Average Error: 29.2 → 29.2
Time: 20.1s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644704999984242022037506103515625\right), 230661.5106160000141244381666183471679688\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, y + a, b \cdot y\right) + c, y, i\right)}\]
\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(\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644704999984242022037506103515625\right), 230661.5106160000141244381666183471679688\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, y + a, b \cdot y\right) + c, y, i\right)}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r46158 = x;
        double r46159 = y;
        double r46160 = r46158 * r46159;
        double r46161 = z;
        double r46162 = r46160 + r46161;
        double r46163 = r46162 * r46159;
        double r46164 = 27464.7644705;
        double r46165 = r46163 + r46164;
        double r46166 = r46165 * r46159;
        double r46167 = 230661.510616;
        double r46168 = r46166 + r46167;
        double r46169 = r46168 * r46159;
        double r46170 = t;
        double r46171 = r46169 + r46170;
        double r46172 = a;
        double r46173 = r46159 + r46172;
        double r46174 = r46173 * r46159;
        double r46175 = b;
        double r46176 = r46174 + r46175;
        double r46177 = r46176 * r46159;
        double r46178 = c;
        double r46179 = r46177 + r46178;
        double r46180 = r46179 * r46159;
        double r46181 = i;
        double r46182 = r46180 + r46181;
        double r46183 = r46171 / r46182;
        return r46183;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r46184 = y;
        double r46185 = x;
        double r46186 = z;
        double r46187 = fma(r46184, r46185, r46186);
        double r46188 = 27464.7644705;
        double r46189 = fma(r46184, r46187, r46188);
        double r46190 = 230661.510616;
        double r46191 = fma(r46184, r46189, r46190);
        double r46192 = t;
        double r46193 = fma(r46191, r46184, r46192);
        double r46194 = r46184 * r46184;
        double r46195 = a;
        double r46196 = r46184 + r46195;
        double r46197 = b;
        double r46198 = r46197 * r46184;
        double r46199 = fma(r46194, r46196, r46198);
        double r46200 = c;
        double r46201 = r46199 + r46200;
        double r46202 = i;
        double r46203 = fma(r46201, r46184, r46202);
        double r46204 = r46193 / r46203;
        return r46204;
}

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

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

  2. Simplified29.2

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

  4. Applied fma-udef29.2

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

  5. Taylor expanded around inf 29.3

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644704999984242022037506103515625\right), 230661.5106160000141244381666183471679688\right), y, t\right)}{\mathsf{fma}\left(\color{blue}{\left(a \cdot {y}^{2} + \left({y}^{3} + y \cdot b\right)\right)} + c, y, i\right)}\]

  6. Simplified29.2

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

  7. Final simplification29.2

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

Reproduce

herbie shell --seed 2019174 +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)))