Average Error: 28.8 → 29.0
Time: 1.2m
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{1}{\frac{(\left((y \cdot \left((\left(y + a\right) \cdot y + b)_*\right) + c)_*\right) \cdot y + i)_*}{(y \cdot \left((y \cdot \left((y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*\right) + 230661.510616)_*\right) + t)_*}}\]
\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}{\frac{(\left((y \cdot \left((\left(y + a\right) \cdot y + b)_*\right) + c)_*\right) \cdot y + i)_*}{(y \cdot \left((y \cdot \left((y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*\right) + 230661.510616)_*\right) + t)_*}}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r8676167 = x;
        double r8676168 = y;
        double r8676169 = r8676167 * r8676168;
        double r8676170 = z;
        double r8676171 = r8676169 + r8676170;
        double r8676172 = r8676171 * r8676168;
        double r8676173 = 27464.7644705;
        double r8676174 = r8676172 + r8676173;
        double r8676175 = r8676174 * r8676168;
        double r8676176 = 230661.510616;
        double r8676177 = r8676175 + r8676176;
        double r8676178 = r8676177 * r8676168;
        double r8676179 = t;
        double r8676180 = r8676178 + r8676179;
        double r8676181 = a;
        double r8676182 = r8676168 + r8676181;
        double r8676183 = r8676182 * r8676168;
        double r8676184 = b;
        double r8676185 = r8676183 + r8676184;
        double r8676186 = r8676185 * r8676168;
        double r8676187 = c;
        double r8676188 = r8676186 + r8676187;
        double r8676189 = r8676188 * r8676168;
        double r8676190 = i;
        double r8676191 = r8676189 + r8676190;
        double r8676192 = r8676180 / r8676191;
        return r8676192;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r8676193 = 1.0;
        double r8676194 = y;
        double r8676195 = a;
        double r8676196 = r8676194 + r8676195;
        double r8676197 = b;
        double r8676198 = fma(r8676196, r8676194, r8676197);
        double r8676199 = c;
        double r8676200 = fma(r8676194, r8676198, r8676199);
        double r8676201 = i;
        double r8676202 = fma(r8676200, r8676194, r8676201);
        double r8676203 = x;
        double r8676204 = z;
        double r8676205 = fma(r8676194, r8676203, r8676204);
        double r8676206 = 27464.7644705;
        double r8676207 = fma(r8676194, r8676205, r8676206);
        double r8676208 = 230661.510616;
        double r8676209 = fma(r8676194, r8676207, r8676208);
        double r8676210 = t;
        double r8676211 = fma(r8676194, r8676209, r8676210);
        double r8676212 = r8676202 / r8676211;
        double r8676213 = r8676193 / r8676212;
        return r8676213;
}

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 28.8

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

  2. Simplified28.8

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

  4. Applied *-un-lft-identity28.8

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

  5. Applied associate-/l*29.0

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

  6. Final simplification29.0

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

Reproduce

herbie shell --seed 2019104 +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)))