Average Error: 28.7 → 28.8
Time: 19.1s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), y, t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}\]
\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}
↓
\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) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, 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 r97112 = x;
        double r97113 = y;
        double r97114 = r97112 * r97113;
        double r97115 = z;
        double r97116 = r97114 + r97115;
        double r97117 = r97116 * r97113;
        double r97118 = 27464.7644705;
        double r97119 = r97117 + r97118;
        double r97120 = r97119 * r97113;
        double r97121 = 230661.510616;
        double r97122 = r97120 + r97121;
        double r97123 = r97122 * r97113;
        double r97124 = t;
        double r97125 = r97123 + r97124;
        double r97126 = a;
        double r97127 = r97113 + r97126;
        double r97128 = r97127 * r97113;
        double r97129 = b;
        double r97130 = r97128 + r97129;
        double r97131 = r97130 * r97113;
        double r97132 = c;
        double r97133 = r97131 + r97132;
        double r97134 = r97133 * r97113;
        double r97135 = i;
        double r97136 = r97134 + r97135;
        double r97137 = r97125 / r97136;
        return r97137;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r97138 = x;
        double r97139 = y;
        double r97140 = z;
        double r97141 = fma(r97138, r97139, r97140);
        double r97142 = 27464.7644705;
        double r97143 = fma(r97141, r97139, r97142);
        double r97144 = 230661.510616;
        double r97145 = fma(r97143, r97139, r97144);
        double r97146 = t;
        double r97147 = fma(r97145, r97139, r97146);
        double r97148 = 1.0;
        double r97149 = a;
        double r97150 = r97139 + r97149;
        double r97151 = b;
        double r97152 = fma(r97150, r97139, r97151);
        double r97153 = c;
        double r97154 = fma(r97152, r97139, r97153);
        double r97155 = i;
        double r97156 = fma(r97154, r97139, r97155);
        double r97157 = r97148 / r97156;
        double r97158 = r97147 * r97157;
        return r97158;
}

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

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

  2. Simplified28.7

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

  4. Applied div-inv28.8

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

  5. Final simplification28.8

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

Reproduce

herbie shell --seed 2020047 +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)))