Average Error: 0.0 → 0.0
Time: 2.6s
Precision: 64
\[\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\]
↓
\[\left(2.30753 + x \cdot 0.27061000000000002\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right) \cdot 1} - x\]
\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x
↓
\left(2.30753 + x \cdot 0.27061000000000002\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right) \cdot 1} - x
double f(double x) {
        double r44461 = 2.30753;
        double r44462 = x;
        double r44463 = 0.27061;
        double r44464 = r44462 * r44463;
        double r44465 = r44461 + r44464;
        double r44466 = 1.0;
        double r44467 = 0.99229;
        double r44468 = 0.04481;
        double r44469 = r44462 * r44468;
        double r44470 = r44467 + r44469;
        double r44471 = r44462 * r44470;
        double r44472 = r44466 + r44471;
        double r44473 = r44465 / r44472;
        double r44474 = r44473 - r44462;
        return r44474;
}
↓
double f(double x) {
        double r44475 = 2.30753;
        double r44476 = x;
        double r44477 = 0.27061;
        double r44478 = r44476 * r44477;
        double r44479 = r44475 + r44478;
        double r44480 = 1.0;
        double r44481 = 0.04481;
        double r44482 = 0.99229;
        double r44483 = fma(r44481, r44476, r44482);
        double r44484 = 1.0;
        double r44485 = fma(r44476, r44483, r44484);
        double r44486 = r44485 * r44480;
        double r44487 = r44480 / r44486;
        double r44488 = r44479 * r44487;
        double r44489 = r44488 - r44476;
        return r44489;
}

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 div-inv0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020024 +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))