Average Error: 0.1 → 0.1
Time: 19.1s
Precision: 64
\[1 - x \cdot \left(0.253 + x \cdot 0.12\right)\]
↓
\[\mathsf{fma}\left(-x, \mathsf{fma}\left(0.12, x, 0.253\right), {\left(\sqrt[3]{1}\right)}^{3}\right) + \mathsf{fma}\left(0.12, x, 0.253\right) \cdot \left(\left(-x\right) + x\right)\]
1 - x \cdot \left(0.253 + x \cdot 0.12\right)
↓
\mathsf{fma}\left(-x, \mathsf{fma}\left(0.12, x, 0.253\right), {\left(\sqrt[3]{1}\right)}^{3}\right) + \mathsf{fma}\left(0.12, x, 0.253\right) \cdot \left(\left(-x\right) + x\right)
double f(double x) {
        double r71511 = 1.0;
        double r71512 = x;
        double r71513 = 0.253;
        double r71514 = 0.12;
        double r71515 = r71512 * r71514;
        double r71516 = r71513 + r71515;
        double r71517 = r71512 * r71516;
        double r71518 = r71511 - r71517;
        return r71518;
}
↓
double f(double x) {
        double r71519 = x;
        double r71520 = -r71519;
        double r71521 = 0.12;
        double r71522 = 0.253;
        double r71523 = fma(r71521, r71519, r71522);
        double r71524 = 1.0;
        double r71525 = cbrt(r71524);
        double r71526 = 3.0;
        double r71527 = pow(r71525, r71526);
        double r71528 = fma(r71520, r71523, r71527);
        double r71529 = r71520 + r71519;
        double r71530 = r71523 * r71529;
        double r71531 = r71528 + r71530;
        return r71531;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.1

    \[1 - x \cdot \left(0.253 + x \cdot 0.12\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{1 - \mathsf{fma}\left(0.12, x, 0.253\right) \cdot x}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.1

    \[\leadsto \color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}} - \mathsf{fma}\left(0.12, x, 0.253\right) \cdot x\]

  5. Applied prod-diff0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -x \cdot \mathsf{fma}\left(0.12, x, 0.253\right)\right) + \mathsf{fma}\left(-x, \mathsf{fma}\left(0.12, x, 0.253\right), x \cdot \mathsf{fma}\left(0.12, x, 0.253\right)\right)}\]

  6. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \mathsf{fma}\left(0.12, x, 0.253\right), {\left(\sqrt[3]{1}\right)}^{3}\right)} + \mathsf{fma}\left(-x, \mathsf{fma}\left(0.12, x, 0.253\right), x \cdot \mathsf{fma}\left(0.12, x, 0.253\right)\right)\]

  7. Simplified0.1

    \[\leadsto \mathsf{fma}\left(-x, \mathsf{fma}\left(0.12, x, 0.253\right), {\left(\sqrt[3]{1}\right)}^{3}\right) + \color{blue}{\mathsf{fma}\left(0.12, x, 0.253\right) \cdot \left(\left(-x\right) + x\right)}\]

  8. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(-x, \mathsf{fma}\left(0.12, x, 0.253\right), {\left(\sqrt[3]{1}\right)}^{3}\right) + \mathsf{fma}\left(0.12, x, 0.253\right) \cdot \left(\left(-x\right) + x\right)\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))