Average Error: 0.0 → 0.0
Time: 1.8s
Precision: 64
\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
↓
\[\mathsf{fma}\left(-\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right), \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}, x\right)\]
x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}
↓
\mathsf{fma}\left(-\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right), \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}, x\right)
double f(double x) {
        double r79423 = x;
        double r79424 = 2.30753;
        double r79425 = 0.27061;
        double r79426 = r79423 * r79425;
        double r79427 = r79424 + r79426;
        double r79428 = 1.0;
        double r79429 = 0.99229;
        double r79430 = 0.04481;
        double r79431 = r79423 * r79430;
        double r79432 = r79429 + r79431;
        double r79433 = r79432 * r79423;
        double r79434 = r79428 + r79433;
        double r79435 = r79427 / r79434;
        double r79436 = r79423 - r79435;
        return r79436;
}
↓
double f(double x) {
        double r79437 = 0.27061;
        double r79438 = x;
        double r79439 = 2.30753;
        double r79440 = fma(r79437, r79438, r79439);
        double r79441 = -r79440;
        double r79442 = 1.0;
        double r79443 = 0.04481;
        double r79444 = 0.99229;
        double r79445 = fma(r79443, r79438, r79444);
        double r79446 = 1.0;
        double r79447 = fma(r79438, r79445, r79446);
        double r79448 = r79442 / r79447;
        double r79449 = fma(r79441, r79448, r79438);
        return r79449;
}

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. Applied fma-def0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020021 +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)))))