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

Average Error: 27.0 → 20.3

Time: 7.7s

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 r838996 = x;
        double r838997 = y;
        double r838998 = r838996 + r838997;
        double r838999 = z;
        double r839000 = r838998 * r838999;
        double r839001 = t;
        double r839002 = r839001 + r838997;
        double r839003 = a;
        double r839004 = r839002 * r839003;
        double r839005 = r839000 + r839004;
        double r839006 = b;
        double r839007 = r838997 * r839006;
        double r839008 = r839005 - r839007;
        double r839009 = r838996 + r839001;
        double r839010 = r839009 + r838997;
        double r839011 = r839008 / r839010;
        return r839011;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r839012 = z;
        double r839013 = -3.0225981366843547e+182;
        bool r839014 = r839012 <= r839013;
        double r839015 = y;
        double r839016 = x;
        double r839017 = t;
        double r839018 = r839016 + r839017;
        double r839019 = r839018 + r839015;
        double r839020 = r839015 / r839019;
        double r839021 = 1.0;
        double r839022 = b;
        double r839023 = r839021 / r839022;
        double r839024 = r839020 / r839023;
        double r839025 = r839012 - r839024;
        double r839026 = -6.580972394505658e-07;
        bool r839027 = r839012 <= r839026;
        double r839028 = r839016 + r839015;
        double r839029 = r839017 + r839015;
        double r839030 = a;
        double r839031 = r839029 * r839030;
        double r839032 = fma(r839028, r839012, r839031);
        double r839033 = r839032 / r839021;
        double r839034 = r839033 / r839019;
        double r839035 = r839034 - r839024;
        double r839036 = -5.561955879913683e-33;
        bool r839037 = r839012 <= r839036;
        double r839038 = r839019 / r839022;
        double r839039 = r839015 / r839038;
        double r839040 = r839030 - r839039;
        double r839041 = 2.1883827135341942e-265;
        bool r839042 = r839012 <= r839041;
        double r839043 = 3.5312042503262674e-186;
        bool r839044 = r839012 <= r839043;
        double r839045 = 9.836792356441502e-126;
        bool r839046 = r839012 <= r839045;
        double r839047 = 7.56984045759666e-75;
        bool r839048 = r839012 <= r839047;
        double r839049 = 4.992279324067696e+56;
        bool r839050 = r839012 <= r839049;
        double r839051 = r839050 ? r839035 : r839025;
        double r839052 = r839048 ? r839040 : r839051;
        double r839053 = r839046 ? r839035 : r839052;
        double r839054 = r839044 ? r839040 : r839053;
        double r839055 = r839042 ? r839035 : r839054;
        double r839056 = r839037 ? r839040 : r839055;
        double r839057 = r839027 ? r839035 : r839056;
        double r839058 = r839014 ? r839025 : r839057;
        return r839058;
}

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)))