Asymptote C

Average Error: 29.3 → 0.0

Time: 7.7s

Precision: binary64

Cost: 968

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+28}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{x \cdot -3 + -1}{x \cdot x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -5e+28)
   (/ -3.0 x)
   (if (<= x 2e+15) (/ (+ (* x -3.0) -1.0) (+ (* x x) -1.0)) (/ -3.0 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 <= -5e+28) {
		tmp = -3.0 / x;
	} else if (x <= 2e+15) {
		tmp = ((x * -3.0) + -1.0) / ((x * x) + -1.0);
	} else {
		tmp = -3.0 / 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 <= (-5d+28)) then
        tmp = (-3.0d0) / x
    else if (x <= 2d+15) then
        tmp = ((x * (-3.0d0)) + (-1.0d0)) / ((x * x) + (-1.0d0))
    else
        tmp = (-3.0d0) / 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 <= -5e+28) {
		tmp = -3.0 / x;
	} else if (x <= 2e+15) {
		tmp = ((x * -3.0) + -1.0) / ((x * x) + -1.0);
	} else {
		tmp = -3.0 / 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 <= -5e+28:
		tmp = -3.0 / x
	elif x <= 2e+15:
		tmp = ((x * -3.0) + -1.0) / ((x * x) + -1.0)
	else:
		tmp = -3.0 / 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 (x <= -5e+28)
		tmp = Float64(-3.0 / x);
	elseif (x <= 2e+15)
		tmp = Float64(Float64(Float64(x * -3.0) + -1.0) / Float64(Float64(x * x) + -1.0));
	else
		tmp = Float64(-3.0 / 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 <= -5e+28)
		tmp = -3.0 / x;
	elseif (x <= 2e+15)
		tmp = ((x * -3.0) + -1.0) / ((x * x) + -1.0);
	else
		tmp = -3.0 / 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[x, -5e+28], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 2e+15], N[(N[(N[(x * -3.0), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(-3.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999957e28 or 2e15 < x

   1. Initial program 60.4

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

   2. Simplified 60.4

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

      [Start] 60.4	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 60.4	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 60.4	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 60.4	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 60.4	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 60.4	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 60.4	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 60.4	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 60.4	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 60.4	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 60.4	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 60.4	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 60.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 60.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 60.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 60.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 60.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 60.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 60.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 60.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 60.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 60.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 60.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]
      distribute-neg-in [=>] 60.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{\left(-x\right) + \left(-1\right)}} \]
      +-commutative [<=] 60.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{\left(-1\right) + \left(-x\right)}} \]
      unsub-neg [=>] 60.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{\left(-1\right) - x}} \]
      metadata-eval [=>] 60.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} - x} \]

   3. Taylor expanded in x around inf 0.0

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

   if -4.99999999999999957e28 < x < 2e15

   1. Initial program 2.2

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

   2. Simplified 2.2

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

      [Start] 2.2	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 2.2	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 2.2	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 2.2	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 2.2	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 2.2	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 2.2	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 2.2	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 2.2	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 2.2	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 2.2	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 2.2	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 2.2	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 2.2	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 2.2	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 2.2	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 2.2	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 2.2	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 2.2	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 2.2	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 2.2	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 2.2	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 2.2	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]
      distribute-neg-in [=>] 2.2	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{\left(-x\right) + \left(-1\right)}} \]
      +-commutative [<=] 2.2	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{\left(-1\right) + \left(-x\right)}} \]
      unsub-neg [=>] 2.2	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{\left(-1\right) - x}} \]
      metadata-eval [=>] 2.2	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} - x} \]

   3. Applied egg-rr 2.2

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

   4. Taylor expanded in x around 0 0.0

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

   5. Taylor expanded in x around 0 0.0

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

   6. Simplified 0.0

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

      [Start] 0.0	\[ \frac{-3 \cdot x - 1}{{x}^{2} - 1} \]
      sub-neg [=>] 0.0	\[ \frac{-3 \cdot x - 1}{\color{blue}{{x}^{2} + \left(-1\right)}} \]
      unpow2 [=>] 0.0	\[ \frac{-3 \cdot x - 1}{\color{blue}{x \cdot x} + \left(-1\right)} \]
      metadata-eval [=>] 0.0	\[ \frac{-3 \cdot x - 1}{x \cdot x + \color{blue}{-1}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+28}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{x \cdot -3 + -1}{x \cdot x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	713

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

Alternative 2
Error	0.7
Cost	712

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

Alternative 3
Error	1.0
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 4
Error	1.3
Cost	456

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

Alternative 5
Error	62.3
Cost	64

\[-2 \]

Alternative 6
Error	31.4
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022356 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))