Average Error: 29.2 → 29.4
Time: 7.3s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\left(\left(\left(\left(\left(\mathsf{fma}\left(x, y, z\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y} + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]
\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(\left(\mathsf{fma}\left(x, y, z\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{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 r44240 = x;
        double r44241 = y;
        double r44242 = r44240 * r44241;
        double r44243 = z;
        double r44244 = r44242 + r44243;
        double r44245 = r44244 * r44241;
        double r44246 = 27464.7644705;
        double r44247 = r44245 + r44246;
        double r44248 = r44247 * r44241;
        double r44249 = 230661.510616;
        double r44250 = r44248 + r44249;
        double r44251 = r44250 * r44241;
        double r44252 = t;
        double r44253 = r44251 + r44252;
        double r44254 = a;
        double r44255 = r44241 + r44254;
        double r44256 = r44255 * r44241;
        double r44257 = b;
        double r44258 = r44256 + r44257;
        double r44259 = r44258 * r44241;
        double r44260 = c;
        double r44261 = r44259 + r44260;
        double r44262 = r44261 * r44241;
        double r44263 = i;
        double r44264 = r44262 + r44263;
        double r44265 = r44253 / r44264;
        return r44265;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r44266 = x;
        double r44267 = y;
        double r44268 = z;
        double r44269 = fma(r44266, r44267, r44268);
        double r44270 = cbrt(r44267);
        double r44271 = r44269 * r44270;
        double r44272 = r44271 * r44270;
        double r44273 = r44272 * r44270;
        double r44274 = 27464.7644705;
        double r44275 = r44273 + r44274;
        double r44276 = r44275 * r44267;
        double r44277 = 230661.510616;
        double r44278 = r44276 + r44277;
        double r44279 = r44278 * r44267;
        double r44280 = t;
        double r44281 = r44279 + r44280;
        double r44282 = 1.0;
        double r44283 = a;
        double r44284 = r44267 + r44283;
        double r44285 = b;
        double r44286 = fma(r44284, r44267, r44285);
        double r44287 = c;
        double r44288 = fma(r44286, r44267, r44287);
        double r44289 = i;
        double r44290 = fma(r44288, r44267, r44289);
        double r44291 = r44290 * r44282;
        double r44292 = r44282 / r44291;
        double r44293 = r44281 * r44292;
        return r44293;
}

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. Using strategy rm

  6. Applied add-cube-cbrt29.4

    \[\leadsto \left(\left(\left(\left(x \cdot y + z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]

  7. Applied associate-*r*29.4

    \[\leadsto \left(\left(\left(\color{blue}{\left(\left(x \cdot y + z\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}} + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]

  8. Simplified29.4

    \[\leadsto \left(\left(\left(\color{blue}{\left(\left(\mathsf{fma}\left(x, y, z\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} \cdot \sqrt[3]{y} + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]

  9. Final simplification29.4

    \[\leadsto \left(\left(\left(\left(\left(\mathsf{fma}\left(x, y, z\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y} + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]

Reproduce

herbie shell --seed 2020060 +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)))