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

Average Error: 26.3 → 20.6

Time: 16.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.329351671379600395696054258455065074018 \cdot 10^{162}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le \frac{2523015234541329}{4.737909217226284592129454005317695786306 \cdot 10^{226}}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\sqrt[3]{\left(x + t\right) + y} \cdot \sqrt[3]{\left(x + t\right) + y}} \cdot \frac{b}{\sqrt[3]{\left(x + t\right) + y}}\\ \mathbf{elif}\;a \le \frac{6133953907186681}{3.213876088517980551083924184682325205044 \cdot 10^{60}}:\\ \;\;\;\;z - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le 1.978965818958528848241507275261864808485 \cdot 10^{164}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{else}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r701394 = x;
        double r701395 = y;
        double r701396 = r701394 + r701395;
        double r701397 = z;
        double r701398 = r701396 * r701397;
        double r701399 = t;
        double r701400 = r701399 + r701395;
        double r701401 = a;
        double r701402 = r701400 * r701401;
        double r701403 = r701398 + r701402;
        double r701404 = b;
        double r701405 = r701395 * r701404;
        double r701406 = r701403 - r701405;
        double r701407 = r701394 + r701399;
        double r701408 = r701407 + r701395;
        double r701409 = r701406 / r701408;
        return r701409;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r701410 = a;
        double r701411 = -1.3293516713796004e+162;
        bool r701412 = r701410 <= r701411;
        double r701413 = y;
        double r701414 = b;
        double r701415 = x;
        double r701416 = t;
        double r701417 = r701415 + r701416;
        double r701418 = r701417 + r701413;
        double r701419 = r701414 / r701418;
        double r701420 = r701413 * r701419;
        double r701421 = r701410 - r701420;
        double r701422 = 2523015234541329.0;
        double r701423 = 4.737909217226285e+226;
        double r701424 = r701422 / r701423;
        bool r701425 = r701410 <= r701424;
        double r701426 = r701415 + r701413;
        double r701427 = z;
        double r701428 = r701426 * r701427;
        double r701429 = r701416 + r701413;
        double r701430 = r701429 * r701410;
        double r701431 = r701428 + r701430;
        double r701432 = r701431 / r701418;
        double r701433 = cbrt(r701418);
        double r701434 = r701433 * r701433;
        double r701435 = r701413 / r701434;
        double r701436 = r701414 / r701433;
        double r701437 = r701435 * r701436;
        double r701438 = r701432 - r701437;
        double r701439 = 6133953907186681.0;
        double r701440 = 3.2138760885179806e+60;
        double r701441 = r701439 / r701440;
        bool r701442 = r701410 <= r701441;
        double r701443 = r701427 - r701420;
        double r701444 = 1.9789658189585288e+164;
        bool r701445 = r701410 <= r701444;
        double r701446 = r701418 / r701414;
        double r701447 = r701413 / r701446;
        double r701448 = r701432 - r701447;
        double r701449 = r701445 ? r701448 : r701421;
        double r701450 = r701442 ? r701443 : r701449;
        double r701451 = r701425 ? r701438 : r701450;
        double r701452 = r701412 ? r701421 : r701451;
        return r701452;
}

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	26.3
Target	11.6
Herbie	20.6

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

Derivation

1. Split input into 4 regimes

2. if a < -1.3293516713796004e+162 or 1.9789658189585288e+164 < a

   1. Initial program 42.9

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

   3. Applied div-sub 42.9

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

   5. Applied *-un-lft-identity 42.9

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

   6. Applied times-frac 43.6

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

   7. Simplified 43.6

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

   8. Taylor expanded around 0 25.0

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

   if -1.3293516713796004e+162 < a < 5.325165846082586e-212

   1. Initial program 20.3

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

   3. Applied div-sub 20.3

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

   5. Applied *-un-lft-identity 20.3

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

   6. Applied times-frac 17.5

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

   7. Simplified 17.5

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

   9. Applied add-cube-cbrt 17.7

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

   10. Applied *-un-lft-identity 17.7

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

   11. Applied times-frac 17.7

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

   12. Applied associate-*r* 16.9

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

   13. Simplified 16.9

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

   if 5.325165846082586e-212 < a < 1.908584443905938e-45

   1. Initial program 18.8

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

   3. Applied div-sub 18.8

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

   5. Applied *-un-lft-identity 18.8

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

   6. Applied times-frac 15.4

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

   7. Simplified 15.4

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

   8. Taylor expanded around inf 22.0

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

   if 1.908584443905938e-45 < a < 1.9789658189585288e+164

   1. Initial program 26.4

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

   3. Applied div-sub 26.4

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

   5. Applied associate-/l* 23.7

      \[\leadsto \frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \color{blue}{\frac{y}{\frac{\left(x + t\right) + y}{b}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 20.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.329351671379600395696054258455065074018 \cdot 10^{162}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le \frac{2523015234541329}{4.737909217226284592129454005317695786306 \cdot 10^{226}}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\sqrt[3]{\left(x + t\right) + y} \cdot \sqrt[3]{\left(x + t\right) + y}} \cdot \frac{b}{\sqrt[3]{\left(x + t\right) + y}}\\ \mathbf{elif}\;a \le \frac{6133953907186681}{3.213876088517980551083924184682325205044 \cdot 10^{60}}:\\ \;\;\;\;z - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le 1.978965818958528848241507275261864808485 \cdot 10^{164}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{else}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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)) -3.5813117084150564e153) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 1.2285964308315609e82) (/ 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)))