
		tmp = ((3.0 + (2.0 / (x ^ 2.0))) - (2.0 / x)) / (1.0 - x);
	else
		tmp = (((x + -1.0) / (x + 1.0)) - ((x + 1.0) / x)) / (1.0 + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-12], N[(N[(N[(3.0 + N[(2.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 3.99999999999999992e-12

   1. Initial program 8.5%

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

      1. remove-double-neg 8.5%

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

      2. distribute-neg-frac 8.5%

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

      3. distribute-neg-in 8.5%

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

      4. sub-neg 8.5%

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

      5. distribute-frac-neg2 8.5%

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

      6. sub-neg 8.5%

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

      7. +-commutative 8.5%

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

      8. unsub-neg 8.5%

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

      9. metadata-eval 8.5%

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

      10. neg-sub0 8.5%

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

      11. associate-+l- 8.5%

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

      12. neg-sub0 8.5%

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

      13. +-commutative 8.5%

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

      14. unsub-neg 8.5%

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

   3. Simplified 8.5%

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

      1. frac-2neg 8.5%

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

      2. clear-num 8.5%

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

      3. frac-sub 9.7%

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

      4. +-commutative 9.7%

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

      5. distribute-neg-in 9.7%

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

      6. metadata-eval 9.7%

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

      7. sub-neg 9.7%

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

      8. *-commutative 9.7%

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

      9. *-un-lft-identity 9.7%

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

      10. +-commutative 9.7%

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

      11. distribute-neg-in 9.7%

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

      12. metadata-eval 9.7%

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

      13. sub-neg 9.7%

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

   6. Applied egg-rr 9.7%

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

   7. Taylor expanded in x around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. sub-neg 9.7%

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

   9. Simplified 9.7%

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

   10. Taylor expanded in x around inf 99.3%

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

       1. associate-*r/ 99.3%

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

       2. metadata-eval 99.3%

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

       3. associate-*r/ 99.3%

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

       4. metadata-eval 99.3%

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

   12. Simplified 99.3%

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

   if 3.99999999999999992e-12 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. distribute-neg-frac 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-frac-neg2 99.9%

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

      6. sub-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. unsub-neg 99.9%

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

      9. metadata-eval 99.9%

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

      10. neg-sub0 99.9%

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

      11. associate-+l- 99.9%

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

      12. neg-sub0 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. clear-num 99.9%

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

      3. frac-sub 99.9%

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

      4. *-un-lft-identity 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. div-sub 99.9%

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

      2. sub-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. metadata-eval 99.9%

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

      6. *-rgt-identity 99.9%

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

      7. +-commutative 99.9%

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

      8. sub-neg 99.9%

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

      9. distribute-neg-frac 99.9%

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

      10. metadata-eval 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. sub-neg 99.9%

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

       2. div-sub 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{x + 1}{1 - x} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(3 + \frac{2}{{x}^{2}}\right) - \frac{2}{x}}{1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + -1}{x + 1} - \frac{x + 1}{x}}{1 + \frac{-1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{x + 1}{1 - x} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\frac{3 - \frac{2}{x}}{1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + -1}{x + 1} - \frac{x + 1}{x}}{1 + \frac{-1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (+ x 1.0) (- 1.0 x))) 4e-12)
   (/ (- 3.0 (/ 2.0 x)) (- 1.0 x))
   (/ (- (/ (+ x -1.0) (+ x 1.0)) (/ (+ x 1.0) x)) (+ 1.0 (/ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((x + 1.0) / (1.0 - x))) <= 4e-12) {
		tmp = (3.0 - (2.0 / x)) / (1.0 - x);
	} else {
		tmp = (((x + -1.0) / (x + 1.0)) - ((x + 1.0) / x)) / (1.0 + (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x / (x + 1.0d0)) + ((x + 1.0d0) / (1.0d0 - x))) <= 4d-12) then
        tmp = (3.0d0 - (2.0d0 / x)) / (1.0d0 - x)
    else
        tmp = (((x + (-1.0d0)) / (x + 1.0d0)) - ((x + 1.0d0) / x)) / (1.0d0 + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((x + 1.0) / (1.0 - x))) <= 4e-12) {
		tmp = (3.0 - (2.0 / x)) / (1.0 - x);
	} else {
		tmp = (((x + -1.0) / (x + 1.0)) - ((x + 1.0) / x)) / (1.0 + (-1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) + ((x + 1.0) / (1.0 - x))) <= 4e-12:
		tmp = (3.0 - (2.0 / x)) / (1.0 - x)
	else:
		tmp = (((x + -1.0) / (x + 1.0)) - ((x + 1.0) / x)) / (1.0 + (-1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(x + 1.0) / Float64(1.0 - x))) <= 4e-12)
		tmp = Float64(Float64(3.0 - Float64(2.0 / x)) / Float64(1.0 - x));
	else
		tmp = Float64(Float64(Float64(Float64(x + -1.0) / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / x)) / Float64(1.0 + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) + ((x + 1.0) / (1.0 - x))) <= 4e-12)
		tmp = (3.0 - (2.0 / x)) / (1.0 - x);
	else
		tmp = (((x + -1.0) / (x + 1.0)) - ((x + 1.0) / x)) / (1.0 + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-12], N[(N[(3.0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 3.99999999999999992e-12

   1. Initial program 8.5%

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

      1. remove-double-neg 8.5%

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

      2. distribute-neg-frac 8.5%

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

      3. distribute-neg-in 8.5%

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

      4. sub-neg 8.5%

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

      5. distribute-frac-neg2 8.5%

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

      6. sub-neg 8.5%

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

      7. +-commutative 8.5%

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

      8. unsub-neg 8.5%

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

      9. metadata-eval 8.5%

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

      10. neg-sub0 8.5%

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

      11. associate-+l- 8.5%

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

      12. neg-sub0 8.5%

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

      13. +-commutative 8.5%

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

      14. unsub-neg 8.5%

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

   3. Simplified 8.5%

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

      1. frac-2neg 8.5%

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

      2. clear-num 8.5%

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

      3. frac-sub 9.7%

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

      4. +-commutative 9.7%

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

      5. distribute-neg-in 9.7%

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

      6. metadata-eval 9.7%

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

      7. sub-neg 9.7%

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

      8. *-commutative 9.7%

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

      9. *-un-lft-identity 9.7%

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

      10. +-commutative 9.7%

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

      11. distribute-neg-in 9.7%

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

      12. metadata-eval 9.7%

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

      13. sub-neg 9.7%

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

   6. Applied egg-rr 9.7%

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

   7. Taylor expanded in x around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. sub-neg 9.7%

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

   9. Simplified 9.7%

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

   10. Taylor expanded in x around inf 99.3%

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

       1. associate-*r/ 99.3%

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

       2. metadata-eval 99.3%

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

   12. Simplified 99.3%

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

   if 3.99999999999999992e-12 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. distribute-neg-frac 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-frac-neg2 99.9%

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

      6. sub-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. unsub-neg 99.9%

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

      9. metadata-eval 99.9%

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

      10. neg-sub0 99.9%

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

      11. associate-+l- 99.9%

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

      12. neg-sub0 99.9%

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

      13. +-commutative 99.9%

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

      14. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. clear-num 99.9%

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

      3. frac-sub 99.9%

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

      4. *-un-lft-identity 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. div-sub 99.9%

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

      2. sub-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. metadata-eval 99.9%

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

      6. *-rgt-identity 99.9%

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

      7. +-commutative 99.9%

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

      8. sub-neg 99.9%

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

      9. distribute-neg-frac 99.9%

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

      10. metadata-eval 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. sub-neg 99.9%

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

       2. div-sub 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{x + 1}{1 - x} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\frac{3 - \frac{2}{x}}{1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + -1}{x + 1} - \frac{x + 1}{x}}{1 + \frac{-1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{x + 1}{1 - x}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\frac{3 - \frac{2}{x}}{1 - x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (+ x 1.0) (- 1.0 x)))))
   (if (<= t_0 4e-12) (/ (- 3.0 (/ 2.0 x)) (- 1.0 x)) t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((x + 1.0) / (1.0 - x));
	double tmp;
	if (t_0 <= 4e-12) {
		tmp = (3.0 - (2.0 / x)) / (1.0 - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (x + 1.0d0)) + ((x + 1.0d0) / (1.0d0 - x))
    if (t_0 <= 4d-12) then
        tmp = (3.0d0 - (2.0d0 / x)) / (1.0d0 - x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((x + 1.0) / (1.0 - x));
	double tmp;
	if (t_0 <= 4e-12) {
		tmp = (3.0 - (2.0 / x)) / (1.0 - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) + ((x + 1.0) / (1.0 - x))
	tmp = 0
	if t_0 <= 4e-12:
		tmp = (3.0 - (2.0 / x)) / (1.0 - x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(x + 1.0) / Float64(1.0 - x)))
	tmp = 0.0
	if (t_0 <= 4e-12)
		tmp = Float64(Float64(3.0 - Float64(2.0 / x)) / Float64(1.0 - x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) + ((x + 1.0) / (1.0 - x));
	tmp = 0.0;
	if (t_0 <= 4e-12)
		tmp = (3.0 - (2.0 / x)) / (1.0 - x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-12], N[(N[(3.0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 - 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))) < 3.99999999999999992e-12

   1. Initial program 8.5%

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

      1. remove-double-neg 8.5%

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

      2. distribute-neg-frac 8.5%

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

      3. distribute-neg-in 8.5%

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

      4. sub-neg 8.5%

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

      5. distribute-frac-neg2 8.5%

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

      6. sub-neg 8.5%

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

      7. +-commutative 8.5%

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

      8. unsub-neg 8.5%

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

      9. metadata-eval 8.5%

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

      10. neg-sub0 8.5%

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

      11. associate-+l- 8.5%

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

      12. neg-sub0 8.5%

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

      13. +-commutative 8.5%

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

      14. unsub-neg 8.5%

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

   3. Simplified 8.5%

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

      1. frac-2neg 8.5%

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

      2. clear-num 8.5%

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

      3. frac-sub 9.7%

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

      4. +-commutative 9.7%

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

      5. distribute-neg-in 9.7%

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

      6. metadata-eval 9.7%

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

      7. sub-neg 9.7%

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

      8. *-commutative 9.7%

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

      9. *-un-lft-identity 9.7%

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

      10. +-commutative 9.7%

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

      11. distribute-neg-in 9.7%

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

      12. metadata-eval 9.7%

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

      13. sub-neg 9.7%

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

   6. Applied egg-rr 9.7%

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

   7. Taylor expanded in x around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. sub-neg 9.7%

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

   9. Simplified 9.7%

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

   10. Taylor expanded in x around inf 99.3%

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

       1. associate-*r/ 99.3%

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

       2. metadata-eval 99.3%

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

   12. Simplified 99.3%

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

   if 3.99999999999999992e-12 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{x + 1}{1 - x} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\frac{3 - \frac{2}{x}}{1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{x + 1}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\frac{3 - \frac{2}{x}}{1 - x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.85)))
   (/ (- 3.0 (/ 2.0 x)) (- 1.0 x))
   (+ 1.0 (* x (+ x 3.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.85)) {
		tmp = (3.0 - (2.0 / x)) / (1.0 - x);
	} else {
		tmp = 1.0 + (x * (x + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.85d0))) then
        tmp = (3.0d0 - (2.0d0 / x)) / (1.0d0 - x)
    else
        tmp = 1.0d0 + (x * (x + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.85)) {
		tmp = (3.0 - (2.0 / x)) / (1.0 - x);
	} else {
		tmp = 1.0 + (x * (x + 3.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.85):
		tmp = (3.0 - (2.0 / x)) / (1.0 - x)
	else:
		tmp = 1.0 + (x * (x + 3.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.85))
		tmp = Float64(Float64(3.0 - Float64(2.0 / x)) / Float64(1.0 - x));
	else
		tmp = Float64(1.0 + Float64(x * Float64(x + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.85)))
		tmp = (3.0 - (2.0 / x)) / (1.0 - x);
	else
		tmp = 1.0 + (x * (x + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.85]], $MachinePrecision]], N[(N[(3.0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.849999999999999978 < x

   1. Initial program 9.2%

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

      1. remove-double-neg 9.2%

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

      2. distribute-neg-frac 9.2%

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

      3. distribute-neg-in 9.2%

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

      4. sub-neg 9.2%

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

      5. distribute-frac-neg2 9.2%

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

      6. sub-neg 9.2%

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

      7. +-commutative 9.2%

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

      8. unsub-neg 9.2%

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

      9. metadata-eval 9.2%

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

      10. neg-sub0 9.2%

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

      11. associate-+l- 9.2%

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

      12. neg-sub0 9.2%

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

      13. +-commutative 9.2%

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

      14. unsub-neg 9.2%

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

   3. Simplified 9.2%

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

      1. frac-2neg 9.2%

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

      2. clear-num 9.2%

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

      3. frac-sub 10.4%

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

      4. +-commutative 10.4%

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

      5. distribute-neg-in 10.4%

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

      6. metadata-eval 10.4%

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

      7. sub-neg 10.4%

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

      8. *-commutative 10.4%

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

      9. *-un-lft-identity 10.4%

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

      10. +-commutative 10.4%

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

      11. distribute-neg-in 10.4%

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

      12. metadata-eval 10.4%

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

      13. sub-neg 10.4%

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

   6. Applied egg-rr 10.4%

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

   7. Taylor expanded in x around 0 10.4%

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

      1. mul-1-neg 10.4%

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

      2. sub-neg 10.4%

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

   9. Simplified 10.4%

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

   10. Taylor expanded in x around inf 98.8%

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

       1. associate-*r/ 98.8%

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

       2. metadata-eval 98.8%

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

   12. Simplified 98.8%

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

   if -1 < x < 0.849999999999999978

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-frac-neg2 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-sub0 100.0%

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

      11. associate-+l- 100.0%

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

      12. neg-sub0 100.0%

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

      13. +-commutative 100.0%

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

      14. unsub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{x + 1} - \frac{-1 - x}{1 - x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.1%

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

      1. unpow2 99.1%

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

      2. distribute-rgt-out 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\frac{3 - \frac{2}{x}}{1 - x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 9.2%

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

      1. remove-double-neg 9.2%

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

      2. distribute-neg-frac 9.2%

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

      3. distribute-neg-in 9.2%

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

      4. sub-neg 9.2%

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

      5. distribute-frac-neg2 9.2%

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

      6. sub-neg 9.2%

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

      7. +-commutative 9.2%

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

      8. unsub-neg 9.2%

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

      9. metadata-eval 9.2%

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

      10. neg-sub0 9.2%

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

      11. associate-+l- 9.2%

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

      12. neg-sub0 9.2%

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

      13. +-commutative 9.2%

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

      14. unsub-neg 9.2%

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

   3. Simplified 9.2%

      \[\leadsto \color{blue}{\frac{x}{x + 1} - \frac{-1 - x}{1 - x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.7%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-frac-neg2 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-sub0 100.0%

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

      11. associate-+l- 100.0%

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

      12. neg-sub0 100.0%

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

      13. +-commutative 100.0%

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

      14. unsub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{x + 1} - \frac{-1 - x}{1 - x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.1%

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

      1. unpow2 99.1%

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

      2. distribute-rgt-out 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 98.5%

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

Alternative 6: 98.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 9.2%

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

      1. remove-double-neg 9.2%

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

      2. distribute-neg-frac 9.2%

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

      3. distribute-neg-in 9.2%

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

      4. sub-neg 9.2%

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

      5. distribute-frac-neg2 9.2%

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

      6. sub-neg 9.2%

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

      7. +-commutative 9.2%

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

      8. unsub-neg 9.2%

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

      9. metadata-eval 9.2%

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

      10. neg-sub0 9.2%

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

      11. associate-+l- 9.2%

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

      12. neg-sub0 9.2%

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

      13. +-commutative 9.2%

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

      14. unsub-neg 9.2%

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

   3. Simplified 9.2%

      \[\leadsto \color{blue}{\frac{x}{x + 1} - \frac{-1 - x}{1 - x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.7%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-frac-neg2 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-sub0 100.0%

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

      11. associate-+l- 100.0%

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

      12. neg-sub0 100.0%

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

      13. +-commutative 100.0%

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

      14. unsub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{x + 1} - \frac{-1 - x}{1 - x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.9%

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

4. Final simplification 98.3%

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

Alternative 7: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 9.2%

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

      1. remove-double-neg 9.2%

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

      2. distribute-neg-frac 9.2%

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

      3. distribute-neg-in 9.2%

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

      4. sub-neg 9.2%

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

      5. distribute-frac-neg2 9.2%

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

      6. sub-neg 9.2%

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

      7. +-commutative 9.2%

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

      8. unsub-neg 9.2%

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

      9. metadata-eval 9.2%

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

      10. neg-sub0 9.2%

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

      11. associate-+l- 9.2%

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

      12. neg-sub0 9.2%

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

      13. +-commutative 9.2%

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

      14. unsub-neg 9.2%

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

   3. Simplified 9.2%

      \[\leadsto \color{blue}{\frac{x}{x + 1} - \frac{-1 - x}{1 - x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.7%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-frac-neg2 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-sub0 100.0%

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

      11. associate-+l- 100.0%

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

      12. neg-sub0 100.0%

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

      13. +-commutative 100.0%

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

      14. unsub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{x + 1} - \frac{-1 - x}{1 - x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.9%

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

   6. Taylor expanded in x around 0 98.5%

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

4. Final simplification 98.1%

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

Alternative 8: 51.6% 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 58.9%

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

   1. remove-double-neg 58.9%

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

   2. distribute-neg-frac 58.9%

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

   3. distribute-neg-in 58.9%

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

   4. sub-neg 58.9%

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

   5. distribute-frac-neg2 58.9%

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

   6. sub-neg 58.9%

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

   7. +-commutative 58.9%

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

   8. unsub-neg 58.9%

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

   9. metadata-eval 58.9%

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

   10. neg-sub0 58.9%

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

   11. associate-+l- 58.9%

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

   12. neg-sub0 58.9%

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

   13. +-commutative 58.9%

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

   14. unsub-neg 58.9%

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

3. Simplified 58.9%

   \[\leadsto \color{blue}{\frac{x}{x + 1} - \frac{-1 - x}{1 - x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 55.4%

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

6. Final simplification 55.4%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))