Asymptote A

Average Error: 14.6 → 0.4

Time: 4.1s

Precision: binary64

Cost: 448

Math

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

↓

\[\frac{2}{1 - x \cdot x} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (/ 2.0 (- 1.0 (* x x))))

C

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

↓

double code(double x) {
	return 2.0 / (1.0 - (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x + 1.0d0)) - (1.0d0 / (x - 1.0d0))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (1.0d0 - (x * x))
end function

Java

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

↓

public static double code(double x) {
	return 2.0 / (1.0 - (x * x));
}

Python

def code(x):
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0))

↓

def code(x):
	return 2.0 / (1.0 - (x * x))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(1.0 / Float64(x - 1.0)))
end

↓

function code(x)
	return Float64(2.0 / Float64(1.0 - Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x + 1.0)) - (1.0 / (x - 1.0));
end

↓

function tmp = code(x)
	tmp = 2.0 / (1.0 - (x * x));
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(2.0 / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.6

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

2. Applied egg-rr 14.0

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

3. Taylor expanded in x around 0 0.4

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

4. Taylor expanded in x around 0 0.4

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

5. Simplified 0.4

   \[\leadsto \frac{2}{\color{blue}{1 - x \cdot x}} \]
   Proof
   (-.f64 1 (*.f64 x x)): 0 points increase in error, 0 points decrease in error
   (-.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 x 2))): 0 points increase in error, 1 points decrease in error
   (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.4

   \[\leadsto \frac{2}{1 - x \cdot x} \]

Alternatives

Alternative 1
Error	0.9
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{-2}{x}}{x}\\ \mathbf{if}\;x \leq -3.902514750335879:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.2150609600684981 \cdot 10^{-5}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	31.8
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022300 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))