Average Error: 0.0 → 0.1
Time: 5.3s
Precision: binary64
\[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\\ x - \frac{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{t_0}}{t_0} \end{array} \]
x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x}
↓
\begin{array}{l}
t_0 := \sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\\
x - \frac{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{t_0}}{t_0}
\end{array}

(FPCore (x)
 :precision binary64
 (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (fma x (fma x 0.04481 0.99229) 1.0))))
   (- x (/ (/ (fma x 0.27061 2.30753) t_0) t_0))))
double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
}
↓
double code(double x) {
	double t_0 = sqrt(fma(x, fma(x, 0.04481, 0.99229), 1.0));
	return x - ((fma(x, 0.27061, 2.30753) / t_0) / t_0);
}

Error

Bits error versus x

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. Simplified0.0

    \[\leadsto \color{blue}{x - \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} \]

  3. Applied expm1-log1p-u_binary6416.2

    \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, 0.27061, 2.30753\right)\right)\right)}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \]

  4. Applied add-sqr-sqrt_binary6416.3

    \[\leadsto x - \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, 0.27061, 2.30753\right)\right)\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot \sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}} \]

  5. Applied associate-/r*_binary6416.3

    \[\leadsto x - \color{blue}{\frac{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, 0.27061, 2.30753\right)\right)\right)}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}} \]

  6. Simplified0.1

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

  7. Final simplification0.1

    \[\leadsto x - \frac{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} \]

Reproduce

herbie shell --seed 2022068 
(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)))))