Average Error: 28.7 → 28.8
Time: 1.1m
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{1}{\mathsf{fma}\left(\left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(a + y\right), y, b\right)\right), c\right)\right), y, i\right)} \cdot \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, x, z\right)\right), 27464.7644705\right)\right), 230661.510616\right) \cdot y + t\right)\]
\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}
↓
\frac{1}{\mathsf{fma}\left(\left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(a + y\right), y, b\right)\right), c\right)\right), y, i\right)} \cdot \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, x, z\right)\right), 27464.7644705\right)\right), 230661.510616\right) \cdot y + t\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r6355499 = x;
        double r6355500 = y;
        double r6355501 = r6355499 * r6355500;
        double r6355502 = z;
        double r6355503 = r6355501 + r6355502;
        double r6355504 = r6355503 * r6355500;
        double r6355505 = 27464.7644705;
        double r6355506 = r6355504 + r6355505;
        double r6355507 = r6355506 * r6355500;
        double r6355508 = 230661.510616;
        double r6355509 = r6355507 + r6355508;
        double r6355510 = r6355509 * r6355500;
        double r6355511 = t;
        double r6355512 = r6355510 + r6355511;
        double r6355513 = a;
        double r6355514 = r6355500 + r6355513;
        double r6355515 = r6355514 * r6355500;
        double r6355516 = b;
        double r6355517 = r6355515 + r6355516;
        double r6355518 = r6355517 * r6355500;
        double r6355519 = c;
        double r6355520 = r6355518 + r6355519;
        double r6355521 = r6355520 * r6355500;
        double r6355522 = i;
        double r6355523 = r6355521 + r6355522;
        double r6355524 = r6355512 / r6355523;
        return r6355524;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r6355525 = 1.0;
        double r6355526 = y;
        double r6355527 = a;
        double r6355528 = r6355527 + r6355526;
        double r6355529 = b;
        double r6355530 = fma(r6355528, r6355526, r6355529);
        double r6355531 = c;
        double r6355532 = fma(r6355526, r6355530, r6355531);
        double r6355533 = i;
        double r6355534 = fma(r6355532, r6355526, r6355533);
        double r6355535 = r6355525 / r6355534;
        double r6355536 = x;
        double r6355537 = z;
        double r6355538 = fma(r6355526, r6355536, r6355537);
        double r6355539 = 27464.7644705;
        double r6355540 = fma(r6355526, r6355538, r6355539);
        double r6355541 = 230661.510616;
        double r6355542 = fma(r6355526, r6355540, r6355541);
        double r6355543 = r6355542 * r6355526;
        double r6355544 = t;
        double r6355545 = r6355543 + r6355544;
        double r6355546 = r6355535 * r6355545;
        return r6355546;
}

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 28.7

    \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

  2. Simplified28.7

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, x, z\right)\right), 27464.7644705\right)\right), 230661.510616\right)\right), t\right)}{\mathsf{fma}\left(\left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(y + a\right), y, b\right)\right), c\right)\right), y, i\right)}}\]
  3. Using strategy rm

  4. Applied fma-udef28.7

    \[\leadsto \frac{\color{blue}{y \cdot \mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, x, z\right)\right), 27464.7644705\right)\right), 230661.510616\right) + t}}{\mathsf{fma}\left(\left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(y + a\right), y, b\right)\right), c\right)\right), y, i\right)}\]
  5. Using strategy rm

  6. Applied div-inv28.8

    \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, x, z\right)\right), 27464.7644705\right)\right), 230661.510616\right) + t\right) \cdot \frac{1}{\mathsf{fma}\left(\left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(y + a\right), y, b\right)\right), c\right)\right), y, i\right)}}\]

  7. Final simplification28.8

    \[\leadsto \frac{1}{\mathsf{fma}\left(\left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(\left(a + y\right), y, b\right)\right), c\right)\right), y, i\right)} \cdot \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(y, x, z\right)\right), 27464.7644705\right)\right), 230661.510616\right) \cdot y + t\right)\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))