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

Average Error: 26.4 → 20.4

Time: 7.2s

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 -6.3766260127845671 \cdot 10^{136}:\\ \;\;\;\;z - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;z \le -7.5553554336738665 \cdot 10^{-50}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{1}{\frac{\frac{\left(x + t\right) + y}{b}}{y}}\\ \mathbf{elif}\;z \le -1.4420526235398581 \cdot 10^{-96}:\\ \;\;\;\;a - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;z \le -5.2560627363695754 \cdot 10^{-198}:\\ \;\;\;\;\left(\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a} \cdot \sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}\right) \cdot \frac{\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;z \le -3.0011721532957425 \cdot 10^{-279}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le -8.4694881761373042 \cdot 10^{-290}:\\ \;\;\;\;\left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right) \cdot \frac{1}{\left(x + t\right) + y}\\ \mathbf{elif}\;z \le 1.829058436943914 \cdot 10^{-282}:\\ \;\;\;\;\frac{a \cdot \left(t + y\right) + z \cdot y}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;z \le 5.14388566176424308 \cdot 10^{-183}:\\ \;\;\;\;a - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;z \le 1.1248158106174079 \cdot 10^{-25}:\\ \;\;\;\;\left(\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a} \cdot \sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}\right) \cdot \frac{\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;z \le 4.44861392459712437 \cdot 10^{27}:\\ \;\;\;\;a - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;z \le 1.07400177686964386 \cdot 10^{86}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{1}{\frac{\frac{\left(x + t\right) + y}{b}}{y}}\\ \mathbf{else}:\\ \;\;\;\;z - \frac{y}{\left(x + t\right) + y} \cdot b\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r891190 = x;
        double r891191 = y;
        double r891192 = r891190 + r891191;
        double r891193 = z;
        double r891194 = r891192 * r891193;
        double r891195 = t;
        double r891196 = r891195 + r891191;
        double r891197 = a;
        double r891198 = r891196 * r891197;
        double r891199 = r891194 + r891198;
        double r891200 = b;
        double r891201 = r891191 * r891200;
        double r891202 = r891199 - r891201;
        double r891203 = r891190 + r891195;
        double r891204 = r891203 + r891191;
        double r891205 = r891202 / r891204;
        return r891205;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r891206 = z;
        double r891207 = -6.376626012784567e+136;
        bool r891208 = r891206 <= r891207;
        double r891209 = y;
        double r891210 = x;
        double r891211 = t;
        double r891212 = r891210 + r891211;
        double r891213 = r891212 + r891209;
        double r891214 = r891209 / r891213;
        double r891215 = b;
        double r891216 = r891214 * r891215;
        double r891217 = r891206 - r891216;
        double r891218 = -7.5553554336738665e-50;
        bool r891219 = r891206 <= r891218;
        double r891220 = r891210 + r891209;
        double r891221 = r891220 * r891206;
        double r891222 = r891211 + r891209;
        double r891223 = a;
        double r891224 = r891222 * r891223;
        double r891225 = r891221 + r891224;
        double r891226 = r891225 / r891213;
        double r891227 = 1.0;
        double r891228 = r891213 / r891215;
        double r891229 = r891228 / r891209;
        double r891230 = r891227 / r891229;
        double r891231 = r891226 - r891230;
        double r891232 = -1.442052623539858e-96;
        bool r891233 = r891206 <= r891232;
        double r891234 = r891223 - r891216;
        double r891235 = -5.2560627363695754e-198;
        bool r891236 = r891206 <= r891235;
        double r891237 = cbrt(r891225);
        double r891238 = r891237 * r891237;
        double r891239 = r891237 / r891213;
        double r891240 = r891238 * r891239;
        double r891241 = r891240 - r891216;
        double r891242 = -3.0011721532957425e-279;
        bool r891243 = r891206 <= r891242;
        double r891244 = r891209 / r891228;
        double r891245 = r891223 - r891244;
        double r891246 = -8.469488176137304e-290;
        bool r891247 = r891206 <= r891246;
        double r891248 = r891209 * r891215;
        double r891249 = r891225 - r891248;
        double r891250 = r891227 / r891213;
        double r891251 = r891249 * r891250;
        double r891252 = 1.829058436943914e-282;
        bool r891253 = r891206 <= r891252;
        double r891254 = r891223 * r891222;
        double r891255 = r891206 * r891209;
        double r891256 = r891254 + r891255;
        double r891257 = r891256 / r891213;
        double r891258 = r891257 - r891216;
        double r891259 = 5.143885661764243e-183;
        bool r891260 = r891206 <= r891259;
        double r891261 = 1.124815810617408e-25;
        bool r891262 = r891206 <= r891261;
        double r891263 = 4.4486139245971244e+27;
        bool r891264 = r891206 <= r891263;
        double r891265 = 1.0740017768696439e+86;
        bool r891266 = r891206 <= r891265;
        double r891267 = r891266 ? r891231 : r891217;
        double r891268 = r891264 ? r891234 : r891267;
        double r891269 = r891262 ? r891241 : r891268;
        double r891270 = r891260 ? r891234 : r891269;
        double r891271 = r891253 ? r891258 : r891270;
        double r891272 = r891247 ? r891251 : r891271;
        double r891273 = r891243 ? r891245 : r891272;
        double r891274 = r891236 ? r891241 : r891273;
        double r891275 = r891233 ? r891234 : r891274;
        double r891276 = r891219 ? r891231 : r891275;
        double r891277 = r891208 ? r891217 : r891276;
        return r891277;
}

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	26.4
Target	11.1
Herbie	20.4

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.5813117084150564 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.2285964308315609 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

1. Split input into 7 regimes

2. if z < -6.376626012784567e+136 or 1.0740017768696439e+86 < z

   1. Initial program 39.4

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

   3. Applied div-sub 39.4

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

   5. Applied associate-/l* 39.7

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

   7. Applied associate-/r/ 38.8

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

   8. Taylor expanded around inf 24.4

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

   if -6.376626012784567e+136 < z < -7.5553554336738665e-50 or 4.4486139245971244e+27 < z < 1.0740017768696439e+86

   1. Initial program 24.2

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

   3. Applied div-sub 24.2

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

   5. Applied associate-/l* 21.5

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

   7. Applied clear-num 21.5

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

   if -7.5553554336738665e-50 < z < -1.442052623539858e-96 or 1.829058436943914e-282 < z < 5.143885661764243e-183 or 1.124815810617408e-25 < z < 4.4486139245971244e+27

   1. Initial program 19.1

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

   3. Applied div-sub 19.1

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

   5. Applied associate-/l* 16.5

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

   7. Applied associate-/r/ 14.8

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

   8. Taylor expanded around 0 21.8

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

   if -1.442052623539858e-96 < z < -5.2560627363695754e-198 or 5.143885661764243e-183 < z < 1.124815810617408e-25

   1. Initial program 17.7

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

   3. Applied div-sub 17.7

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

   5. Applied associate-/l* 14.1

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

   7. Applied associate-/r/ 13.0

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

   9. Applied *-un-lft-identity 13.0

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

   10. Applied add-cube-cbrt 13.6

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a} \cdot \sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}\right) \cdot \sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}}{1 \cdot \left(\left(x + t\right) + y\right)} - \frac{y}{\left(x + t\right) + y} \cdot b\]

   11. Applied times-frac 13.6

       \[\leadsto \color{blue}{\frac{\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a} \cdot \sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}{1} \cdot \frac{\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}{\left(x + t\right) + y}} - \frac{y}{\left(x + t\right) + y} \cdot b\]

   12. Simplified 13.6

       \[\leadsto \color{blue}{\left(\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a} \cdot \sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}\right)} \cdot \frac{\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\]

   if -5.2560627363695754e-198 < z < -3.0011721532957425e-279

   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 20.1

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

   5. Applied associate-/l* 16.2

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

   6. Taylor expanded around 0 18.7

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

   if -3.0011721532957425e-279 < z < -8.469488176137304e-290

   1. Initial program 17.9

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

   3. Applied div-inv 17.9

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

   if -8.469488176137304e-290 < z < 1.829058436943914e-282

   1. Initial program 18.4

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

   3. Applied div-sub 18.3

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

   5. Applied associate-/l* 15.2

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

   7. Applied associate-/r/ 13.4

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

   8. Taylor expanded around inf 15.8

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

   9. Simplified 15.8

      \[\leadsto \frac{\color{blue}{a \cdot \left(t + y\right) + z \cdot y}}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\]
3. Recombined 7 regimes into one program.

4. Final simplification 20.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.3766260127845671 \cdot 10^{136}:\\ \;\;\;\;z - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;z \le -7.5553554336738665 \cdot 10^{-50}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{1}{\frac{\frac{\left(x + t\right) + y}{b}}{y}}\\ \mathbf{elif}\;z \le -1.4420526235398581 \cdot 10^{-96}:\\ \;\;\;\;a - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;z \le -5.2560627363695754 \cdot 10^{-198}:\\ \;\;\;\;\left(\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a} \cdot \sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}\right) \cdot \frac{\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;z \le -3.0011721532957425 \cdot 10^{-279}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le -8.4694881761373042 \cdot 10^{-290}:\\ \;\;\;\;\left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right) \cdot \frac{1}{\left(x + t\right) + y}\\ \mathbf{elif}\;z \le 1.829058436943914 \cdot 10^{-282}:\\ \;\;\;\;\frac{a \cdot \left(t + y\right) + z \cdot y}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;z \le 5.14388566176424308 \cdot 10^{-183}:\\ \;\;\;\;a - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;z \le 1.1248158106174079 \cdot 10^{-25}:\\ \;\;\;\;\left(\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a} \cdot \sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}\right) \cdot \frac{\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;z \le 4.44861392459712437 \cdot 10^{27}:\\ \;\;\;\;a - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;z \le 1.07400177686964386 \cdot 10^{86}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{1}{\frac{\frac{\left(x + t\right) + y}{b}}{y}}\\ \mathbf{else}:\\ \;\;\;\;z - \frac{y}{\left(x + t\right) + y} \cdot b\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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)))