Average Error: 28.7 → 28.8
Time: 25.1s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right), y, t\right)}}\]
\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{\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, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right), y, t\right)}}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r85014 = x;
        double r85015 = y;
        double r85016 = r85014 * r85015;
        double r85017 = z;
        double r85018 = r85016 + r85017;
        double r85019 = r85018 * r85015;
        double r85020 = 27464.7644705;
        double r85021 = r85019 + r85020;
        double r85022 = r85021 * r85015;
        double r85023 = 230661.510616;
        double r85024 = r85022 + r85023;
        double r85025 = r85024 * r85015;
        double r85026 = t;
        double r85027 = r85025 + r85026;
        double r85028 = a;
        double r85029 = r85015 + r85028;
        double r85030 = r85029 * r85015;
        double r85031 = b;
        double r85032 = r85030 + r85031;
        double r85033 = r85032 * r85015;
        double r85034 = c;
        double r85035 = r85033 + r85034;
        double r85036 = r85035 * r85015;
        double r85037 = i;
        double r85038 = r85036 + r85037;
        double r85039 = r85027 / r85038;
        return r85039;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r85040 = 1.0;
        double r85041 = y;
        double r85042 = a;
        double r85043 = r85041 + r85042;
        double r85044 = b;
        double r85045 = fma(r85043, r85041, r85044);
        double r85046 = c;
        double r85047 = fma(r85045, r85041, r85046);
        double r85048 = i;
        double r85049 = fma(r85047, r85041, r85048);
        double r85050 = r85040 / r85049;
        double r85051 = x;
        double r85052 = z;
        double r85053 = fma(r85051, r85041, r85052);
        double r85054 = 27464.7644705;
        double r85055 = fma(r85053, r85041, r85054);
        double r85056 = 230661.510616;
        double r85057 = fma(r85055, r85041, r85056);
        double r85058 = t;
        double r85059 = fma(r85057, r85041, r85058);
        double r85060 = r85040 / r85059;
        double r85061 = r85050 / r85060;
        return r85061;
}

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

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

  3. Applied clear-num28.9

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

  4. Simplified28.9

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

  6. Applied div-inv28.9

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

  7. Applied associate-/r*28.8

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

  8. Final simplification28.8

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

Reproduce

herbie shell --seed 2019212 +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) 27464.764470499998) y) 230661.510616000014) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))