Average Error: 0.0 → 0.0
Time: 4.6s
Precision: 64
\[\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\]
↓
\[\sqrt[3]{{\left(\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)}^{3}} - x\]
\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x
↓
\sqrt[3]{{\left(\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)}^{3}} - x
double f(double x) {
        double r67281 = 2.30753;
        double r67282 = x;
        double r67283 = 0.27061;
        double r67284 = r67282 * r67283;
        double r67285 = r67281 + r67284;
        double r67286 = 1.0;
        double r67287 = 0.99229;
        double r67288 = 0.04481;
        double r67289 = r67282 * r67288;
        double r67290 = r67287 + r67289;
        double r67291 = r67282 * r67290;
        double r67292 = r67286 + r67291;
        double r67293 = r67285 / r67292;
        double r67294 = r67293 - r67282;
        return r67294;
}
↓
double f(double x) {
        double r67295 = 0.27061;
        double r67296 = x;
        double r67297 = 2.30753;
        double r67298 = fma(r67295, r67296, r67297);
        double r67299 = 1.0;
        double r67300 = 0.04481;
        double r67301 = 0.99229;
        double r67302 = fma(r67300, r67296, r67301);
        double r67303 = 1.0;
        double r67304 = fma(r67296, r67302, r67303);
        double r67305 = r67299 / r67304;
        double r67306 = r67298 * r67305;
        double r67307 = 3.0;
        double r67308 = pow(r67306, r67307);
        double r67309 = cbrt(r67308);
        double r67310 = r67309 - r67296;
        return r67310;
}

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-cbrt-cube0.0

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

  4. Applied add-cbrt-cube21.7

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

  5. Applied cbrt-undiv21.7

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

  6. Simplified0.0

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)}^{3}}} - x\]
  7. Using strategy rm

  8. Applied div-inv0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +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))