Average Error: 0.0 → 0.0
Time: 9.5s
Precision: 64
\[\frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\]
↓
\[\frac{\mathsf{fma}\left(\sqrt[3]{2.307529999999999859028321225196123123169 \cdot 2.307529999999999859028321225196123123169}, \sqrt[3]{2.307529999999999859028321225196123123169}, x \cdot 0.2706100000000000171951342053944244980812\right)}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x\]
\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)}{1 + x \cdot \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right)} - x
double f(double x) {
        double r4559228 = 2.30753;
        double r4559229 = x;
        double r4559230 = 0.27061;
        double r4559231 = r4559229 * r4559230;
        double r4559232 = r4559228 + r4559231;
        double r4559233 = 1.0;
        double r4559234 = 0.99229;
        double r4559235 = 0.04481;
        double r4559236 = r4559229 * r4559235;
        double r4559237 = r4559234 + r4559236;
        double r4559238 = r4559229 * r4559237;
        double r4559239 = r4559233 + r4559238;
        double r4559240 = r4559232 / r4559239;
        double r4559241 = r4559240 - r4559229;
        return r4559241;
}
↓
double f(double x) {
        double r4559242 = 2.30753;
        double r4559243 = r4559242 * r4559242;
        double r4559244 = cbrt(r4559243);
        double r4559245 = cbrt(r4559242);
        double r4559246 = x;
        double r4559247 = 0.27061;
        double r4559248 = r4559246 * r4559247;
        double r4559249 = fma(r4559244, r4559245, r4559248);
        double r4559250 = 1.0;
        double r4559251 = 0.99229;
        double r4559252 = 0.04481;
        double r4559253 = r4559246 * r4559252;
        double r4559254 = r4559251 + r4559253;
        double r4559255 = r4559246 * r4559254;
        double r4559256 = r4559250 + r4559255;
        double r4559257 = r4559249 / r4559256;
        double r4559258 = r4559257 - r4559246;
        return r4559258;
}

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

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

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))