Average Error: 0.0 → 0.0
Time: 4.4s
Precision: 64
\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
↓
\[\left(-\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} + x\]
x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}
↓
\left(-\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} + x
double f(double x) {
        double r100072 = x;
        double r100073 = 2.30753;
        double r100074 = 0.27061;
        double r100075 = r100072 * r100074;
        double r100076 = r100073 + r100075;
        double r100077 = 1.0;
        double r100078 = 0.99229;
        double r100079 = 0.04481;
        double r100080 = r100072 * r100079;
        double r100081 = r100078 + r100080;
        double r100082 = r100081 * r100072;
        double r100083 = r100077 + r100082;
        double r100084 = r100076 / r100083;
        double r100085 = r100072 - r100084;
        return r100085;
}
↓
double f(double x) {
        double r100086 = 0.27061;
        double r100087 = x;
        double r100088 = 2.30753;
        double r100089 = fma(r100086, r100087, r100088);
        double r100090 = -r100089;
        double r100091 = 1.0;
        double r100092 = 0.04481;
        double r100093 = 0.99229;
        double r100094 = fma(r100092, r100087, r100093);
        double r100095 = 1.0;
        double r100096 = fma(r100087, r100094, r100095);
        double r100097 = r100091 / r100096;
        double r100098 = r100090 * r100097;
        double r100099 = r100098 + r100087;
        return r100099;
}

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. Simplified0.0

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

  4. Applied div-inv0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020024 +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)))))