Average Error: 29.0 → 29.1
Time: 8.5s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]
\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}
↓
\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot 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) \cdot 1}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r75538 = x;
        double r75539 = y;
        double r75540 = r75538 * r75539;
        double r75541 = z;
        double r75542 = r75540 + r75541;
        double r75543 = r75542 * r75539;
        double r75544 = 27464.7644705;
        double r75545 = r75543 + r75544;
        double r75546 = r75545 * r75539;
        double r75547 = 230661.510616;
        double r75548 = r75546 + r75547;
        double r75549 = r75548 * r75539;
        double r75550 = t;
        double r75551 = r75549 + r75550;
        double r75552 = a;
        double r75553 = r75539 + r75552;
        double r75554 = r75553 * r75539;
        double r75555 = b;
        double r75556 = r75554 + r75555;
        double r75557 = r75556 * r75539;
        double r75558 = c;
        double r75559 = r75557 + r75558;
        double r75560 = r75559 * r75539;
        double r75561 = i;
        double r75562 = r75560 + r75561;
        double r75563 = r75551 / r75562;
        return r75563;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r75564 = x;
        double r75565 = y;
        double r75566 = r75564 * r75565;
        double r75567 = z;
        double r75568 = r75566 + r75567;
        double r75569 = r75568 * r75565;
        double r75570 = 27464.7644705;
        double r75571 = r75569 + r75570;
        double r75572 = r75571 * r75565;
        double r75573 = 230661.510616;
        double r75574 = r75572 + r75573;
        double r75575 = r75574 * r75565;
        double r75576 = t;
        double r75577 = r75575 + r75576;
        double r75578 = 1.0;
        double r75579 = a;
        double r75580 = r75565 + r75579;
        double r75581 = b;
        double r75582 = fma(r75580, r75565, r75581);
        double r75583 = c;
        double r75584 = fma(r75582, r75565, r75583);
        double r75585 = i;
        double r75586 = fma(r75584, r75565, r75585);
        double r75587 = r75586 * r75578;
        double r75588 = r75578 / r75587;
        double r75589 = r75577 * r75588;
        return r75589;
}

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 29.0

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

  3. Applied div-inv29.1

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

  4. Simplified29.1

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

  5. Final simplification29.1

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

Reproduce

herbie shell --seed 2020100 +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)))