Average Error: 29.1 → 29.2
Time: 1.3m
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y + a, y, b\right), c\right), y, i\right)} \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)\]
\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}{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y + a, y, b\right), c\right), y, i\right)} \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r2337120 = x;
        double r2337121 = y;
        double r2337122 = r2337120 * r2337121;
        double r2337123 = z;
        double r2337124 = r2337122 + r2337123;
        double r2337125 = r2337124 * r2337121;
        double r2337126 = 27464.7644705;
        double r2337127 = r2337125 + r2337126;
        double r2337128 = r2337127 * r2337121;
        double r2337129 = 230661.510616;
        double r2337130 = r2337128 + r2337129;
        double r2337131 = r2337130 * r2337121;
        double r2337132 = t;
        double r2337133 = r2337131 + r2337132;
        double r2337134 = a;
        double r2337135 = r2337121 + r2337134;
        double r2337136 = r2337135 * r2337121;
        double r2337137 = b;
        double r2337138 = r2337136 + r2337137;
        double r2337139 = r2337138 * r2337121;
        double r2337140 = c;
        double r2337141 = r2337139 + r2337140;
        double r2337142 = r2337141 * r2337121;
        double r2337143 = i;
        double r2337144 = r2337142 + r2337143;
        double r2337145 = r2337133 / r2337144;
        return r2337145;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r2337146 = 1.0;
        double r2337147 = y;
        double r2337148 = a;
        double r2337149 = r2337147 + r2337148;
        double r2337150 = b;
        double r2337151 = fma(r2337149, r2337147, r2337150);
        double r2337152 = c;
        double r2337153 = fma(r2337147, r2337151, r2337152);
        double r2337154 = i;
        double r2337155 = fma(r2337153, r2337147, r2337154);
        double r2337156 = r2337146 / r2337155;
        double r2337157 = x;
        double r2337158 = z;
        double r2337159 = fma(r2337147, r2337157, r2337158);
        double r2337160 = 27464.7644705;
        double r2337161 = fma(r2337147, r2337159, r2337160);
        double r2337162 = 230661.510616;
        double r2337163 = fma(r2337147, r2337161, r2337162);
        double r2337164 = t;
        double r2337165 = fma(r2337147, r2337163, r2337164);
        double r2337166 = r2337156 * r2337165;
        return r2337166;
}

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 29.1

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

  2. Simplified29.1

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

  4. Applied div-inv29.2

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

  5. Final simplification29.2

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

Reproduce

herbie shell --seed 2019165 +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)))