Average Error: 0.0 → 0.0
Time: 9.7s
Precision: 64
\[\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\]
↓
\[\frac{\mathsf{fma}\left(\sqrt[3]{2.307529999999999859028321225196123123169 \cdot 2.307529999999999859028321225196123123169}, \sqrt[3]{2.307529999999999859028321225196123123169}, x \cdot 0.2706100000000000171951342053944244980812\right)}{x \cdot \left(x \cdot 0.04481000000000000260680366181986755691469 + 0.992290000000000005364597654988756403327\right) + 1} - x\]
\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x
↓
\frac{\mathsf{fma}\left(\sqrt[3]{2.307529999999999859028321225196123123169 \cdot 2.307529999999999859028321225196123123169}, \sqrt[3]{2.307529999999999859028321225196123123169}, x \cdot 0.2706100000000000171951342053944244980812\right)}{x \cdot \left(x \cdot 0.04481000000000000260680366181986755691469 + 0.992290000000000005364597654988756403327\right) + 1} - x
double f(double x) {
        double r3160160 = 2.30753;
        double r3160161 = x;
        double r3160162 = 0.27061;
        double r3160163 = r3160161 * r3160162;
        double r3160164 = r3160160 + r3160163;
        double r3160165 = 1.0;
        double r3160166 = 0.99229;
        double r3160167 = 0.04481;
        double r3160168 = r3160161 * r3160167;
        double r3160169 = r3160166 + r3160168;
        double r3160170 = r3160161 * r3160169;
        double r3160171 = r3160165 + r3160170;
        double r3160172 = r3160164 / r3160171;
        double r3160173 = r3160172 - r3160161;
        return r3160173;
}
↓
double f(double x) {
        double r3160174 = 2.30753;
        double r3160175 = r3160174 * r3160174;
        double r3160176 = cbrt(r3160175);
        double r3160177 = cbrt(r3160174);
        double r3160178 = x;
        double r3160179 = 0.27061;
        double r3160180 = r3160178 * r3160179;
        double r3160181 = fma(r3160176, r3160177, r3160180);
        double r3160182 = 0.04481;
        double r3160183 = r3160178 * r3160182;
        double r3160184 = 0.99229;
        double r3160185 = r3160183 + r3160184;
        double r3160186 = r3160178 * r3160185;
        double r3160187 = 1.0;
        double r3160188 = r3160186 + r3160187;
        double r3160189 = r3160181 / r3160188;
        double r3160190 = r3160189 - r3160178;
        return r3160190;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.5

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{2.307529999999999859028321225196123123169} \cdot \sqrt[3]{2.307529999999999859028321225196123123169}\right) \cdot \sqrt[3]{2.307529999999999859028321225196123123169}} + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\]

  4. Applied fma-def0.5

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

  6. Applied cbrt-unprod0.0

    \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt[3]{2.307529999999999859028321225196123123169 \cdot 2.307529999999999859028321225196123123169}}, \sqrt[3]{2.307529999999999859028321225196123123169}, x \cdot 0.2706100000000000171951342053944244980812\right)}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\]

  7. Final simplification0.0

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{2.307529999999999859028321225196123123169 \cdot 2.307529999999999859028321225196123123169}, \sqrt[3]{2.307529999999999859028321225196123123169}, x \cdot 0.2706100000000000171951342053944244980812\right)}{x \cdot \left(x \cdot 0.04481000000000000260680366181986755691469 + 0.992290000000000005364597654988756403327\right) + 1} - x\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))