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

Average Error: 26.5 → 7.8

Time: 6.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} = -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 7.898361285701560830746917301273604408641 \cdot 10^{198}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r1172974 = x;
        double r1172975 = y;
        double r1172976 = r1172974 + r1172975;
        double r1172977 = z;
        double r1172978 = r1172976 * r1172977;
        double r1172979 = t;
        double r1172980 = r1172979 + r1172975;
        double r1172981 = a;
        double r1172982 = r1172980 * r1172981;
        double r1172983 = r1172978 + r1172982;
        double r1172984 = b;
        double r1172985 = r1172975 * r1172984;
        double r1172986 = r1172983 - r1172985;
        double r1172987 = r1172974 + r1172979;
        double r1172988 = r1172987 + r1172975;
        double r1172989 = r1172986 / r1172988;
        return r1172989;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r1172990 = x;
        double r1172991 = y;
        double r1172992 = r1172990 + r1172991;
        double r1172993 = z;
        double r1172994 = r1172992 * r1172993;
        double r1172995 = t;
        double r1172996 = r1172995 + r1172991;
        double r1172997 = a;
        double r1172998 = r1172996 * r1172997;
        double r1172999 = r1172994 + r1172998;
        double r1173000 = b;
        double r1173001 = r1172991 * r1173000;
        double r1173002 = r1172999 - r1173001;
        double r1173003 = r1172990 + r1172995;
        double r1173004 = r1173003 + r1172991;
        double r1173005 = r1173002 / r1173004;
        double r1173006 = -inf.0;
        bool r1173007 = r1173005 <= r1173006;
        double r1173008 = 7.898361285701561e+198;
        bool r1173009 = r1173005 <= r1173008;
        double r1173010 = !r1173009;
        bool r1173011 = r1173007 || r1173010;
        double r1173012 = r1172997 + r1172993;
        double r1173013 = r1173012 - r1173000;
        double r1173014 = r1173011 ? r1173013 : r1173005;
        return r1173014;
}

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	26.5
Target	11.1
Herbie	7.8

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

Derivation

1. Split input into 2 regimes

2. if (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < -inf.0 or 7.898361285701561e+198 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))

   1. Initial program 59.3

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

   3. Applied clear-num 59.4

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

   4. Simplified 59.4

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

   5. Taylor expanded around 0 17.2

      \[\leadsto \color{blue}{\left(a + z\right) - b}\]

   if -inf.0 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < 7.898361285701561e+198

   1. Initial program 0.3

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
3. Recombined 2 regimes into one program.

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} = -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 7.898361285701560830746917301273604408641 \cdot 10^{198}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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)))