Average Error: 0.1 → 0.1
Time: 2.3s
Precision: 64
\[1 - x \cdot \left(0.253 + x \cdot 0.12\right)\]
↓
\[\left(\left({\left(\sqrt[3]{1}\right)}^{3} - x \cdot 0.253\right) - {x}^{2} \cdot 0.12\right) + \mathsf{fma}\left(0.12, x, 0.253\right) \cdot \left(\left(-x\right) + x\right)\]
1 - x \cdot \left(0.253 + x \cdot 0.12\right)
↓
\left(\left({\left(\sqrt[3]{1}\right)}^{3} - x \cdot 0.253\right) - {x}^{2} \cdot 0.12\right) + \mathsf{fma}\left(0.12, x, 0.253\right) \cdot \left(\left(-x\right) + x\right)
double f(double x) {
        double r76345 = 1.0;
        double r76346 = x;
        double r76347 = 0.253;
        double r76348 = 0.12;
        double r76349 = r76346 * r76348;
        double r76350 = r76347 + r76349;
        double r76351 = r76346 * r76350;
        double r76352 = r76345 - r76351;
        return r76352;
}
↓
double f(double x) {
        double r76353 = 1.0;
        double r76354 = cbrt(r76353);
        double r76355 = 3.0;
        double r76356 = pow(r76354, r76355);
        double r76357 = x;
        double r76358 = 0.253;
        double r76359 = r76357 * r76358;
        double r76360 = r76356 - r76359;
        double r76361 = 2.0;
        double r76362 = pow(r76357, r76361);
        double r76363 = 0.12;
        double r76364 = r76362 * r76363;
        double r76365 = r76360 - r76364;
        double r76366 = fma(r76363, r76357, r76358);
        double r76367 = -r76357;
        double r76368 = r76367 + r76357;
        double r76369 = r76366 * r76368;
        double r76370 = r76365 + r76369;
        return r76370;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.1

    \[1 - x \cdot \left(0.253 + x \cdot 0.12\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.1

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

  4. Applied prod-diff0.1

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

  5. Simplified0.1

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

  6. Simplified0.1

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

  8. Applied associate-*r*0.1

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

  9. Simplified0.1

    \[\leadsto \left(\left({\left(\sqrt[3]{1}\right)}^{3} - x \cdot 0.253\right) - \color{blue}{{x}^{2}} \cdot 0.12\right) + \mathsf{fma}\left(0.12, x, 0.253\right) \cdot \left(\left(-x\right) + x\right)\]

  10. Final simplification0.1

    \[\leadsto \left(\left({\left(\sqrt[3]{1}\right)}^{3} - x \cdot 0.253\right) - {x}^{2} \cdot 0.12\right) + \mathsf{fma}\left(0.12, x, 0.253\right) \cdot \left(\left(-x\right) + x\right)\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1 (* x (+ 0.253 (* x 0.12)))))