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

Average Error: 26.4 → 20.4

Time: 6.5s

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 -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 r920448 = x;
        double r920449 = y;
        double r920450 = r920448 + r920449;
        double r920451 = z;
        double r920452 = r920450 * r920451;
        double r920453 = t;
        double r920454 = r920453 + r920449;
        double r920455 = a;
        double r920456 = r920454 * r920455;
        double r920457 = r920452 + r920456;
        double r920458 = b;
        double r920459 = r920449 * r920458;
        double r920460 = r920457 - r920459;
        double r920461 = r920448 + r920453;
        double r920462 = r920461 + r920449;
        double r920463 = r920460 / r920462;
        return r920463;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r920464 = z;
        double r920465 = -6.376626012784567e+136;
        bool r920466 = r920464 <= r920465;
        double r920467 = y;
        double r920468 = x;
        double r920469 = t;
        double r920470 = r920468 + r920469;
        double r920471 = r920470 + r920467;
        double r920472 = r920467 / r920471;
        double r920473 = b;
        double r920474 = r920472 * r920473;
        double r920475 = r920464 - r920474;
        double r920476 = -7.5553554336738665e-50;
        bool r920477 = r920464 <= r920476;
        double r920478 = r920468 + r920467;
        double r920479 = r920478 * r920464;
        double r920480 = r920469 + r920467;
        double r920481 = a;
        double r920482 = r920480 * r920481;
        double r920483 = r920479 + r920482;
        double r920484 = r920483 / r920471;
        double r920485 = 1.0;
        double r920486 = r920471 / r920473;
        double r920487 = r920486 / r920467;
        double r920488 = r920485 / r920487;
        double r920489 = r920484 - r920488;
        double r920490 = -1.442052623539858e-96;
        bool r920491 = r920464 <= r920490;
        double r920492 = r920481 - r920474;
        double r920493 = -5.2560627363695754e-198;
        bool r920494 = r920464 <= r920493;
        double r920495 = cbrt(r920483);
        double r920496 = r920495 * r920495;
        double r920497 = r920495 / r920471;
        double r920498 = r920496 * r920497;
        double r920499 = r920498 - r920474;
        double r920500 = -3.0011721532957425e-279;
        bool r920501 = r920464 <= r920500;
        double r920502 = r920467 / r920486;
        double r920503 = r920481 - r920502;
        double r920504 = -8.469488176137304e-290;
        bool r920505 = r920464 <= r920504;
        double r920506 = r920467 * r920473;
        double r920507 = r920483 - r920506;
        double r920508 = r920485 / r920471;
        double r920509 = r920507 * r920508;
        double r920510 = 1.829058436943914e-282;
        bool r920511 = r920464 <= r920510;
        double r920512 = r920481 * r920480;
        double r920513 = r920464 * r920467;
        double r920514 = r920512 + r920513;
        double r920515 = r920514 / r920471;
        double r920516 = r920515 - r920474;
        double r920517 = 5.143885661764243e-183;
        bool r920518 = r920464 <= r920517;
        double r920519 = 1.124815810617408e-25;
        bool r920520 = r920464 <= r920519;
        double r920521 = 4.4486139245971244e+27;
        bool r920522 = r920464 <= r920521;
        double r920523 = 1.0740017768696439e+86;
        bool r920524 = r920464 <= r920523;
        double r920525 = r920524 ? r920489 : r920475;
        double r920526 = r920522 ? r920492 : r920525;
        double r920527 = r920520 ? r920499 : r920526;
        double r920528 = r920518 ? r920492 : r920527;
        double r920529 = r920511 ? r920516 : r920528;
        double r920530 = r920505 ? r920509 : r920529;
        double r920531 = r920501 ? r920503 : r920530;
        double r920532 = r920494 ? r920499 : r920531;
        double r920533 = r920491 ? r920492 : r920532;
        double r920534 = r920477 ? r920489 : r920533;
        double r920535 = r920466 ? r920475 : r920534;
        return r920535;
}

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)))