Average Error: 0.0 → 0.0
Time: 4.9s
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 r68449 = 2.30753;
        double r68450 = x;
        double r68451 = 0.27061;
        double r68452 = r68450 * r68451;
        double r68453 = r68449 + r68452;
        double r68454 = 1.0;
        double r68455 = 0.99229;
        double r68456 = 0.04481;
        double r68457 = r68450 * r68456;
        double r68458 = r68455 + r68457;
        double r68459 = r68450 * r68458;
        double r68460 = r68454 + r68459;
        double r68461 = r68453 / r68460;
        double r68462 = r68461 - r68450;
        return r68462;
}
↓
double f(double x) {
        double r68463 = 0.27061;
        double r68464 = x;
        double r68465 = 2.30753;
        double r68466 = fma(r68463, r68464, r68465);
        double r68467 = 1.0;
        double r68468 = 0.04481;
        double r68469 = 0.99229;
        double r68470 = fma(r68468, r68464, r68469);
        double r68471 = 1.0;
        double r68472 = fma(r68464, r68470, r68471);
        double r68473 = r68467 / r68472;
        double r68474 = r68466 * r68473;
        double r68475 = 3.0;
        double r68476 = pow(r68474, r68475);
        double r68477 = cbrt(r68476);
        double r68478 = r68477 - r68464;
        return r68478;
}

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))