Average Error: 29.2 → 29.3
Time: 7.5s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), y, t\right)}}\]
\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}
↓
\frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}}{\frac{1}{\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)}}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r87506 = x;
        double r87507 = y;
        double r87508 = r87506 * r87507;
        double r87509 = z;
        double r87510 = r87508 + r87509;
        double r87511 = r87510 * r87507;
        double r87512 = 27464.7644705;
        double r87513 = r87511 + r87512;
        double r87514 = r87513 * r87507;
        double r87515 = 230661.510616;
        double r87516 = r87514 + r87515;
        double r87517 = r87516 * r87507;
        double r87518 = t;
        double r87519 = r87517 + r87518;
        double r87520 = a;
        double r87521 = r87507 + r87520;
        double r87522 = r87521 * r87507;
        double r87523 = b;
        double r87524 = r87522 + r87523;
        double r87525 = r87524 * r87507;
        double r87526 = c;
        double r87527 = r87525 + r87526;
        double r87528 = r87527 * r87507;
        double r87529 = i;
        double r87530 = r87528 + r87529;
        double r87531 = r87519 / r87530;
        return r87531;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r87532 = 1.0;
        double r87533 = y;
        double r87534 = a;
        double r87535 = r87533 + r87534;
        double r87536 = b;
        double r87537 = fma(r87535, r87533, r87536);
        double r87538 = c;
        double r87539 = fma(r87537, r87533, r87538);
        double r87540 = i;
        double r87541 = fma(r87539, r87533, r87540);
        double r87542 = r87532 / r87541;
        double r87543 = x;
        double r87544 = z;
        double r87545 = fma(r87543, r87533, r87544);
        double r87546 = 27464.7644705;
        double r87547 = fma(r87545, r87533, r87546);
        double r87548 = 230661.510616;
        double r87549 = fma(r87547, r87533, r87548);
        double r87550 = t;
        double r87551 = fma(r87549, r87533, r87550);
        double r87552 = r87532 / r87551;
        double r87553 = r87542 / r87552;
        return r87553;
}

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 29.2

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

  3. Applied *-un-lft-identity29.2

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

  4. Applied *-un-lft-identity29.2

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

  5. Applied times-frac29.2

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

  6. Simplified29.2

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

  7. Simplified29.2

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

  9. Applied clear-num29.4

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

  10. Simplified29.4

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

  12. Applied div-inv29.4

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

  13. Applied associate-/r*29.3

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

  14. Final simplification29.3

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

Reproduce

herbie shell --seed 2020064 +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)))