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

Average Error: 26.4 → 20.4

Time: 8.2s

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{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\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]{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}\right) \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}}{\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 -3.11318533103244395 \cdot 10^{-288}:\\ \;\;\;\;\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{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - \frac{1}{\frac{\frac{\left(x + t\right) + y}{b}}{y}}\\ \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]{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}\right) \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}}{\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{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\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 r958379 = x;
        double r958380 = y;
        double r958381 = r958379 + r958380;
        double r958382 = z;
        double r958383 = r958381 * r958382;
        double r958384 = t;
        double r958385 = r958384 + r958380;
        double r958386 = a;
        double r958387 = r958385 * r958386;
        double r958388 = r958383 + r958387;
        double r958389 = b;
        double r958390 = r958380 * r958389;
        double r958391 = r958388 - r958390;
        double r958392 = r958379 + r958384;
        double r958393 = r958392 + r958380;
        double r958394 = r958391 / r958393;
        return r958394;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r958395 = z;
        double r958396 = -6.376626012784567e+136;
        bool r958397 = r958395 <= r958396;
        double r958398 = y;
        double r958399 = x;
        double r958400 = t;
        double r958401 = r958399 + r958400;
        double r958402 = r958401 + r958398;
        double r958403 = r958398 / r958402;
        double r958404 = b;
        double r958405 = r958403 * r958404;
        double r958406 = r958395 - r958405;
        double r958407 = -7.5553554336738665e-50;
        bool r958408 = r958395 <= r958407;
        double r958409 = r958399 + r958398;
        double r958410 = r958400 + r958398;
        double r958411 = a;
        double r958412 = r958410 * r958411;
        double r958413 = fma(r958409, r958395, r958412);
        double r958414 = 1.0;
        double r958415 = r958413 / r958414;
        double r958416 = r958415 / r958402;
        double r958417 = r958402 / r958404;
        double r958418 = r958417 / r958398;
        double r958419 = r958414 / r958418;
        double r958420 = r958416 - r958419;
        double r958421 = -1.442052623539858e-96;
        bool r958422 = r958395 <= r958421;
        double r958423 = r958411 - r958405;
        double r958424 = -5.2560627363695754e-198;
        bool r958425 = r958395 <= r958424;
        double r958426 = cbrt(r958413);
        double r958427 = r958426 * r958426;
        double r958428 = r958426 / r958402;
        double r958429 = r958427 * r958428;
        double r958430 = r958429 - r958405;
        double r958431 = -3.0011721532957425e-279;
        bool r958432 = r958395 <= r958431;
        double r958433 = r958398 / r958417;
        double r958434 = r958411 - r958433;
        double r958435 = -3.113185331032444e-288;
        bool r958436 = r958395 <= r958435;
        double r958437 = r958409 * r958395;
        double r958438 = r958437 + r958412;
        double r958439 = r958398 * r958404;
        double r958440 = r958438 - r958439;
        double r958441 = r958414 / r958402;
        double r958442 = r958440 * r958441;
        double r958443 = 1.829058436943914e-282;
        bool r958444 = r958395 <= r958443;
        double r958445 = 5.143885661764243e-183;
        bool r958446 = r958395 <= r958445;
        double r958447 = 1.124815810617408e-25;
        bool r958448 = r958395 <= r958447;
        double r958449 = 4.4486139245971244e+27;
        bool r958450 = r958395 <= r958449;
        double r958451 = 1.0740017768696439e+86;
        bool r958452 = r958395 <= r958451;
        double r958453 = r958452 ? r958420 : r958406;
        double r958454 = r958450 ? r958423 : r958453;
        double r958455 = r958448 ? r958430 : r958454;
        double r958456 = r958446 ? r958423 : r958455;
        double r958457 = r958444 ? r958420 : r958456;
        double r958458 = r958436 ? r958442 : r958457;
        double r958459 = r958432 ? r958434 : r958458;
        double r958460 = r958425 ? r958430 : r958459;
        double r958461 = r958422 ? r958423 : r958460;
        double r958462 = r958408 ? r958420 : r958461;
        double r958463 = r958397 ? r958406 : r958462;
        return r958463;
}

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 6 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. Simplified 39.4

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

   6. Applied associate-/l* 39.7

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

   8. Applied associate-/r/ 38.8

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

   9. 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 -3.113185331032444e-288 < z < 1.829058436943914e-282 or 4.4486139245971244e+27 < z < 1.0740017768696439e+86

   1. Initial program 23.1

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

   3. Applied div-sub 23.1

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

   4. Simplified 23.1

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

   6. Applied associate-/l* 20.4

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

   8. Applied clear-num 20.4

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

   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. Simplified 19.1

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

   6. Applied associate-/l* 16.5

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

   8. Applied associate-/r/ 14.8

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

   9. 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. Simplified 17.7

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

   6. Applied associate-/l* 14.1

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

   8. Applied associate-/r/ 13.0

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

   10. Applied *-un-lft-identity 13.0

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

   11. Applied add-cube-cbrt 13.6

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

   12. Applied times-frac 13.6

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

   13. Simplified 13.6

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

   14. Simplified 13.6

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

   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. Simplified 20.1

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

   6. Applied associate-/l* 16.2

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

   7. Taylor expanded around 0 18.7

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

   if -3.0011721532957425e-279 < z < -3.113185331032444e-288

   1. Initial program 19.4

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

   3. Applied div-inv 19.4

      \[\leadsto \color{blue}{\left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right) \cdot \frac{1}{\left(x + t\right) + y}}\]
3. Recombined 6 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{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\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]{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}\right) \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}}{\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 -3.11318533103244395 \cdot 10^{-288}:\\ \;\;\;\;\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{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - \frac{1}{\frac{\frac{\left(x + t\right) + y}{b}}{y}}\\ \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]{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}\right) \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}}{\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{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\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 +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)))