Average Error: 0.0 → 0.0
Time: 3.3s
Precision: 64
\[\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\]
↓
\[\frac{1}{\sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} \cdot \sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}} \cdot \frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}} - x\]
\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x
↓
\frac{1}{\sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} \cdot \sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}} \cdot \frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}} - x
double f(double x) {
        double r64227 = 2.30753;
        double r64228 = x;
        double r64229 = 0.27061;
        double r64230 = r64228 * r64229;
        double r64231 = r64227 + r64230;
        double r64232 = 1.0;
        double r64233 = 0.99229;
        double r64234 = 0.04481;
        double r64235 = r64228 * r64234;
        double r64236 = r64233 + r64235;
        double r64237 = r64228 * r64236;
        double r64238 = r64232 + r64237;
        double r64239 = r64231 / r64238;
        double r64240 = r64239 - r64228;
        return r64240;
}
↓
double f(double x) {
        double r64241 = 1.0;
        double r64242 = 1.0;
        double r64243 = x;
        double r64244 = 0.99229;
        double r64245 = 0.04481;
        double r64246 = r64243 * r64245;
        double r64247 = r64244 + r64246;
        double r64248 = r64243 * r64247;
        double r64249 = r64242 + r64248;
        double r64250 = cbrt(r64249);
        double r64251 = r64250 * r64250;
        double r64252 = r64241 / r64251;
        double r64253 = 0.27061;
        double r64254 = 2.30753;
        double r64255 = fma(r64253, r64243, r64254);
        double r64256 = r64255 / r64250;
        double r64257 = r64252 * r64256;
        double r64258 = r64257 - r64243;
        return r64258;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.0

    \[\leadsto \frac{2.30753 + x \cdot 0.27061000000000002}{\color{blue}{\left(\sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} \cdot \sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}\right) \cdot \sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}}} - x\]

  4. Applied *-un-lft-identity0.0

    \[\leadsto \frac{\color{blue}{1 \cdot \left(2.30753 + x \cdot 0.27061000000000002\right)}}{\left(\sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} \cdot \sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}\right) \cdot \sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}} - x\]

  5. Applied times-frac0.0

    \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} \cdot \sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}} \cdot \frac{2.30753 + x \cdot 0.27061000000000002}{\sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}}} - x\]

  6. Simplified0.0

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

  7. Final simplification0.0

    \[\leadsto \frac{1}{\sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} \cdot \sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}} \cdot \frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\sqrt[3]{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}} - x\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  :precision binary64
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* x (+ 0.99229 (* x 0.04481))))) x))