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

Average Error: 27.0 → 7.5

Time: 13.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.392895250809098 \cdot 10^{+202} \lor \neg \left(y \leq 9.384766001373891 \cdot 10^{+133}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y \cdot a}{y + \left(t + x\right)} + \left(z \cdot \frac{x}{t + \left(y + x\right)} + \left(a \cdot \frac{t}{t + \left(y + x\right)} + \frac{y}{\sqrt[3]{y + \left(t + x\right)} \cdot \sqrt[3]{y + \left(t + x\right)}} \cdot \frac{z}{\sqrt[3]{y + \left(t + x\right)}}\right)\right)\right) - \frac{y \cdot b}{y + \left(t + x\right)}\\ \end{array}\]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))

↓

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.392895250809098e+202) (not (<= y 9.384766001373891e+133)))
   (- (+ a z) b)
   (-
    (+
     (/ (* y a) (+ y (+ t x)))
     (+
      (* z (/ x (+ t (+ y x))))
      (+
       (* a (/ t (+ t (+ y x))))
       (*
        (/ y (* (cbrt (+ y (+ t x))) (cbrt (+ y (+ t x)))))
        (/ z (cbrt (+ y (+ t x))))))))
    (/ (* y b) (+ y (+ t x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.392895250809098e+202) || !(y <= 9.384766001373891e+133)) {
		tmp = (a + z) - b;
	} else {
		tmp = (((y * a) / (y + (t + x))) + ((z * (x / (t + (y + x)))) + ((a * (t / (t + (y + x)))) + ((y / (cbrt(y + (t + x)) * cbrt(y + (t + x)))) * (z / cbrt(y + (t + x))))))) - ((y * b) / (y + (t + x)));
	}
	return tmp;
}

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	11.1
Herbie	7.5

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} < -3.5813117084150564 \cdot 10^{+153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} < 1.2285964308315609 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -1.3928952508090981e202 or 9.3847660013738911e133 < y

   1. Initial program 49.7

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

   2. Taylor expanded around inf 10.2

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

   if -1.3928952508090981e202 < y < 9.3847660013738911e133

   1. Initial program 20.1

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

   2. Taylor expanded around 0 20.1

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

   3. Simplified 20.1

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

   5. Applied *-un-lft-identity_binary64_24630 20.1

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

   6. Applied times-frac_binary64_24636 14.4

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

   7. Simplified 14.4

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

   8. Simplified 14.4

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

   10. Applied *-un-lft-identity_binary64_24630 14.4

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

   11. Applied times-frac_binary64_24636 8.2

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

   12. Simplified 8.2

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

   13. Simplified 8.2

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

   15. Applied add-cube-cbrt_binary64_24665 8.3

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

   16. Applied times-frac_binary64_24636 6.7

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

4. Final simplification 7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.392895250809098 \cdot 10^{+202} \lor \neg \left(y \leq 9.384766001373891 \cdot 10^{+133}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y \cdot a}{y + \left(t + x\right)} + \left(z \cdot \frac{x}{t + \left(y + x\right)} + \left(a \cdot \frac{t}{t + \left(y + x\right)} + \frac{y}{\sqrt[3]{y + \left(t + x\right)} \cdot \sqrt[3]{y + \left(t + x\right)}} \cdot \frac{z}{\sqrt[3]{y + \left(t + x\right)}}\right)\right)\right) - \frac{y \cdot b}{y + \left(t + x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(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) (/ 1.0 (/ (+ (+ 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)))