Average Error: 29.2 → 29.3
Time: 19.3s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\left(\left(\left(y \cdot \mathsf{fma}\left(x, y, z\right) + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]
\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(y \cdot \mathsf{fma}\left(x, y, z\right) + 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 r297 = x;
        double r298 = y;
        double r299 = r297 * r298;
        double r300 = z;
        double r301 = r299 + r300;
        double r302 = r301 * r298;
        double r303 = 27464.7644705;
        double r304 = r302 + r303;
        double r305 = r304 * r298;
        double r306 = 230661.510616;
        double r307 = r305 + r306;
        double r308 = r307 * r298;
        double r309 = t;
        double r310 = r308 + r309;
        double r311 = a;
        double r312 = r298 + r311;
        double r313 = r312 * r298;
        double r314 = b;
        double r315 = r313 + r314;
        double r316 = r315 * r298;
        double r317 = c;
        double r318 = r316 + r317;
        double r319 = r318 * r298;
        double r320 = i;
        double r321 = r319 + r320;
        double r322 = r310 / r321;
        return r322;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r323 = y;
        double r324 = x;
        double r325 = z;
        double r326 = fma(r324, r323, r325);
        double r327 = r323 * r326;
        double r328 = 27464.7644705;
        double r329 = r327 + r328;
        double r330 = r329 * r323;
        double r331 = 230661.510616;
        double r332 = r330 + r331;
        double r333 = r332 * r323;
        double r334 = t;
        double r335 = r333 + r334;
        double r336 = 1.0;
        double r337 = a;
        double r338 = r323 + r337;
        double r339 = b;
        double r340 = fma(r338, r323, r339);
        double r341 = c;
        double r342 = fma(r340, r323, r341);
        double r343 = i;
        double r344 = fma(r342, r323, r343);
        double r345 = r344 * r336;
        double r346 = r336 / r345;
        double r347 = r335 * r346;
        return r347;
}

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 29.2

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

  3. Applied div-inv29.3

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

  4. Simplified29.3

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

  5. Taylor expanded around 0 29.3

    \[\leadsto \left(\left(\left(\color{blue}{\left(x \cdot {y}^{2} + z \cdot y\right)} + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]

  6. Simplified29.3

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

  7. Final simplification29.3

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

Reproduce

herbie shell --seed 2020025 +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)))