Asymptote C

Average Error: 29.4 → 0.1

Time: 7.0s

Precision: binary64

Cost: 7496

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-3 - \frac{1 + \frac{3}{x}}{x}}{x}\\ \mathbf{if}\;x \leq -80162780.40680923:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 90.93146838917038:\\ \;\;\;\;\mathsf{fma}\left(x + 1, \frac{1}{1 - x}, \frac{x}{x + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- -3.0 (/ (+ 1.0 (/ 3.0 x)) x)) x)))
   (if (<= x -80162780.40680923)
     t_0
     (if (<= x 90.93146838917038)
       (fma (+ x 1.0) (/ 1.0 (- 1.0 x)) (/ x (+ x 1.0)))
       t_0))))

C

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

↓

double code(double x) {
	double t_0 = (-3.0 - ((1.0 + (3.0 / x)) / x)) / x;
	double tmp;
	if (x <= -80162780.40680923) {
		tmp = t_0;
	} else if (x <= 90.93146838917038) {
		tmp = fma((x + 1.0), (1.0 / (1.0 - x)), (x / (x + 1.0)));
	} else {
		tmp = t_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)
	t_0 = Float64(Float64(-3.0 - Float64(Float64(1.0 + Float64(3.0 / x)) / x)) / x)
	tmp = 0.0
	if (x <= -80162780.40680923)
		tmp = t_0;
	elseif (x <= 90.93146838917038)
		tmp = fma(Float64(x + 1.0), Float64(1.0 / Float64(1.0 - x)), Float64(x / Float64(x + 1.0)));
	else
		tmp = t_0;
	end
	return 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_] := Block[{t$95$0 = N[(N[(-3.0 - N[(N[(1.0 + N[(3.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -80162780.40680923], t$95$0, If[LessEqual[x, 90.93146838917038], N[(N[(x + 1.0), $MachinePrecision] * N[(1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] + N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -80162780.4068092257 or 90.9314683891703766 < x

   1. Initial program 59.4

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

   2. Applied egg-rr 59.4

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

   3. Taylor expanded in x around inf 0.4

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

   4. Simplified 0.1

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

   5. Taylor expanded in x around 0 0.4

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

   6. Simplified 0.1

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

   7. Applied egg-rr 0.1

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

   if -80162780.4068092257 < x < 90.9314683891703766

   1. Initial program 0.2

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

   2. Applied egg-rr 0.2

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

   3. Applied egg-rr 0.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -80162780.40680923:\\ \;\;\;\;\frac{-3 - \frac{1 + \frac{3}{x}}{x}}{x}\\ \mathbf{elif}\;x \leq 90.93146838917038:\\ \;\;\;\;\mathsf{fma}\left(x + 1, \frac{1}{1 - x}, \frac{x}{x + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 - \frac{1 + \frac{3}{x}}{x}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	1736

\[\begin{array}{l} t_0 := \frac{-3 - \frac{1 + \frac{3}{x}}{x}}{x}\\ \mathbf{if}\;x \leq -80162780.40680923:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 90.93146838917038:\\ \;\;\;\;\frac{\left(x + -1\right) + \left(x + 1\right) \cdot \frac{-1 - x}{x}}{\left(x + -1\right) \cdot \frac{x + 1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1224

\[\begin{array}{l} t_0 := \frac{-3 - \frac{1 + \frac{3}{x}}{x}}{x}\\ \mathbf{if}\;x \leq -80162780.40680923:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 90.93146838917038:\\ \;\;\;\;\frac{x}{x + 1} - \frac{1}{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.8
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -80162780.40680923:\\ \;\;\;\;\frac{-3 - \frac{1 + \frac{3}{x}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.17384801702489983:\\ \;\;\;\;x - \left(-1 + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \frac{-1 - \frac{3}{x}}{x \cdot x}\\ \end{array} \]

Alternative 4
Error	0.1
Cost	1096

\[\begin{array}{l} t_0 := \frac{-3 - \frac{1 + \frac{3}{x}}{x}}{x}\\ \mathbf{if}\;x \leq -80162780.40680923:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 90.93146838917038:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.8
Cost	968

\[\begin{array}{l} t_0 := \frac{-3 - \frac{1 + \frac{3}{x}}{x}}{x}\\ \mathbf{if}\;x \leq -80162780.40680923:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.17384801702489983:\\ \;\;\;\;x - \left(-1 + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.9
Cost	840

\[\begin{array}{l} t_0 := \frac{-3}{x} + \frac{-1}{x \cdot x}\\ \mathbf{if}\;x \leq -80162780.40680923:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.17384801702489983:\\ \;\;\;\;x - \left(-1 + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -80162780.40680923:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 0.17384801702489983:\\ \;\;\;\;x - \left(-1 + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 8
Error	1.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -80162780.40680923:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 0.17384801702489983:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 9
Error	1.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -80162780.40680923:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 0.17384801702489983:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 10
Error	31.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022217 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))