AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1

Average Error: 27.0 → 20.3

Time: 7.5s

Precision: 64

Math

\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.022598136684354663412245344550126028397 \cdot 10^{182}:\\ \;\;\;\;z - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;z \le -6.58097239450565769445350948474438013136 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;z \le -5.561955879913683304630071073532749837049 \cdot 10^{-33}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 2.188382713534194168391592823268801705419 \cdot 10^{-265}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;z \le 3.531204250326267435165734577025015017466 \cdot 10^{-186}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 9.836792356441501745440262935078248892751 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;z \le 7.569840457596660380072714567516066108349 \cdot 10^{-75}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 4.992279324067695797541286300663075135612 \cdot 10^{56}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{else}:\\ \;\;\;\;z - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r800229 = x;
        double r800230 = y;
        double r800231 = r800229 + r800230;
        double r800232 = z;
        double r800233 = r800231 * r800232;
        double r800234 = t;
        double r800235 = r800234 + r800230;
        double r800236 = a;
        double r800237 = r800235 * r800236;
        double r800238 = r800233 + r800237;
        double r800239 = b;
        double r800240 = r800230 * r800239;
        double r800241 = r800238 - r800240;
        double r800242 = r800229 + r800234;
        double r800243 = r800242 + r800230;
        double r800244 = r800241 / r800243;
        return r800244;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r800245 = z;
        double r800246 = -3.0225981366843547e+182;
        bool r800247 = r800245 <= r800246;
        double r800248 = y;
        double r800249 = x;
        double r800250 = t;
        double r800251 = r800249 + r800250;
        double r800252 = r800251 + r800248;
        double r800253 = r800248 / r800252;
        double r800254 = 1.0;
        double r800255 = b;
        double r800256 = r800254 / r800255;
        double r800257 = r800253 / r800256;
        double r800258 = r800245 - r800257;
        double r800259 = -6.580972394505658e-07;
        bool r800260 = r800245 <= r800259;
        double r800261 = r800249 + r800248;
        double r800262 = r800250 + r800248;
        double r800263 = a;
        double r800264 = r800262 * r800263;
        double r800265 = fma(r800261, r800245, r800264);
        double r800266 = r800265 / r800254;
        double r800267 = r800266 / r800252;
        double r800268 = r800267 - r800257;
        double r800269 = -5.561955879913683e-33;
        bool r800270 = r800245 <= r800269;
        double r800271 = r800252 / r800255;
        double r800272 = r800248 / r800271;
        double r800273 = r800263 - r800272;
        double r800274 = 2.1883827135341942e-265;
        bool r800275 = r800245 <= r800274;
        double r800276 = 3.5312042503262674e-186;
        bool r800277 = r800245 <= r800276;
        double r800278 = 9.836792356441502e-126;
        bool r800279 = r800245 <= r800278;
        double r800280 = 7.56984045759666e-75;
        bool r800281 = r800245 <= r800280;
        double r800282 = 4.992279324067696e+56;
        bool r800283 = r800245 <= r800282;
        double r800284 = r800283 ? r800268 : r800258;
        double r800285 = r800281 ? r800273 : r800284;
        double r800286 = r800279 ? r800268 : r800285;
        double r800287 = r800277 ? r800273 : r800286;
        double r800288 = r800275 ? r800268 : r800287;
        double r800289 = r800270 ? r800273 : r800288;
        double r800290 = r800260 ? r800268 : r800289;
        double r800291 = r800247 ? r800258 : r800290;
        return r800291;
}

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

Target

Original	27.0
Target	11.3
Herbie	20.3

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.581311708415056427521064305370896655752 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.228596430831560895857110658734089400289 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -3.0225981366843547e+182 or 4.992279324067696e+56 < z

   1. Initial program 40.3

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

   3. Applied div-sub 40.3

      \[\leadsto \color{blue}{\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y \cdot b}{\left(x + t\right) + y}}\]

   4. Simplified 40.3

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y}} - \frac{y \cdot b}{\left(x + t\right) + y}\]
   5. Using strategy rm

   6. Applied associate-/l* 40.4

      \[\leadsto \frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - \color{blue}{\frac{y}{\frac{\left(x + t\right) + y}{b}}}\]
   7. Using strategy rm

   8. Applied div-inv 40.4

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

   9. Applied associate-/r* 39.6

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

   10. Taylor expanded around inf 25.0

       \[\leadsto \color{blue}{z} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\]

   if -3.0225981366843547e+182 < z < -6.580972394505658e-07 or -5.561955879913683e-33 < z < 2.1883827135341942e-265 or 3.5312042503262674e-186 < z < 9.836792356441502e-126 or 7.56984045759666e-75 < z < 4.992279324067696e+56

   1. Initial program 21.4

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

   3. Applied div-sub 21.4

      \[\leadsto \color{blue}{\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y \cdot b}{\left(x + t\right) + y}}\]

   4. Simplified 21.4

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y}} - \frac{y \cdot b}{\left(x + t\right) + y}\]
   5. Using strategy rm

   6. Applied associate-/l* 18.2

      \[\leadsto \frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - \color{blue}{\frac{y}{\frac{\left(x + t\right) + y}{b}}}\]
   7. Using strategy rm

   8. Applied div-inv 18.3

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

   9. Applied associate-/r* 17.6

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

   if -6.580972394505658e-07 < z < -5.561955879913683e-33 or 2.1883827135341942e-265 < z < 3.5312042503262674e-186 or 9.836792356441502e-126 < z < 7.56984045759666e-75

   1. Initial program 20.2

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

   3. Applied div-sub 20.2

      \[\leadsto \color{blue}{\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y \cdot b}{\left(x + t\right) + y}}\]

   4. Simplified 20.2

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y}} - \frac{y \cdot b}{\left(x + t\right) + y}\]
   5. Using strategy rm

   6. Applied associate-/l* 16.1

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

   7. Taylor expanded around 0 21.0

      \[\leadsto \color{blue}{a} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 20.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.022598136684354663412245344550126028397 \cdot 10^{182}:\\ \;\;\;\;z - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;z \le -6.58097239450565769445350948474438013136 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;z \le -5.561955879913683304630071073532749837049 \cdot 10^{-33}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 2.188382713534194168391592823268801705419 \cdot 10^{-265}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;z \le 3.531204250326267435165734577025015017466 \cdot 10^{-186}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 9.836792356441501745440262935078248892751 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;z \le 7.569840457596660380072714567516066108349 \cdot 10^{-75}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 4.992279324067695797541286300663075135612 \cdot 10^{56}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{else}:\\ \;\;\;\;z - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64

  :herbie-target
  (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) -3.5813117084150564e+153) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 1.2285964308315609e+82) (/ 1 (/ (+ (+ x t) y) (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)))) (- (+ z a) b)))

  (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))