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

Average Error: 27.0 → 20.3

Time: 7.0s

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 r754361 = x;
        double r754362 = y;
        double r754363 = r754361 + r754362;
        double r754364 = z;
        double r754365 = r754363 * r754364;
        double r754366 = t;
        double r754367 = r754366 + r754362;
        double r754368 = a;
        double r754369 = r754367 * r754368;
        double r754370 = r754365 + r754369;
        double r754371 = b;
        double r754372 = r754362 * r754371;
        double r754373 = r754370 - r754372;
        double r754374 = r754361 + r754366;
        double r754375 = r754374 + r754362;
        double r754376 = r754373 / r754375;
        return r754376;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r754377 = z;
        double r754378 = -3.0225981366843547e+182;
        bool r754379 = r754377 <= r754378;
        double r754380 = y;
        double r754381 = x;
        double r754382 = t;
        double r754383 = r754381 + r754382;
        double r754384 = r754383 + r754380;
        double r754385 = r754380 / r754384;
        double r754386 = 1.0;
        double r754387 = b;
        double r754388 = r754386 / r754387;
        double r754389 = r754385 / r754388;
        double r754390 = r754377 - r754389;
        double r754391 = -6.580972394505658e-07;
        bool r754392 = r754377 <= r754391;
        double r754393 = r754381 + r754380;
        double r754394 = r754382 + r754380;
        double r754395 = a;
        double r754396 = r754394 * r754395;
        double r754397 = fma(r754393, r754377, r754396);
        double r754398 = r754397 / r754386;
        double r754399 = r754398 / r754384;
        double r754400 = r754399 - r754389;
        double r754401 = -5.561955879913683e-33;
        bool r754402 = r754377 <= r754401;
        double r754403 = r754384 / r754387;
        double r754404 = r754380 / r754403;
        double r754405 = r754395 - r754404;
        double r754406 = 2.1883827135341942e-265;
        bool r754407 = r754377 <= r754406;
        double r754408 = 3.5312042503262674e-186;
        bool r754409 = r754377 <= r754408;
        double r754410 = 9.836792356441502e-126;
        bool r754411 = r754377 <= r754410;
        double r754412 = 7.56984045759666e-75;
        bool r754413 = r754377 <= r754412;
        double r754414 = 4.992279324067696e+56;
        bool r754415 = r754377 <= r754414;
        double r754416 = r754415 ? r754400 : r754390;
        double r754417 = r754413 ? r754405 : r754416;
        double r754418 = r754411 ? r754400 : r754417;
        double r754419 = r754409 ? r754405 : r754418;
        double r754420 = r754407 ? r754400 : r754419;
        double r754421 = r754402 ? r754405 : r754420;
        double r754422 = r754392 ? r754400 : r754421;
        double r754423 = r754379 ? r754390 : r754422;
        return r754423;
}

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)))