Average Error: 29.4 → 29.5
Time: 18.3s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), y, t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}\]
\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}
↓
\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), y, t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r75527 = x;
        double r75528 = y;
        double r75529 = r75527 * r75528;
        double r75530 = z;
        double r75531 = r75529 + r75530;
        double r75532 = r75531 * r75528;
        double r75533 = 27464.7644705;
        double r75534 = r75532 + r75533;
        double r75535 = r75534 * r75528;
        double r75536 = 230661.510616;
        double r75537 = r75535 + r75536;
        double r75538 = r75537 * r75528;
        double r75539 = t;
        double r75540 = r75538 + r75539;
        double r75541 = a;
        double r75542 = r75528 + r75541;
        double r75543 = r75542 * r75528;
        double r75544 = b;
        double r75545 = r75543 + r75544;
        double r75546 = r75545 * r75528;
        double r75547 = c;
        double r75548 = r75546 + r75547;
        double r75549 = r75548 * r75528;
        double r75550 = i;
        double r75551 = r75549 + r75550;
        double r75552 = r75540 / r75551;
        return r75552;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r75553 = x;
        double r75554 = y;
        double r75555 = z;
        double r75556 = fma(r75553, r75554, r75555);
        double r75557 = 27464.7644705;
        double r75558 = fma(r75556, r75554, r75557);
        double r75559 = 230661.510616;
        double r75560 = fma(r75558, r75554, r75559);
        double r75561 = t;
        double r75562 = fma(r75560, r75554, r75561);
        double r75563 = 1.0;
        double r75564 = a;
        double r75565 = r75554 + r75564;
        double r75566 = b;
        double r75567 = fma(r75565, r75554, r75566);
        double r75568 = c;
        double r75569 = fma(r75567, r75554, r75568);
        double r75570 = i;
        double r75571 = fma(r75569, r75554, r75570);
        double r75572 = r75563 / r75571;
        double r75573 = r75562 * r75572;
        return r75573;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Derivation

  1. Initial program 29.4

    \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

  2. Simplified29.4

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}}\]
  3. Using strategy rm

  4. Applied div-inv29.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), y, t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}}\]

  5. Final simplification29.5

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), y, t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))