Average Error: 0.0 → 0.0
Time: 2.4s
Precision: 64
\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
↓
\[x - \left(2.30753 + x \cdot 0.27061000000000002\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right) \cdot 1}\]
x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}
↓
x - \left(2.30753 + x \cdot 0.27061000000000002\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right) \cdot 1}
double f(double x) {
        double r112118 = x;
        double r112119 = 2.30753;
        double r112120 = 0.27061;
        double r112121 = r112118 * r112120;
        double r112122 = r112119 + r112121;
        double r112123 = 1.0;
        double r112124 = 0.99229;
        double r112125 = 0.04481;
        double r112126 = r112118 * r112125;
        double r112127 = r112124 + r112126;
        double r112128 = r112127 * r112118;
        double r112129 = r112123 + r112128;
        double r112130 = r112122 / r112129;
        double r112131 = r112118 - r112130;
        return r112131;
}
↓
double f(double x) {
        double r112132 = x;
        double r112133 = 2.30753;
        double r112134 = 0.27061;
        double r112135 = r112132 * r112134;
        double r112136 = r112133 + r112135;
        double r112137 = 1.0;
        double r112138 = 0.04481;
        double r112139 = 0.99229;
        double r112140 = fma(r112138, r112132, r112139);
        double r112141 = 1.0;
        double r112142 = fma(r112132, r112140, r112141);
        double r112143 = r112142 * r112137;
        double r112144 = r112137 / r112143;
        double r112145 = r112136 * r112144;
        double r112146 = r112132 - r112145;
        return r112146;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
  2. Using strategy rm

  3. Applied div-inv0.0

    \[\leadsto x - \color{blue}{\left(2.30753 + x \cdot 0.27061000000000002\right) \cdot \frac{1}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}}\]

  4. Simplified0.0

    \[\leadsto x - \left(2.30753 + x \cdot 0.27061000000000002\right) \cdot \color{blue}{\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right) \cdot 1}}\]

  5. Final simplification0.0

    \[\leadsto x - \left(2.30753 + x \cdot 0.27061000000000002\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right) \cdot 1}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* (+ 0.99229 (* x 0.04481)) x)))))