Asymptote C

Average Error: 29.2 → 0.0

Time: 9.2s

Precision: binary64

Cost: 968

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{x \cdot 3 + 1}{1 - x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -2e+41)
   (/ -3.0 x)
   (if (<= x 2e+16) (/ (+ (* x 3.0) 1.0) (- 1.0 (* x x))) (/ -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 <= -2e+41) {
		tmp = -3.0 / x;
	} else if (x <= 2e+16) {
		tmp = ((x * 3.0) + 1.0) / (1.0 - (x * x));
	} 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 <= (-2d+41)) then
        tmp = (-3.0d0) / x
    else if (x <= 2d+16) then
        tmp = ((x * 3.0d0) + 1.0d0) / (1.0d0 - (x * x))
    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 <= -2e+41) {
		tmp = -3.0 / x;
	} else if (x <= 2e+16) {
		tmp = ((x * 3.0) + 1.0) / (1.0 - (x * x));
	} 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 <= -2e+41:
		tmp = -3.0 / x
	elif x <= 2e+16:
		tmp = ((x * 3.0) + 1.0) / (1.0 - (x * x))
	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 <= -2e+41)
		tmp = Float64(-3.0 / x);
	elseif (x <= 2e+16)
		tmp = Float64(Float64(Float64(x * 3.0) + 1.0) / Float64(1.0 - Float64(x * x)));
	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 <= -2e+41)
		tmp = -3.0 / x;
	elseif (x <= 2e+16)
		tmp = ((x * 3.0) + 1.0) / (1.0 - (x * x));
	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, -2e+41], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 2e+16], N[(N[(N[(x * 3.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-3.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000001e41 or 2e16 < 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	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 0

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

   if -2.00000000000000001e41 < x < 2e16

   1. Initial program 3.4

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

   2. Simplified 3.4

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

      [Start] 3.4	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 3.4	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 3.4	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 3.4	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 3.4	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 3.4	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 3.4	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 3.4	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 3.4	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 3.4	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 3.4	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 3.4	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 3.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 3.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 3.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 3.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 3.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 3.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 3.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 3.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 3.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 3.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 3.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 3.4

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

   4. Simplified 3.3

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

      [Start] 3.4	\[ \left({\left(-1 - x\right)}^{2} + \left(1 - x\right) \cdot x\right) \cdot \frac{1}{\left(-1 - x\right) \cdot \left(-1 + x\right)} \]
      associate-*r/ [=>] 3.4	\[ \color{blue}{\frac{\left({\left(-1 - x\right)}^{2} + \left(1 - x\right) \cdot x\right) \cdot 1}{\left(-1 - x\right) \cdot \left(-1 + x\right)}} \]
      *-rgt-identity [=>] 3.4	\[ \frac{\color{blue}{{\left(-1 - x\right)}^{2} + \left(1 - x\right) \cdot x}}{\left(-1 - x\right) \cdot \left(-1 + x\right)} \]
      +-commutative [=>] 3.4	\[ \frac{\color{blue}{\left(1 - x\right) \cdot x + {\left(-1 - x\right)}^{2}}}{\left(-1 - x\right) \cdot \left(-1 + x\right)} \]
      *-commutative [=>] 3.4	\[ \frac{\color{blue}{x \cdot \left(1 - x\right)} + {\left(-1 - x\right)}^{2}}{\left(-1 - x\right) \cdot \left(-1 + x\right)} \]
      fma-def [=>] 3.3	\[ \frac{\color{blue}{\mathsf{fma}\left(x, 1 - x, {\left(-1 - x\right)}^{2}\right)}}{\left(-1 - x\right) \cdot \left(-1 + x\right)} \]
      +-commutative [=>] 3.3	\[ \frac{\mathsf{fma}\left(x, 1 - x, {\left(-1 - x\right)}^{2}\right)}{\left(-1 - x\right) \cdot \color{blue}{\left(x + -1\right)}} \]

   5. Taylor expanded in x around 0 0.1

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

   6. Taylor expanded in x around 0 0.0

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

   7. Simplified 0.0

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

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

3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{x \cdot 3 + 1}{1 - x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	712

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

Alternative 2
Error	1.1
Cost	584

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

Alternative 3
Error	1.5
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 4
Error	62.3
Cost	64

\[-2 \]

Alternative 5
Error	31.5
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023056 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))