Average Error: 29.0 → 29.0
Time: 28.9s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644704999984242022037506103515625\right), 230661.5106160000141244381666183471679688\right), t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\]
\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}
↓
\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644704999984242022037506103515625\right), 230661.5106160000141244381666183471679688\right), t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, 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 r72737 = x;
        double r72738 = y;
        double r72739 = r72737 * r72738;
        double r72740 = z;
        double r72741 = r72739 + r72740;
        double r72742 = r72741 * r72738;
        double r72743 = 27464.7644705;
        double r72744 = r72742 + r72743;
        double r72745 = r72744 * r72738;
        double r72746 = 230661.510616;
        double r72747 = r72745 + r72746;
        double r72748 = r72747 * r72738;
        double r72749 = t;
        double r72750 = r72748 + r72749;
        double r72751 = a;
        double r72752 = r72738 + r72751;
        double r72753 = r72752 * r72738;
        double r72754 = b;
        double r72755 = r72753 + r72754;
        double r72756 = r72755 * r72738;
        double r72757 = c;
        double r72758 = r72756 + r72757;
        double r72759 = r72758 * r72738;
        double r72760 = i;
        double r72761 = r72759 + r72760;
        double r72762 = r72750 / r72761;
        return r72762;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r72763 = y;
        double r72764 = x;
        double r72765 = z;
        double r72766 = fma(r72763, r72764, r72765);
        double r72767 = 27464.7644705;
        double r72768 = fma(r72763, r72766, r72767);
        double r72769 = 230661.510616;
        double r72770 = fma(r72763, r72768, r72769);
        double r72771 = t;
        double r72772 = fma(r72763, r72770, r72771);
        double r72773 = a;
        double r72774 = r72773 + r72763;
        double r72775 = b;
        double r72776 = fma(r72774, r72763, r72775);
        double r72777 = c;
        double r72778 = fma(r72776, r72763, r72777);
        double r72779 = i;
        double r72780 = fma(r72778, r72763, r72779);
        double r72781 = r72772 / r72780;
        return r72781;
}

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

    \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt29.1

    \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\color{blue}{\left(\left(\sqrt[3]{\left(y + a\right) \cdot y + b} \cdot \sqrt[3]{\left(y + a\right) \cdot y + b}\right) \cdot \sqrt[3]{\left(y + a\right) \cdot y + b}\right)} \cdot y + c\right) \cdot y + i}\]

  4. Applied associate-*l*29.1

    \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\color{blue}{\left(\sqrt[3]{\left(y + a\right) \cdot y + b} \cdot \sqrt[3]{\left(y + a\right) \cdot y + b}\right) \cdot \left(\sqrt[3]{\left(y + a\right) \cdot y + b} \cdot y\right)} + c\right) \cdot y + i}\]

  5. Simplified29.1

    \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\sqrt[3]{\left(y + a\right) \cdot y + b} \cdot \sqrt[3]{\left(y + a\right) \cdot y + b}\right) \cdot \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(y, a + y, b\right)} \cdot y\right)} + c\right) \cdot y + i}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity29.1

    \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\color{blue}{1 \cdot \left(\left(\left(\sqrt[3]{\left(y + a\right) \cdot y + b} \cdot \sqrt[3]{\left(y + a\right) \cdot y + b}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(y, a + y, b\right)} \cdot y\right) + c\right) \cdot y + i\right)}}\]

  8. Applied *-un-lft-identity29.1

    \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t\right)}}{1 \cdot \left(\left(\left(\sqrt[3]{\left(y + a\right) \cdot y + b} \cdot \sqrt[3]{\left(y + a\right) \cdot y + b}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(y, a + y, b\right)} \cdot y\right) + c\right) \cdot y + i\right)}\]

  9. Applied times-frac29.1

    \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\sqrt[3]{\left(y + a\right) \cdot y + b} \cdot \sqrt[3]{\left(y + a\right) \cdot y + b}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(y, a + y, b\right)} \cdot y\right) + c\right) \cdot y + i}}\]

  10. Simplified29.1

    \[\leadsto \color{blue}{1} \cdot \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\sqrt[3]{\left(y + a\right) \cdot y + b} \cdot \sqrt[3]{\left(y + a\right) \cdot y + b}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(y, a + y, b\right)} \cdot y\right) + c\right) \cdot y + i}\]

  11. Simplified29.0

    \[\leadsto 1 \cdot \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644704999984242022037506103515625\right), 230661.5106160000141244381666183471679688\right), t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}}\]

  12. Final simplification29.0

    \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644704999984242022037506103515625\right), 230661.5106160000141244381666183471679688\right), t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\]

Reproduce

herbie shell --seed 2019196 +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)))