Asymptote C

Average Accuracy: 54.1% → 99.7%

Time: 11.8s

Precision: binary64

Cost: 1732

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 2e-13) {
		tmp = ((x / (x + -1.0)) + 2.0) / -x;
	} else {
		tmp = (1.0 / (1.0 - (x * x))) * (1.0 + (x * 3.0));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x / (x + 1.0d0)) + (((-1.0d0) - x) / (x + (-1.0d0)))) <= 2d-13) then
        tmp = ((x / (x + (-1.0d0))) + 2.0d0) / -x
    else
        tmp = (1.0d0 / (1.0d0 - (x * x))) * (1.0d0 + (x * 3.0d0))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 2e-13) {
		tmp = ((x / (x + -1.0)) + 2.0) / -x;
	} else {
		tmp = (1.0 / (1.0 - (x * x))) * (1.0 + (x * 3.0));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 2e-13:
		tmp = ((x / (x + -1.0)) + 2.0) / -x
	else:
		tmp = (1.0 / (1.0 - (x * x))) * (1.0 + (x * 3.0))
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 2e-13)
		tmp = Float64(Float64(Float64(x / Float64(x + -1.0)) + 2.0) / Float64(-x));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 - Float64(x * x))) * Float64(1.0 + Float64(x * 3.0)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 2e-13)
		tmp = ((x / (x + -1.0)) + 2.0) / -x;
	else
		tmp = (1.0 / (1.0 - (x * x))) * (1.0 + (x * 3.0));
	end
	tmp_2 = tmp;
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]

↓

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-13], N[(N[(N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / (-x)), $MachinePrecision], N[(N[(1.0 / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.2%

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

   2. Simplified 7.2%

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

      [Start] 7.2	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 7.2	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 7.2	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 7.2	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 7.2	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 7.2	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 7.2	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 7.2	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 7.2	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 7.2	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 7.2	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 7.2	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 7.2	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 7.2	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 7.2	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 19.0%

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

      [Start] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{x}{-1 - x} \]
      div-sub [=>] 7.2	\[ \color{blue}{\left(\frac{-1}{x + -1} - \frac{x}{x + -1}\right)} - \frac{x}{-1 - x} \]
      sub-neg [=>] 7.2	\[ \color{blue}{\left(\frac{-1}{x + -1} + \left(-\frac{x}{x + -1}\right)\right)} - \frac{x}{-1 - x} \]
      associate--l+ [=>] 19.0	\[ \color{blue}{\frac{-1}{x + -1} + \left(\left(-\frac{x}{x + -1}\right) - \frac{x}{-1 - x}\right)} \]
      +-commutative [=>] 19.0	\[ \frac{-1}{\color{blue}{-1 + x}} + \left(\left(-\frac{x}{x + -1}\right) - \frac{x}{-1 - x}\right) \]
      +-commutative [=>] 19.0	\[ \frac{-1}{-1 + x} + \left(\left(-\frac{x}{\color{blue}{-1 + x}}\right) - \frac{x}{-1 - x}\right) \]

   4. Applied egg-rr 19.0%

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

      [Start] 19.0	\[ \frac{-1}{-1 + x} + \left(\left(-\frac{x}{-1 + x}\right) - \frac{x}{-1 - x}\right) \]
      associate-+r- [=>] 7.2	\[ \color{blue}{\left(\frac{-1}{-1 + x} + \left(-\frac{x}{-1 + x}\right)\right) - \frac{x}{-1 - x}} \]
      unsub-neg [=>] 7.2	\[ \color{blue}{\left(\frac{-1}{-1 + x} - \frac{x}{-1 + x}\right)} - \frac{x}{-1 - x} \]
      add-sqr-sqrt [=>] 7.1	\[ \left(\frac{-1}{-1 + x} - \color{blue}{\sqrt{\frac{x}{-1 + x}} \cdot \sqrt{\frac{x}{-1 + x}}}\right) - \frac{x}{-1 - x} \]
      sqrt-unprod [=>] 7.2	\[ \left(\frac{-1}{-1 + x} - \color{blue}{\sqrt{\frac{x}{-1 + x} \cdot \frac{x}{-1 + x}}}\right) - \frac{x}{-1 - x} \]
      sqr-neg [<=] 7.2	\[ \left(\frac{-1}{-1 + x} - \sqrt{\color{blue}{\left(-\frac{x}{-1 + x}\right) \cdot \left(-\frac{x}{-1 + x}\right)}}\right) - \frac{x}{-1 - x} \]
      sqrt-unprod [<=] 0.0	\[ \left(\frac{-1}{-1 + x} - \color{blue}{\sqrt{-\frac{x}{-1 + x}} \cdot \sqrt{-\frac{x}{-1 + x}}}\right) - \frac{x}{-1 - x} \]
      add-sqr-sqrt [<=] 3.6	\[ \left(\frac{-1}{-1 + x} - \color{blue}{\left(-\frac{x}{-1 + x}\right)}\right) - \frac{x}{-1 - x} \]
      associate--l- [=>] 3.6	\[ \color{blue}{\frac{-1}{-1 + x} - \left(\left(-\frac{x}{-1 + x}\right) + \frac{x}{-1 - x}\right)} \]
      +-commutative [=>] 3.6	\[ \frac{-1}{\color{blue}{x + -1}} - \left(\left(-\frac{x}{-1 + x}\right) + \frac{x}{-1 - x}\right) \]
      add-sqr-sqrt [=>] 0.0	\[ \frac{-1}{x + -1} - \left(\color{blue}{\sqrt{-\frac{x}{-1 + x}} \cdot \sqrt{-\frac{x}{-1 + x}}} + \frac{x}{-1 - x}\right) \]
      sqrt-unprod [=>] 19.0	\[ \frac{-1}{x + -1} - \left(\color{blue}{\sqrt{\left(-\frac{x}{-1 + x}\right) \cdot \left(-\frac{x}{-1 + x}\right)}} + \frac{x}{-1 - x}\right) \]
      sqr-neg [=>] 19.0	\[ \frac{-1}{x + -1} - \left(\sqrt{\color{blue}{\frac{x}{-1 + x} \cdot \frac{x}{-1 + x}}} + \frac{x}{-1 - x}\right) \]
      sqrt-unprod [<=] 19.0	\[ \frac{-1}{x + -1} - \left(\color{blue}{\sqrt{\frac{x}{-1 + x}} \cdot \sqrt{\frac{x}{-1 + x}}} + \frac{x}{-1 - x}\right) \]
      add-sqr-sqrt [<=] 19.0	\[ \frac{-1}{x + -1} - \left(\color{blue}{\frac{x}{-1 + x}} + \frac{x}{-1 - x}\right) \]

   5. Taylor expanded in x around inf 98.9%

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

   6. Applied egg-rr 98.9%

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

      [Start] 98.9	\[ \frac{-1}{x + -1} - \frac{2}{x} \]
      metadata-eval [<=] 98.9	\[ \frac{-1}{x + -1} - \frac{\color{blue}{\frac{-2}{-1}}}{x} \]
      metadata-eval [<=] 98.9	\[ \frac{-1}{x + -1} - \frac{\frac{\color{blue}{-2}}{-1}}{x} \]
      associate-/r* [<=] 98.9	\[ \frac{-1}{x + -1} - \color{blue}{\frac{-2}{-1 \cdot x}} \]
      frac-sub [=>] 50.9	\[ \color{blue}{\frac{-1 \cdot \left(-1 \cdot x\right) - \left(x + -1\right) \cdot \left(-2\right)}{\left(x + -1\right) \cdot \left(-1 \cdot x\right)}} \]
      associate-/r* [=>] 98.9	\[ \color{blue}{\frac{\frac{-1 \cdot \left(-1 \cdot x\right) - \left(x + -1\right) \cdot \left(-2\right)}{x + -1}}{-1 \cdot x}} \]
      metadata-eval [<=] 98.9	\[ \frac{\frac{\color{blue}{\left(-1\right)} \cdot \left(-1 \cdot x\right) - \left(x + -1\right) \cdot \left(-2\right)}{x + -1}}{-1 \cdot x} \]
      distribute-lft-neg-in [<=] 98.9	\[ \frac{\frac{\color{blue}{\left(-1 \cdot \left(-1 \cdot x\right)\right)} - \left(x + -1\right) \cdot \left(-2\right)}{x + -1}}{-1 \cdot x} \]
      *-un-lft-identity [<=] 98.9	\[ \frac{\frac{\left(-\color{blue}{-1 \cdot x}\right) - \left(x + -1\right) \cdot \left(-2\right)}{x + -1}}{-1 \cdot x} \]
      distribute-lft-neg-in [=>] 98.9	\[ \frac{\frac{\color{blue}{\left(--1\right) \cdot x} - \left(x + -1\right) \cdot \left(-2\right)}{x + -1}}{-1 \cdot x} \]
      metadata-eval [=>] 98.9	\[ \frac{\frac{\color{blue}{1} \cdot x - \left(x + -1\right) \cdot \left(-2\right)}{x + -1}}{-1 \cdot x} \]
      *-un-lft-identity [<=] 98.9	\[ \frac{\frac{\color{blue}{x} - \left(x + -1\right) \cdot \left(-2\right)}{x + -1}}{-1 \cdot x} \]
      +-commutative [=>] 98.9	\[ \frac{\frac{x - \color{blue}{\left(-1 + x\right)} \cdot \left(-2\right)}{x + -1}}{-1 \cdot x} \]
      metadata-eval [=>] 98.9	\[ \frac{\frac{x - \left(-1 + x\right) \cdot \color{blue}{-2}}{x + -1}}{-1 \cdot x} \]
      +-commutative [=>] 98.9	\[ \frac{\frac{x - \left(-1 + x\right) \cdot -2}{\color{blue}{-1 + x}}}{-1 \cdot x} \]
      mul-1-neg [=>] 98.9	\[ \frac{\frac{x - \left(-1 + x\right) \cdot -2}{-1 + x}}{\color{blue}{-x}} \]

   7. Simplified 99.4%

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

      [Start] 98.9	\[ \frac{\frac{x - \left(-1 + x\right) \cdot -2}{-1 + x}}{-x} \]
      div-sub [=>] 99.3	\[ \frac{\color{blue}{\frac{x}{-1 + x} - \frac{\left(-1 + x\right) \cdot -2}{-1 + x}}}{-x} \]
      sub-neg [=>] 99.3	\[ \frac{\color{blue}{\frac{x}{-1 + x} + \left(-\frac{\left(-1 + x\right) \cdot -2}{-1 + x}\right)}}{-x} \]
      +-commutative [=>] 99.3	\[ \frac{\frac{x}{\color{blue}{x + -1}} + \left(-\frac{\left(-1 + x\right) \cdot -2}{-1 + x}\right)}{-x} \]
      *-commutative [=>] 99.3	\[ \frac{\frac{x}{x + -1} + \left(-\frac{\color{blue}{-2 \cdot \left(-1 + x\right)}}{-1 + x}\right)}{-x} \]
      associate-/l* [=>] 99.4	\[ \frac{\frac{x}{x + -1} + \left(-\color{blue}{\frac{-2}{\frac{-1 + x}{-1 + x}}}\right)}{-x} \]
      *-inverses [=>] 99.4	\[ \frac{\frac{x}{x + -1} + \left(-\frac{-2}{\color{blue}{1}}\right)}{-x} \]
      metadata-eval [=>] 99.4	\[ \frac{\frac{x}{x + -1} + \left(-\color{blue}{-2}\right)}{-x} \]
      metadata-eval [=>] 99.4	\[ \frac{\frac{x}{x + -1} + \color{blue}{2}}{-x} \]

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

   1. Initial program 99.4%

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

   2. Simplified 99.4%

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

      [Start] 99.4	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 99.4	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 99.4	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 99.4	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 99.4	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 99.4	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 99.4	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 99.4	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 99.4	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 99.4	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 99.4	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 99.4	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 99.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 99.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 99.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 99.4%

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

      [Start] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{-1 - x} \]
      frac-2neg [=>] 99.4	\[ \color{blue}{\frac{-\left(-1 - x\right)}{-\left(x + -1\right)}} - \frac{x}{-1 - x} \]
      frac-sub [=>] 99.4	\[ \color{blue}{\frac{\left(-\left(-1 - x\right)\right) \cdot \left(-1 - x\right) - \left(-\left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)}} \]
      neg-sub0 [=>] 99.4	\[ \frac{\color{blue}{\left(0 - \left(-1 - x\right)\right)} \cdot \left(-1 - x\right) - \left(-\left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      metadata-eval [<=] 99.4	\[ \frac{\left(\color{blue}{\log 1} - \left(-1 - x\right)\right) \cdot \left(-1 - x\right) - \left(-\left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      associate--r- [=>] 99.4	\[ \frac{\color{blue}{\left(\left(\log 1 - -1\right) + x\right)} \cdot \left(-1 - x\right) - \left(-\left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      metadata-eval [=>] 99.4	\[ \frac{\left(\left(\color{blue}{0} - -1\right) + x\right) \cdot \left(-1 - x\right) - \left(-\left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      metadata-eval [=>] 99.4	\[ \frac{\left(\color{blue}{1} + x\right) \cdot \left(-1 - x\right) - \left(-\left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      +-commutative [=>] 99.4	\[ \frac{\color{blue}{\left(x + 1\right)} \cdot \left(-1 - x\right) - \left(-\left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      neg-sub0 [=>] 99.4	\[ \frac{\left(x + 1\right) \cdot \left(-1 - x\right) - \color{blue}{\left(0 - \left(x + -1\right)\right)} \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      metadata-eval [<=] 99.4	\[ \frac{\left(x + 1\right) \cdot \left(-1 - x\right) - \left(\color{blue}{\log 1} - \left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      +-commutative [=>] 99.4	\[ \frac{\left(x + 1\right) \cdot \left(-1 - x\right) - \left(\log 1 - \color{blue}{\left(-1 + x\right)}\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      associate--r+ [=>] 99.4	\[ \frac{\left(x + 1\right) \cdot \left(-1 - x\right) - \color{blue}{\left(\left(\log 1 - -1\right) - x\right)} \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      metadata-eval [=>] 99.4	\[ \frac{\left(x + 1\right) \cdot \left(-1 - x\right) - \left(\left(\color{blue}{0} - -1\right) - x\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      metadata-eval [=>] 99.4	\[ \frac{\left(x + 1\right) \cdot \left(-1 - x\right) - \left(\color{blue}{1} - x\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]

   4. Taylor expanded in x around 0 100.0%

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

   5. Applied egg-rr 99.9%

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

      [Start] 100.0	\[ \frac{-3 \cdot x - 1}{\left(1 - x\right) \cdot \left(-1 - x\right)} \]
      frac-2neg [=>] 100.0	\[ \color{blue}{\frac{-\left(-3 \cdot x - 1\right)}{-\left(1 - x\right) \cdot \left(-1 - x\right)}} \]
      clear-num [=>] 99.9	\[ \color{blue}{\frac{1}{\frac{-\left(1 - x\right) \cdot \left(-1 - x\right)}{-\left(-3 \cdot x - 1\right)}}} \]
      associate-/r/ [=>] 99.9	\[ \color{blue}{\frac{1}{-\left(1 - x\right) \cdot \left(-1 - x\right)} \cdot \left(-\left(-3 \cdot x - 1\right)\right)} \]
      distribute-rgt-neg-in [=>] 99.9	\[ \frac{1}{\color{blue}{\left(1 - x\right) \cdot \left(-\left(-1 - x\right)\right)}} \cdot \left(-\left(-3 \cdot x - 1\right)\right) \]
      neg-sub0 [=>] 99.9	\[ \frac{1}{\left(1 - x\right) \cdot \left(-\left(-1 - x\right)\right)} \cdot \color{blue}{\left(0 - \left(-3 \cdot x - 1\right)\right)} \]
      metadata-eval [<=] 99.9	\[ \frac{1}{\left(1 - x\right) \cdot \left(-\left(-1 - x\right)\right)} \cdot \left(\color{blue}{\log 1} - \left(-3 \cdot x - 1\right)\right) \]
      sub-neg [=>] 99.9	\[ \frac{1}{\left(1 - x\right) \cdot \left(-\left(-1 - x\right)\right)} \cdot \left(\log 1 - \color{blue}{\left(-3 \cdot x + \left(-1\right)\right)}\right) \]
      metadata-eval [=>] 99.9	\[ \frac{1}{\left(1 - x\right) \cdot \left(-\left(-1 - x\right)\right)} \cdot \left(\log 1 - \left(-3 \cdot x + \color{blue}{-1}\right)\right) \]
      +-commutative [=>] 99.9	\[ \frac{1}{\left(1 - x\right) \cdot \left(-\left(-1 - x\right)\right)} \cdot \left(\log 1 - \color{blue}{\left(-1 + -3 \cdot x\right)}\right) \]
      associate--r+ [=>] 99.9	\[ \frac{1}{\left(1 - x\right) \cdot \left(-\left(-1 - x\right)\right)} \cdot \color{blue}{\left(\left(\log 1 - -1\right) - -3 \cdot x\right)} \]
      metadata-eval [=>] 99.9	\[ \frac{1}{\left(1 - x\right) \cdot \left(-\left(-1 - x\right)\right)} \cdot \left(\left(\color{blue}{0} - -1\right) - -3 \cdot x\right) \]
      metadata-eval [=>] 99.9	\[ \frac{1}{\left(1 - x\right) \cdot \left(-\left(-1 - x\right)\right)} \cdot \left(\color{blue}{1} - -3 \cdot x\right) \]

   6. Taylor expanded in x around 0 100.0%

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

   7. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{1}{1 + -1 \cdot {x}^{2}} \cdot \left(1 - -3 \cdot x\right) \]
      mul-1-neg [=>] 100.0	\[ \frac{1}{1 + \color{blue}{\left(-{x}^{2}\right)}} \cdot \left(1 - -3 \cdot x\right) \]
      unpow2 [=>] 100.0	\[ \frac{1}{1 + \left(-\color{blue}{x \cdot x}\right)} \cdot \left(1 - -3 \cdot x\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	1732

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

Alternative 2
Accuracy	99.7%
Cost	1732

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

Alternative 3
Accuracy	99.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.88\right):\\ \;\;\;\;\frac{\frac{x}{x + -1} + 2}{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot 3\right) \cdot \left(1 + x \cdot x\right)\\ \end{array} \]

Alternative 4
Accuracy	99.0%
Cost	905

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

Alternative 5
Accuracy	98.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x + -1} + x \cdot 2\\ \end{array} \]

Alternative 6
Accuracy	98.8%
Cost	841

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

Alternative 7
Accuracy	99.0%
Cost	841

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

Alternative 8
Accuracy	98.5%
Cost	584

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

Alternative 9
Accuracy	97.9%
Cost	456

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

Alternative 10
Accuracy	50.6%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023142 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))