Average Error: 28.7 → 28.8
Time: 8.3s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]
\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}
↓
\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot 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) \cdot 1}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r64258 = x;
        double r64259 = y;
        double r64260 = r64258 * r64259;
        double r64261 = z;
        double r64262 = r64260 + r64261;
        double r64263 = r64262 * r64259;
        double r64264 = 27464.7644705;
        double r64265 = r64263 + r64264;
        double r64266 = r64265 * r64259;
        double r64267 = 230661.510616;
        double r64268 = r64266 + r64267;
        double r64269 = r64268 * r64259;
        double r64270 = t;
        double r64271 = r64269 + r64270;
        double r64272 = a;
        double r64273 = r64259 + r64272;
        double r64274 = r64273 * r64259;
        double r64275 = b;
        double r64276 = r64274 + r64275;
        double r64277 = r64276 * r64259;
        double r64278 = c;
        double r64279 = r64277 + r64278;
        double r64280 = r64279 * r64259;
        double r64281 = i;
        double r64282 = r64280 + r64281;
        double r64283 = r64271 / r64282;
        return r64283;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r64284 = x;
        double r64285 = y;
        double r64286 = r64284 * r64285;
        double r64287 = z;
        double r64288 = r64286 + r64287;
        double r64289 = r64288 * r64285;
        double r64290 = 27464.7644705;
        double r64291 = r64289 + r64290;
        double r64292 = r64291 * r64285;
        double r64293 = 230661.510616;
        double r64294 = r64292 + r64293;
        double r64295 = r64294 * r64285;
        double r64296 = t;
        double r64297 = r64295 + r64296;
        double r64298 = 1.0;
        double r64299 = a;
        double r64300 = r64285 + r64299;
        double r64301 = b;
        double r64302 = fma(r64300, r64285, r64301);
        double r64303 = c;
        double r64304 = fma(r64302, r64285, r64303);
        double r64305 = i;
        double r64306 = fma(r64304, r64285, r64305);
        double r64307 = r64306 * r64298;
        double r64308 = r64298 / r64307;
        double r64309 = r64297 * r64308;
        return r64309;
}

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. Using strategy rm

  3. Applied div-inv28.8

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

  4. Simplified28.8

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

  5. Final simplification28.8

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

Reproduce

herbie shell --seed 2020065 +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)))