Average Error: 0.0 → 0.0
Time: 23.8s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}
double f(double x) {
        double r80943 = x;
        double r80944 = 2.30753;
        double r80945 = 0.27061;
        double r80946 = r80943 * r80945;
        double r80947 = r80944 + r80946;
        double r80948 = 1.0;
        double r80949 = 0.99229;
        double r80950 = 0.04481;
        double r80951 = r80943 * r80950;
        double r80952 = r80949 + r80951;
        double r80953 = r80952 * r80943;
        double r80954 = r80948 + r80953;
        double r80955 = r80947 / r80954;
        double r80956 = r80943 - r80955;
        return r80956;
}
↓
double f(double x) {
        double r80957 = x;
        double r80958 = 0.27061;
        double r80959 = 2.30753;
        double r80960 = fma(r80957, r80958, r80959);
        double r80961 = 1.0;
        double r80962 = 0.04481;
        double r80963 = 0.99229;
        double r80964 = fma(r80962, r80957, r80963);
        double r80965 = 1.0;
        double r80966 = fma(r80964, r80957, r80965);
        double r80967 = r80961 / r80966;
        double r80968 = r80960 * r80967;
        double r80969 = r80957 - r80968;
        return r80969;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{x - \frac{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}}\]
  3. Using strategy rm

  4. Applied div-inv0.0

    \[\leadsto x - \color{blue}{\mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}}\]

  5. Final simplification0.0

    \[\leadsto x - \mathsf{fma}\left(x, 0.2706100000000000171951342053944244980812, 2.307529999999999859028321225196123123169\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), x, 1\right)}\]

Reproduce

herbie shell --seed 2019323 +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)))))