Asymptote C

Average Error: 28.7 → 0.9

Time: 9.5s

Precision: binary64

Cost: 1988

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1 - \left(2 + \left(\frac{2}{x \cdot x} + \frac{-2}{x}\right)\right)}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + x \cdot \left(-2 - x \cdot -2\right)}{x + -1}\\ \end{array} \]

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-11)
   (/ (- -1.0 (+ 2.0 (+ (/ 2.0 (* x x)) (/ -2.0 x)))) (+ x -1.0))
   (/ (+ -1.0 (* x (- -2.0 (* x -2.0)))) (+ x -1.0))))

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)) - ((x + 1.0) / (x + -1.0))) <= 5e-11) {
		tmp = (-1.0 - (2.0 + ((2.0 / (x * x)) + (-2.0 / x)))) / (x + -1.0);
	} else {
		tmp = (-1.0 + (x * (-2.0 - (x * -2.0)))) / (x + -1.0);
	}
	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)) - ((x + 1.0d0) / (x + (-1.0d0)))) <= 5d-11) then
        tmp = ((-1.0d0) - (2.0d0 + ((2.0d0 / (x * x)) + ((-2.0d0) / x)))) / (x + (-1.0d0))
    else
        tmp = ((-1.0d0) + (x * ((-2.0d0) - (x * (-2.0d0))))) / (x + (-1.0d0))
    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)) - ((x + 1.0) / (x + -1.0))) <= 5e-11) {
		tmp = (-1.0 - (2.0 + ((2.0 / (x * x)) + (-2.0 / x)))) / (x + -1.0);
	} else {
		tmp = (-1.0 + (x * (-2.0 - (x * -2.0)))) / (x + -1.0);
	}
	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)) - ((x + 1.0) / (x + -1.0))) <= 5e-11:
		tmp = (-1.0 - (2.0 + ((2.0 / (x * x)) + (-2.0 / x)))) / (x + -1.0)
	else:
		tmp = (-1.0 + (x * (-2.0 - (x * -2.0)))) / (x + -1.0)
	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(x + 1.0) / Float64(x + -1.0))) <= 5e-11)
		tmp = Float64(Float64(-1.0 - Float64(2.0 + Float64(Float64(2.0 / Float64(x * x)) + Float64(-2.0 / x)))) / Float64(x + -1.0));
	else
		tmp = Float64(Float64(-1.0 + Float64(x * Float64(-2.0 - Float64(x * -2.0)))) / Float64(x + -1.0));
	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)) - ((x + 1.0) / (x + -1.0))) <= 5e-11)
		tmp = (-1.0 - (2.0 + ((2.0 / (x * x)) + (-2.0 / x)))) / (x + -1.0);
	else
		tmp = (-1.0 + (x * (-2.0 - (x * -2.0)))) / (x + -1.0);
	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[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-11], N[(N[(-1.0 - N[(2.0 + N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + N[(x * N[(-2.0 - N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.00000000000000018e-11

   1. Initial program 59.4

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

   2. Simplified 59.4

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

      [Start] 59.4	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 59.4	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 59.4	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 59.4	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 59.4	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 59.4	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 59.4	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 59.4	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 59.4	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 59.4	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 59.4	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 59.4	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 59.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 59.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 59.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 59.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 59.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 59.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 59.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 59.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 59.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 59.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 59.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 51.4

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

   4. Taylor expanded in x around 0 51.4

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

   5. Taylor expanded in x around inf 0.3

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

   6. Simplified 0.3

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

      [Start] 0.3	\[ \frac{-1 - \left(\left(2 + 2 \cdot \frac{1}{{x}^{2}}\right) - 2 \cdot \frac{1}{x}\right)}{x - 1} \]
      associate--l+ [=>] 0.3	\[ \frac{-1 - \color{blue}{\left(2 + \left(2 \cdot \frac{1}{{x}^{2}} - 2 \cdot \frac{1}{x}\right)\right)}}{x - 1} \]
      associate-*r/ [=>] 0.3	\[ \frac{-1 - \left(2 + \left(\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} - 2 \cdot \frac{1}{x}\right)\right)}{x - 1} \]
      metadata-eval [=>] 0.3	\[ \frac{-1 - \left(2 + \left(\frac{\color{blue}{2}}{{x}^{2}} - 2 \cdot \frac{1}{x}\right)\right)}{x - 1} \]
      unpow2 [=>] 0.3	\[ \frac{-1 - \left(2 + \left(\frac{2}{\color{blue}{x \cdot x}} - 2 \cdot \frac{1}{x}\right)\right)}{x - 1} \]
      associate-*r/ [=>] 0.3	\[ \frac{-1 - \left(2 + \left(\frac{2}{x \cdot x} - \color{blue}{\frac{2 \cdot 1}{x}}\right)\right)}{x - 1} \]
      metadata-eval [=>] 0.3	\[ \frac{-1 - \left(2 + \left(\frac{2}{x \cdot x} - \frac{\color{blue}{2}}{x}\right)\right)}{x - 1} \]

   if 5.00000000000000018e-11 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 0.3

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

   2. Simplified 0.3

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

      [Start] 0.3	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 0.3	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 0.3	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 0.3	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 0.3	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 0.3	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 0.3	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 0.3	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 0.3	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 0.3	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 0.3	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 0.3	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 0.3	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 0.3	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 0.3	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 0.3	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 0.3	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 0.3	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 0.3	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 0.3	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 0.3	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 0.3	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 0.3	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 0.3

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

   4. Taylor expanded in x around 0 0.3

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

   5. Taylor expanded in x around 0 1.5

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

   6. Simplified 1.5

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

      [Start] 1.5	\[ \frac{-1 - \left(x + \left(1 + -2 \cdot x\right) \cdot x\right)}{x - 1} \]
      *-commutative [=>] 1.5	\[ \frac{-1 - \left(x + \left(1 + \color{blue}{x \cdot -2}\right) \cdot x\right)}{x - 1} \]

   7. Applied egg-rr 1.5

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1 - \left(2 + \left(\frac{2}{x \cdot x} + \frac{-2}{x}\right)\right)}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + x \cdot \left(-2 - x \cdot -2\right)}{x + -1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1732

\[\begin{array}{l} t_0 := \frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3 - \frac{\frac{2}{x} + -2}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.9
Cost	1732

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3 - \frac{\frac{2}{x} + -2}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + x \cdot \left(-2 - x \cdot -2\right)}{x + -1}\\ \end{array} \]

Alternative 3
Error	0.1
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -390000:\\ \;\;\;\;\frac{-1}{x \cdot x} + \frac{-3}{x}\\ \mathbf{elif}\;x \leq 270000:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \left(\frac{2}{x} + -2\right)}{x + -1}\\ \end{array} \]

Alternative 4
Error	0.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x \cdot x} + \frac{-3}{x}\\ \mathbf{elif}\;x \leq 0.82:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \left(\frac{2}{x} + -2\right)}{x + -1}\\ \end{array} \]

Alternative 5
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 6
Error	0.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x \cdot x} + \frac{-3}{x}\\ \mathbf{elif}\;x \leq 0.82:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x} + -3}{x + -1}\\ \end{array} \]

Alternative 7
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 8
Error	1.4
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 9
Error	31.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023027 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))