Asymptote C

Average Error: 29.2 → 0.2

Time: 8.2s

Precision: binary64

Cost: 8196

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \frac{-3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x + -1} - \left(\frac{x}{x + -1} + \frac{x}{-1 - x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0))) 5e-8)
   (+ (+ (/ -1.0 (* x x)) (/ -3.0 x)) (/ -3.0 (pow x 3.0)))
   (- (/ -1.0 (+ x -1.0)) (+ (/ x (+ x -1.0)) (/ x (- -1.0 x))))))

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)) - ((x + 1.0) / (x + -1.0))) <= 5e-8) {
		tmp = ((-1.0 / (x * x)) + (-3.0 / x)) + (-3.0 / pow(x, 3.0));
	} else {
		tmp = (-1.0 / (x + -1.0)) - ((x / (x + -1.0)) + (x / (-1.0 - x)));
	}
	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)) - ((x + 1.0d0) / (x + (-1.0d0)))) <= 5d-8) then
        tmp = (((-1.0d0) / (x * x)) + ((-3.0d0) / x)) + ((-3.0d0) / (x ** 3.0d0))
    else
        tmp = ((-1.0d0) / (x + (-1.0d0))) - ((x / (x + (-1.0d0))) + (x / ((-1.0d0) - x)))
    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)) - ((x + 1.0) / (x + -1.0))) <= 5e-8) {
		tmp = ((-1.0 / (x * x)) + (-3.0 / x)) + (-3.0 / Math.pow(x, 3.0));
	} else {
		tmp = (-1.0 / (x + -1.0)) - ((x / (x + -1.0)) + (x / (-1.0 - x)));
	}
	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)) - ((x + 1.0) / (x + -1.0))) <= 5e-8:
		tmp = ((-1.0 / (x * x)) + (-3.0 / x)) + (-3.0 / math.pow(x, 3.0))
	else:
		tmp = (-1.0 / (x + -1.0)) - ((x / (x + -1.0)) + (x / (-1.0 - x)))
	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(x + 1.0) / Float64(x + -1.0))) <= 5e-8)
		tmp = Float64(Float64(Float64(-1.0 / Float64(x * x)) + Float64(-3.0 / x)) + Float64(-3.0 / (x ^ 3.0)));
	else
		tmp = Float64(Float64(-1.0 / Float64(x + -1.0)) - Float64(Float64(x / Float64(x + -1.0)) + Float64(x / Float64(-1.0 - x))));
	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)) - ((x + 1.0) / (x + -1.0))) <= 5e-8)
		tmp = ((-1.0 / (x * x)) + (-3.0 / x)) + (-3.0 / (x ^ 3.0));
	else
		tmp = (-1.0 / (x + -1.0)) - ((x / (x + -1.0)) + (x / (-1.0 - x)));
	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[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-3.0 / x), $MachinePrecision]), $MachinePrecision] + N[(-3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 4.9999999999999998e-8

   1. Initial program 59.3

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

   2. Simplified 59.3

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

      [Start] 59.3	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 59.3	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 59.3	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 59.3	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 59.3	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 59.3	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 59.3	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 59.3	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 59.3	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 59.3	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 59.3	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 59.3	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 59.3	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 59.3	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 59.3	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 59.3	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 59.3	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 59.3	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 59.3	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 59.3	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 59.3	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 59.3	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 59.3	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 0.5

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

   4. Simplified 0.2

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

      [Start] 0.5	\[ -\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right) \]
      distribute-neg-in [=>] 0.5	\[ \color{blue}{\left(-3 \cdot \frac{1}{{x}^{3}}\right) + \left(-\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)} \]
      +-commutative [=>] 0.5	\[ \color{blue}{\left(-\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right)} \]
      distribute-neg-in [=>] 0.5	\[ \color{blue}{\left(\left(-\frac{1}{{x}^{2}}\right) + \left(-3 \cdot \frac{1}{x}\right)\right)} + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      unpow2 [=>] 0.5	\[ \left(\left(-\frac{1}{\color{blue}{x \cdot x}}\right) + \left(-3 \cdot \frac{1}{x}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      distribute-neg-frac [=>] 0.5	\[ \left(\color{blue}{\frac{-1}{x \cdot x}} + \left(-3 \cdot \frac{1}{x}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      metadata-eval [=>] 0.5	\[ \left(\frac{\color{blue}{-1}}{x \cdot x} + \left(-3 \cdot \frac{1}{x}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      associate-*r/ [=>] 0.2	\[ \left(\frac{-1}{x \cdot x} + \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      metadata-eval [=>] 0.2	\[ \left(\frac{-1}{x \cdot x} + \left(-\frac{\color{blue}{3}}{x}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      distribute-neg-frac [=>] 0.2	\[ \left(\frac{-1}{x \cdot x} + \color{blue}{\frac{-3}{x}}\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      metadata-eval [=>] 0.2	\[ \left(\frac{-1}{x \cdot x} + \frac{\color{blue}{-3}}{x}\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      associate-*r/ [=>] 0.2	\[ \left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \left(-\color{blue}{\frac{3 \cdot 1}{{x}^{3}}}\right) \]
      metadata-eval [=>] 0.2	\[ \left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \left(-\frac{\color{blue}{3}}{{x}^{3}}\right) \]
      distribute-neg-frac [=>] 0.2	\[ \left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \color{blue}{\frac{-3}{{x}^{3}}} \]
      metadata-eval [=>] 0.2	\[ \left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \frac{\color{blue}{-3}}{{x}^{3}} \]

   if 4.9999999999999998e-8 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 0.2

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

   2. Simplified 0.2

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

      [Start] 0.2	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 0.2	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 0.2	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 0.2	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 0.2	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 0.2	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 0.2	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 0.2	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 0.2	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 0.2	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 0.2	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 0.2	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 0.2	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 0.2	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 0.2	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 0.2	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 0.2	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 0.2	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 0.2	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 0.2	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 0.2	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 0.2	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 0.2	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 0.2

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

   4. Applied egg-rr 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \frac{-3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x + -1} - \left(\frac{x}{x + -1} + \frac{x}{-1 - x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1988

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x + -1} - \left(\frac{x}{x + -1} + \frac{x}{-1 - x}\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1732

\[\begin{array}{l} t_0 := \frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.7
Cost	841

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

Alternative 4
Error	0.6
Cost	841

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

Alternative 5
Error	0.9
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 6
Error	1.2
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 7
Error	31.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023066 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))