Average Error: 0.0 → 0.1
Time: 3.9s
Precision: binary64
\[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x}\]
↓
\[x - \frac{1}{\sqrt{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} \cdot \frac{2.30753 + x \cdot 0.27061}{\sqrt{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}}\]
x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x}
↓
x - \frac{1}{\sqrt{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} \cdot \frac{2.30753 + x \cdot 0.27061}{\sqrt{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}}
(FPCore (x)
 :precision binary64
 (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))
↓
(FPCore (x)
 :precision binary64
 (-
  x
  (*
   (/ 1.0 (sqrt (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))))
   (/
    (+ 2.30753 (* x 0.27061))
    (sqrt (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))))))
double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
}
↓
double code(double x) {
	return x - ((1.0 / sqrt(1.0 + (x * (0.99229 + (x * 0.04481))))) * ((2.30753 + (x * 0.27061)) / sqrt(1.0 + (x * (0.99229 + (x * 0.04481))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt_binary64_28280.1

    \[\leadsto x - \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\sqrt{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \cdot \sqrt{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x}}}\]

  4. Applied *-un-lft-identity_binary64_28060.1

    \[\leadsto x - \frac{\color{blue}{1 \cdot \left(2.30753 + x \cdot 0.27061\right)}}{\sqrt{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \cdot \sqrt{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x}}\]

  5. Applied times-frac_binary64_28120.1

    \[\leadsto x - \color{blue}{\frac{1}{\sqrt{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x}} \cdot \frac{2.30753 + x \cdot 0.27061}{\sqrt{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x}}}\]

  6. Final simplification0.1

    \[\leadsto x - \frac{1}{\sqrt{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} \cdot \frac{2.30753 + x \cdot 0.27061}{\sqrt{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}}\]

Reproduce

herbie shell --seed 2021093 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))