Asymptote A

Average Error: 14.0 → 0.4

Time: 5.9s

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 14.0

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

2. Applied egg-rr 14.0

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

3. Simplified 0.4

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

   [Start] 14.0	\[ \frac{1}{1 + x} + \left(-\frac{-1}{1 - x}\right) \]
   rational.json-simplify-10 [=>] 14.0	\[ \frac{1}{1 + x} + \color{blue}{\frac{\frac{-1}{1 - x}}{-1}} \]
   rational.json-simplify-44 [=>] 14.0	\[ \frac{1}{1 + x} + \color{blue}{\frac{\frac{-1}{-1}}{1 - x}} \]
   metadata-eval [=>] 14.0	\[ \frac{1}{1 + x} + \frac{\color{blue}{1}}{1 - x} \]
   rational.json-simplify-28 [<=] 13.4	\[ \color{blue}{\frac{\left(1 + x\right) + \left(1 - x\right)}{\left(1 + x\right) \cdot \left(1 - x\right)}} \]
   rational.json-simplify-17 [=>] 13.4	\[ \frac{\color{blue}{\left(x - -1\right)} + \left(1 - x\right)}{\left(1 + x\right) \cdot \left(1 - x\right)} \]
   rational.json-simplify-27 [=>] 0.4	\[ \frac{\color{blue}{1 - -1}}{\left(1 + x\right) \cdot \left(1 - x\right)} \]
   metadata-eval [=>] 0.4	\[ \frac{\color{blue}{2}}{\left(1 + x\right) \cdot \left(1 - x\right)} \]
   rational.json-simplify-1 [=>] 0.4	\[ \frac{2}{\color{blue}{\left(x + 1\right)} \cdot \left(1 - x\right)} \]
   rational.json-simplify-33 [=>] 0.4	\[ \frac{2}{\color{blue}{1 \cdot 1 - x \cdot x}} \]
   metadata-eval [=>] 0.4	\[ \frac{2}{\color{blue}{1} - x \cdot x} \]

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	31.3
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023064 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))