Asymptote C

Percentage Accurate: 55.4% → 99.5%

Time: 7.7s

Alternatives: 11

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: 55.4% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{1 + \frac{6}{x}}{{x}^{2}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0)))))
   (if (<= t_0 2e-13)
     (/
      (/
       (- (/ (+ 1.0 (/ 6.0 x)) (pow x 2.0)) 9.0)
       (+ (/ (+ -1.0 (/ (+ -3.0 (/ -1.0 x)) x)) x) 3.0))
      x)
     t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	double tmp;
	if (t_0 <= 2e-13) {
		tmp = ((((1.0 + (6.0 / x)) / pow(x, 2.0)) - 9.0) / (((-1.0 + ((-3.0 + (-1.0 / x)) / x)) / x) + 3.0)) / 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) / (x + (-1.0d0)))
    if (t_0 <= 2d-13) then
        tmp = ((((1.0d0 + (6.0d0 / x)) / (x ** 2.0d0)) - 9.0d0) / ((((-1.0d0) + (((-3.0d0) + ((-1.0d0) / x)) / x)) / x) + 3.0d0)) / 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) / (x + -1.0));
	double tmp;
	if (t_0 <= 2e-13) {
		tmp = ((((1.0 + (6.0 / x)) / Math.pow(x, 2.0)) - 9.0) / (((-1.0 + ((-3.0 + (-1.0 / x)) / x)) / x) + 3.0)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))
	tmp = 0
	if t_0 <= 2e-13:
		tmp = ((((1.0 + (6.0 / x)) / math.pow(x, 2.0)) - 9.0) / (((-1.0 + ((-3.0 + (-1.0 / x)) / x)) / x) + 3.0)) / 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(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 2e-13)
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 + Float64(6.0 / x)) / (x ^ 2.0)) - 9.0) / Float64(Float64(Float64(-1.0 + Float64(Float64(-3.0 + Float64(-1.0 / x)) / x)) / x) + 3.0)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 2e-13)
		tmp = ((((1.0 + (6.0 / x)) / (x ^ 2.0)) - 9.0) / (((-1.0 + ((-3.0 + (-1.0 / x)) / x)) / x) + 3.0)) / 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[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-13], N[(N[(N[(N[(N[(1.0 + N[(6.0 / x), $MachinePrecision]), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 9.0), $MachinePrecision] / N[(N[(N[(-1.0 + N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 2.0000000000000001e-13

   1. Initial program 8.4%

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

      1. remove-double-neg 8.4%

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

      2. distribute-neg-in 8.4%

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

      3. sub-neg 8.4%

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

      4. distribute-frac-neg 8.4%

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

      5. distribute-frac-neg2 8.4%

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

      6. sub-neg 8.4%

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

      7. +-commutative 8.4%

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

      8. unsub-neg 8.4%

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

      9. metadata-eval 8.4%

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

      10. neg-sub0 8.4%

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

      11. associate-+l- 8.4%

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

      12. neg-sub0 8.4%

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

      13. +-commutative 8.4%

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

      14. unsub-neg 8.4%

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

   3. Simplified 8.4%

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

   5. Taylor expanded in x around inf 99.3%

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

   7. Simplified 99.3%

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

      1. flip-- 99.3%

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

      2. pow2 99.3%

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

      3. metadata-eval 99.3%

         \[\leadsto \frac{\frac{{\left(\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x}\right)}^{2} - \color{blue}{9}}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

   9. Applied egg-rr 99.3%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x}\right)}^{2} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}}{x} \]

   10. Taylor expanded in x around inf 99.3%

       \[\leadsto \frac{\frac{\color{blue}{\frac{1 + 6 \cdot \frac{1}{x}}{{x}^{2}}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]
   11. Step-by-step derivation

       1. associate-*r/ 99.3%

          \[\leadsto \frac{\frac{\frac{1 + \color{blue}{\frac{6 \cdot 1}{x}}}{{x}^{2}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

       2. metadata-eval 99.3%

          \[\leadsto \frac{\frac{\frac{1 + \frac{\color{blue}{6}}{x}}{{x}^{2}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

   12. Simplified 99.3%

       \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \frac{6}{x}}{{x}^{2}}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

   if 2.0000000000000001e-13 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{1 + \frac{6}{x}}{{x}^{2}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{-3 + \frac{1}{x}}{x}\\ t_1 := \frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{\frac{t\_0}{x}}{\frac{x}{t\_0}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ (+ -3.0 (/ 1.0 x)) x)))
        (t_1 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0)))))
   (if (<= t_1 2e-13)
     (/
      (/
       (- (/ (/ t_0 x) (/ x t_0)) 9.0)
       (+ (/ (+ -1.0 (/ (+ -3.0 (/ -1.0 x)) x)) x) 3.0))
      x)
     t_1)))

C

double code(double x) {
	double t_0 = -1.0 + ((-3.0 + (1.0 / x)) / x);
	double t_1 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	double tmp;
	if (t_1 <= 2e-13) {
		tmp = ((((t_0 / x) / (x / t_0)) - 9.0) / (((-1.0 + ((-3.0 + (-1.0 / x)) / x)) / x) + 3.0)) / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) + (((-3.0d0) + (1.0d0 / x)) / x)
    t_1 = (x / (x + 1.0d0)) - ((x + 1.0d0) / (x + (-1.0d0)))
    if (t_1 <= 2d-13) then
        tmp = ((((t_0 / x) / (x / t_0)) - 9.0d0) / ((((-1.0d0) + (((-3.0d0) + ((-1.0d0) / x)) / x)) / x) + 3.0d0)) / x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -1.0 + ((-3.0 + (1.0 / x)) / x);
	double t_1 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	double tmp;
	if (t_1 <= 2e-13) {
		tmp = ((((t_0 / x) / (x / t_0)) - 9.0) / (((-1.0 + ((-3.0 + (-1.0 / x)) / x)) / x) + 3.0)) / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = -1.0 + ((-3.0 + (1.0 / x)) / x)
	t_1 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))
	tmp = 0
	if t_1 <= 2e-13:
		tmp = ((((t_0 / x) / (x / t_0)) - 9.0) / (((-1.0 + ((-3.0 + (-1.0 / x)) / x)) / x) + 3.0)) / x
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(-1.0 + Float64(Float64(-3.0 + Float64(1.0 / x)) / x))
	t_1 = Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= 2e-13)
		tmp = Float64(Float64(Float64(Float64(Float64(t_0 / x) / Float64(x / t_0)) - 9.0) / Float64(Float64(Float64(-1.0 + Float64(Float64(-3.0 + Float64(-1.0 / x)) / x)) / x) + 3.0)) / x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -1.0 + ((-3.0 + (1.0 / x)) / x);
	t_1 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	tmp = 0.0;
	if (t_1 <= 2e-13)
		tmp = ((((t_0 / x) / (x / t_0)) - 9.0) / (((-1.0 + ((-3.0 + (-1.0 / x)) / x)) / x) + 3.0)) / x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-1.0 + N[(N[(-3.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-13], N[(N[(N[(N[(N[(t$95$0 / x), $MachinePrecision] / N[(x / t$95$0), $MachinePrecision]), $MachinePrecision] - 9.0), $MachinePrecision] / N[(N[(N[(-1.0 + N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 2.0000000000000001e-13

   1. Initial program 8.4%

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

      1. remove-double-neg 8.4%

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

      2. distribute-neg-in 8.4%

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

      3. sub-neg 8.4%

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

      4. distribute-frac-neg 8.4%

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

      5. distribute-frac-neg2 8.4%

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

      6. sub-neg 8.4%

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

      7. +-commutative 8.4%

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

      8. unsub-neg 8.4%

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

      9. metadata-eval 8.4%

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

      10. neg-sub0 8.4%

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

      11. associate-+l- 8.4%

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

      12. neg-sub0 8.4%

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

      13. +-commutative 8.4%

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

      14. unsub-neg 8.4%

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

   3. Simplified 8.4%

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

   5. Taylor expanded in x around inf 99.3%

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

   7. Simplified 99.3%

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

      1. flip-- 99.3%

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

      2. pow2 99.3%

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

      3. metadata-eval 99.3%

         \[\leadsto \frac{\frac{{\left(\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x}\right)}^{2} - \color{blue}{9}}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

   9. Applied egg-rr 99.3%

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

       1. unpow2 99.3%

          \[\leadsto \frac{\frac{\color{blue}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} \cdot \frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

       2. clear-num 99.3%

          \[\leadsto \frac{\frac{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} \cdot \color{blue}{\frac{1}{\frac{x}{-1 + \frac{-3 + \frac{-1}{x}}{x}}}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

       3. un-div-inv 99.3%

          \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x}}{\frac{x}{-1 + \frac{-3 + \frac{-1}{x}}{x}}}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

       4. add-sqr-sqrt 50.0%

          \[\leadsto \frac{\frac{\frac{\frac{-1 + \frac{-3 + \color{blue}{\sqrt{\frac{-1}{x}} \cdot \sqrt{\frac{-1}{x}}}}{x}}{x}}{\frac{x}{-1 + \frac{-3 + \frac{-1}{x}}{x}}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

       5. sqrt-unprod 99.3%

          \[\leadsto \frac{\frac{\frac{\frac{-1 + \frac{-3 + \color{blue}{\sqrt{\frac{-1}{x} \cdot \frac{-1}{x}}}}{x}}{x}}{\frac{x}{-1 + \frac{-3 + \frac{-1}{x}}{x}}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

       6. frac-times 99.3%

          \[\leadsto \frac{\frac{\frac{\frac{-1 + \frac{-3 + \sqrt{\color{blue}{\frac{-1 \cdot -1}{x \cdot x}}}}{x}}{x}}{\frac{x}{-1 + \frac{-3 + \frac{-1}{x}}{x}}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

       7. metadata-eval 99.3%

          \[\leadsto \frac{\frac{\frac{\frac{-1 + \frac{-3 + \sqrt{\frac{\color{blue}{1}}{x \cdot x}}}{x}}{x}}{\frac{x}{-1 + \frac{-3 + \frac{-1}{x}}{x}}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

       8. metadata-eval 99.3%

          \[\leadsto \frac{\frac{\frac{\frac{-1 + \frac{-3 + \sqrt{\frac{\color{blue}{1 \cdot 1}}{x \cdot x}}}{x}}{x}}{\frac{x}{-1 + \frac{-3 + \frac{-1}{x}}{x}}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

       9. frac-times 99.3%

          \[\leadsto \frac{\frac{\frac{\frac{-1 + \frac{-3 + \sqrt{\color{blue}{\frac{1}{x} \cdot \frac{1}{x}}}}{x}}{x}}{\frac{x}{-1 + \frac{-3 + \frac{-1}{x}}{x}}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

       10. sqrt-unprod 49.3%

           \[\leadsto \frac{\frac{\frac{\frac{-1 + \frac{-3 + \color{blue}{\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}}}{x}}{x}}{\frac{x}{-1 + \frac{-3 + \frac{-1}{x}}{x}}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

       11. add-sqr-sqrt 99.3%

           \[\leadsto \frac{\frac{\frac{\frac{-1 + \frac{-3 + \color{blue}{\frac{1}{x}}}{x}}{x}}{\frac{x}{-1 + \frac{-3 + \frac{-1}{x}}{x}}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

   11. Applied egg-rr 99.3%

       \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{-1 + \frac{-3 + \frac{1}{x}}{x}}{x}}{\frac{x}{-1 + \frac{-3 + \frac{1}{x}}{x}}}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x} \]

   if 2.0000000000000001e-13 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{\frac{-1 + \frac{-3 + \frac{1}{x}}{x}}{x}}{\frac{x}{-1 + \frac{-3 + \frac{1}{x}}{x}}} - 9}{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} + 3}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{-1 + \frac{-3 + \frac{-1}{x}}{x}}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	double tmp;
	if (t_0 <= 2e-13) {
		tmp = (((-1.0 + ((-3.0 + (-1.0 / x)) / x)) / x) - 3.0) / 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) / (x + (-1.0d0)))
    if (t_0 <= 2d-13) then
        tmp = ((((-1.0d0) + (((-3.0d0) + ((-1.0d0) / x)) / x)) / x) - 3.0d0) / 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) / (x + -1.0));
	double tmp;
	if (t_0 <= 2e-13) {
		tmp = (((-1.0 + ((-3.0 + (-1.0 / x)) / x)) / x) - 3.0) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))
	tmp = 0
	if t_0 <= 2e-13:
		tmp = (((-1.0 + ((-3.0 + (-1.0 / x)) / x)) / x) - 3.0) / 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(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 2e-13)
		tmp = Float64(Float64(Float64(Float64(-1.0 + Float64(Float64(-3.0 + Float64(-1.0 / x)) / x)) / x) - 3.0) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 2e-13)
		tmp = (((-1.0 + ((-3.0 + (-1.0 / x)) / x)) / x) - 3.0) / 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[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-13], N[(N[(N[(N[(-1.0 + N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 3.0), $MachinePrecision] / x), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 2.0000000000000001e-13

   1. Initial program 8.4%

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

      1. remove-double-neg 8.4%

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

      2. distribute-neg-in 8.4%

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

      3. sub-neg 8.4%

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

      4. distribute-frac-neg 8.4%

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

      5. distribute-frac-neg2 8.4%

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

      6. sub-neg 8.4%

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

      7. +-commutative 8.4%

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

      8. unsub-neg 8.4%

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

      9. metadata-eval 8.4%

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

      10. neg-sub0 8.4%

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

      11. associate-+l- 8.4%

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

      12. neg-sub0 8.4%

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

      13. +-commutative 8.4%

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

      14. unsub-neg 8.4%

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

   3. Simplified 8.4%

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

   5. Taylor expanded in x around inf 99.3%

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

   7. Simplified 99.3%

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

   if 2.0000000000000001e-13 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

4. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{-3 + \frac{-1 - \frac{3}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 2.0000000000000001e-13

   1. Initial program 8.4%

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

      1. remove-double-neg 8.4%

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

      2. distribute-neg-in 8.4%

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

      3. sub-neg 8.4%

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

      4. distribute-frac-neg 8.4%

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

      5. distribute-frac-neg2 8.4%

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

      6. sub-neg 8.4%

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

      7. +-commutative 8.4%

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

      8. unsub-neg 8.4%

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

      9. metadata-eval 8.4%

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

      10. neg-sub0 8.4%

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

      11. associate-+l- 8.4%

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

      12. neg-sub0 8.4%

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

      13. +-commutative 8.4%

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

      14. unsub-neg 8.4%

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

   3. Simplified 8.4%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. sub-neg 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

      4. mul-1-neg 99.1%

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

      5. unsub-neg 99.1%

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

      6. associate-*r/ 99.1%

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

      7. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

   if 2.0000000000000001e-13 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

4. Final simplification 99.5%

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

Alternative 5: 99.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 5.7%

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

      1. remove-double-neg 5.7%

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

      2. distribute-neg-in 5.7%

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

      3. sub-neg 5.7%

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

      4. distribute-frac-neg 5.7%

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

      5. distribute-frac-neg2 5.7%

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

      6. sub-neg 5.7%

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

      7. +-commutative 5.7%

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

      8. unsub-neg 5.7%

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

      9. metadata-eval 5.7%

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

      10. neg-sub0 5.7%

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

      11. associate-+l- 5.7%

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

      12. neg-sub0 5.7%

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

      13. +-commutative 5.7%

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

      14. unsub-neg 5.7%

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

   3. Simplified 5.7%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-in 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      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 99.8%

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

   if 1 < x

   1. Initial program 11.1%

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

      1. remove-double-neg 11.1%

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

      2. distribute-neg-in 11.1%

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

      3. sub-neg 11.1%

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

      4. distribute-frac-neg 11.1%

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

      5. distribute-frac-neg2 11.1%

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

      6. sub-neg 11.1%

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

      7. +-commutative 11.1%

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

      8. unsub-neg 11.1%

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

      9. metadata-eval 11.1%

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

      10. neg-sub0 11.1%

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

      11. associate-+l- 11.1%

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

      12. neg-sub0 11.1%

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

      13. +-commutative 11.1%

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

      14. unsub-neg 11.1%

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

   3. Simplified 11.1%

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

   5. Taylor expanded in x around inf 98.1%

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

      1. sub-neg 98.1%

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

      2. metadata-eval 98.1%

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

      3. +-commutative 98.1%

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

      4. mul-1-neg 98.1%

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

      5. unsub-neg 98.1%

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

      6. associate-*r/ 98.1%

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

      7. metadata-eval 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 99.4%

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

Alternative 6: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 5.7%

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

      1. remove-double-neg 5.7%

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

      2. distribute-neg-in 5.7%

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

      3. sub-neg 5.7%

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

      4. distribute-frac-neg 5.7%

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

      5. distribute-frac-neg2 5.7%

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

      6. sub-neg 5.7%

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

      7. +-commutative 5.7%

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

      8. unsub-neg 5.7%

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

      9. metadata-eval 5.7%

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

      10. neg-sub0 5.7%

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

      11. associate-+l- 5.7%

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

      12. neg-sub0 5.7%

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

      13. +-commutative 5.7%

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

      14. unsub-neg 5.7%

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

   3. Simplified 5.7%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-in 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      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 99.8%

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

   if 1 < x

   1. Initial program 11.1%

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

      1. remove-double-neg 11.1%

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

      2. distribute-neg-in 11.1%

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

      3. sub-neg 11.1%

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

      4. distribute-frac-neg 11.1%

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

      5. distribute-frac-neg2 11.1%

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

      6. sub-neg 11.1%

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

      7. +-commutative 11.1%

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

      8. unsub-neg 11.1%

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

      9. metadata-eval 11.1%

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

      10. neg-sub0 11.1%

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

      11. associate-+l- 11.1%

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

      12. neg-sub0 11.1%

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

      13. +-commutative 11.1%

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

      14. unsub-neg 11.1%

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

   3. Simplified 11.1%

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

   5. Taylor expanded in x around inf 98.1%

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

      1. sub-neg 98.1%

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

      2. metadata-eval 98.1%

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

      3. +-commutative 98.1%

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

      4. mul-1-neg 98.1%

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

      5. unsub-neg 98.1%

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

      6. associate-*r/ 98.1%

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

      7. metadata-eval 98.1%

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

   7. Simplified 98.1%

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

      1. div-sub 98.2%

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

      2. div-inv 98.2%

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

      3. metadata-eval 98.2%

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

      4. associate-*r* 98.2%

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

      5. div-inv 98.2%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 97.2%

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

      8. frac-times 97.2%

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

      9. metadata-eval 97.2%

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

      10. metadata-eval 97.2%

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

      11. frac-times 97.2%

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

      12. sqrt-unprod 97.2%

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

      13. add-sqr-sqrt 97.2%

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

      14. div-inv 97.2%

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

   9. Applied egg-rr 97.2%

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

   10. Taylor expanded in x around inf 97.3%

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

4. Final simplification 99.2%

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

Alternative 7: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.4%

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

      1. remove-double-neg 8.4%

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

      2. distribute-neg-in 8.4%

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

      3. sub-neg 8.4%

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

      4. distribute-frac-neg 8.4%

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

      5. distribute-frac-neg2 8.4%

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

      6. sub-neg 8.4%

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

      7. +-commutative 8.4%

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

      8. unsub-neg 8.4%

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

      9. metadata-eval 8.4%

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

      10. neg-sub0 8.4%

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

      11. associate-+l- 8.4%

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

      12. neg-sub0 8.4%

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

      13. +-commutative 8.4%

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

      14. unsub-neg 8.4%

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

   3. Simplified 8.4%

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

   5. Taylor expanded in x around inf 98.6%

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

      1. associate-*r/ 98.6%

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

      2. neg-mul-1 98.6%

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

      3. distribute-neg-in 98.6%

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

      4. metadata-eval 98.6%

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

      5. distribute-neg-frac 98.6%

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

      6. metadata-eval 98.6%

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

   7. Simplified 98.6%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-in 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      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 99.8%

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

4. Final simplification 99.2%

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

Alternative 8: 98.5% 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 8.4%

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

      1. remove-double-neg 8.4%

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

      2. distribute-neg-in 8.4%

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

      3. sub-neg 8.4%

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

      4. distribute-frac-neg 8.4%

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

      5. distribute-frac-neg2 8.4%

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

      6. sub-neg 8.4%

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

      7. +-commutative 8.4%

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

      8. unsub-neg 8.4%

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

      9. metadata-eval 8.4%

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

      10. neg-sub0 8.4%

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

      11. associate-+l- 8.4%

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

      12. neg-sub0 8.4%

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

      13. +-commutative 8.4%

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

      14. unsub-neg 8.4%

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

   3. Simplified 8.4%

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

   5. Taylor expanded in x around inf 97.9%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-in 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      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 99.8%

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

4. Final simplification 98.8%

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

Alternative 9: 98.4% 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 8.4%

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

      1. remove-double-neg 8.4%

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

      2. distribute-neg-in 8.4%

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

      3. sub-neg 8.4%

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

      4. distribute-frac-neg 8.4%

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

      5. distribute-frac-neg2 8.4%

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

      6. sub-neg 8.4%

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

      7. +-commutative 8.4%

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

      8. unsub-neg 8.4%

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

      9. metadata-eval 8.4%

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

      10. neg-sub0 8.4%

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

      11. associate-+l- 8.4%

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

      12. neg-sub0 8.4%

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

      13. +-commutative 8.4%

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

      14. unsub-neg 8.4%

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

   3. Simplified 8.4%

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

   5. Taylor expanded in x around inf 97.9%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-in 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      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 99.4%

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

4. Final simplification 98.6%

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

Alternative 10: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.4%

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

      1. remove-double-neg 8.4%

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

      2. distribute-neg-in 8.4%

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

      3. sub-neg 8.4%

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

      4. distribute-frac-neg 8.4%

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

      5. distribute-frac-neg2 8.4%

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

      6. sub-neg 8.4%

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

      7. +-commutative 8.4%

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

      8. unsub-neg 8.4%

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

      9. metadata-eval 8.4%

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

      10. neg-sub0 8.4%

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

      11. associate-+l- 8.4%

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

      12. neg-sub0 8.4%

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

      13. +-commutative 8.4%

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

      14. unsub-neg 8.4%

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

   3. Simplified 8.4%

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

   5. Taylor expanded in x around inf 97.9%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-in 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      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 97.8%

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

4. Final simplification 97.9%

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

Alternative 11: 51.8% 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 49.9%

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

   1. remove-double-neg 49.9%

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

   2. distribute-neg-in 49.9%

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

   3. sub-neg 49.9%

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

   4. distribute-frac-neg 49.9%

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

   5. distribute-frac-neg2 49.9%

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

   6. sub-neg 49.9%

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

   7. +-commutative 49.9%

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

   8. unsub-neg 49.9%

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

   9. metadata-eval 49.9%

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

   10. neg-sub0 49.9%

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

   11. associate-+l- 49.9%

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

   12. neg-sub0 49.9%

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

   13. +-commutative 49.9%

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

   14. unsub-neg 49.9%

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

3. Simplified 49.9%

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

5. Taylor expanded in x around 0 46.3%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024182 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))