Average Error: 29.3 → 29.3
Time: 14.1s
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 r74257 = x;
        double r74258 = y;
        double r74259 = r74257 * r74258;
        double r74260 = z;
        double r74261 = r74259 + r74260;
        double r74262 = r74261 * r74258;
        double r74263 = 27464.7644705;
        double r74264 = r74262 + r74263;
        double r74265 = r74264 * r74258;
        double r74266 = 230661.510616;
        double r74267 = r74265 + r74266;
        double r74268 = r74267 * r74258;
        double r74269 = t;
        double r74270 = r74268 + r74269;
        double r74271 = a;
        double r74272 = r74258 + r74271;
        double r74273 = r74272 * r74258;
        double r74274 = b;
        double r74275 = r74273 + r74274;
        double r74276 = r74275 * r74258;
        double r74277 = c;
        double r74278 = r74276 + r74277;
        double r74279 = r74278 * r74258;
        double r74280 = i;
        double r74281 = r74279 + r74280;
        double r74282 = r74270 / r74281;
        return r74282;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r74283 = x;
        double r74284 = y;
        double r74285 = z;
        double r74286 = fma(r74283, r74284, r74285);
        double r74287 = 27464.7644705;
        double r74288 = fma(r74286, r74284, r74287);
        double r74289 = 230661.510616;
        double r74290 = fma(r74288, r74284, r74289);
        double r74291 = t;
        double r74292 = fma(r74290, r74284, r74291);
        double r74293 = 1.0;
        double r74294 = a;
        double r74295 = r74284 + r74294;
        double r74296 = b;
        double r74297 = fma(r74295, r74284, r74296);
        double r74298 = c;
        double r74299 = fma(r74297, r74284, r74298);
        double r74300 = i;
        double r74301 = fma(r74299, r74284, r74300);
        double r74302 = r74293 / r74301;
        double r74303 = r74292 * r74302;
        return r74303;
}

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 29.3

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

  2. Simplified29.3

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

  4. Applied div-inv29.3

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

  5. Final simplification29.3

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

Reproduce

herbie shell --seed 2020045 +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)))