Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.4 → 1.6

Time: 24.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.362884853252186351003430209899103823374 \cdot 10^{-200}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;t \le 8.698046239578131913729902310956849143645 \cdot 10^{-72}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r217957 = x;
        double r217958 = y;
        double r217959 = r217958 - r217957;
        double r217960 = z;
        double r217961 = r217959 * r217960;
        double r217962 = t;
        double r217963 = r217961 / r217962;
        double r217964 = r217957 + r217963;
        return r217964;
}

↓

double f(double x, double y, double z, double t) {
        double r217965 = t;
        double r217966 = -2.3628848532521864e-200;
        bool r217967 = r217965 <= r217966;
        double r217968 = x;
        double r217969 = y;
        double r217970 = r217969 - r217968;
        double r217971 = z;
        double r217972 = r217971 / r217965;
        double r217973 = r217970 * r217972;
        double r217974 = r217968 + r217973;
        double r217975 = 8.698046239578132e-72;
        bool r217976 = r217965 <= r217975;
        double r217977 = r217970 * r217971;
        double r217978 = r217977 / r217965;
        double r217979 = r217968 + r217978;
        double r217980 = r217965 / r217971;
        double r217981 = r217970 / r217980;
        double r217982 = r217968 + r217981;
        double r217983 = r217976 ? r217979 : r217982;
        double r217984 = r217967 ? r217974 : r217983;
        return r217984;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.4
Target	2.1
Herbie	1.6

\[\begin{array}{l} \mathbf{if}\;x \lt -9.025511195533004570453352523209034680317 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.275032163700714748507147332551979944314 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if t < -2.3628848532521864e-200

   1. Initial program 7.1

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

   3. Applied *-un-lft-identity 7.1

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

   4. Applied times-frac 1.6

      \[\leadsto x + \color{blue}{\frac{y - x}{1} \cdot \frac{z}{t}}\]

   5. Simplified 1.6

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

   if -2.3628848532521864e-200 < t < 8.698046239578132e-72

   1. Initial program 2.7

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

   if 8.698046239578132e-72 < t

   1. Initial program 7.5

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

   3. Applied associate-/l* 1.1

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.362884853252186351003430209899103823374 \cdot 10^{-200}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;t \le 8.698046239578131913729902310956849143645 \cdot 10^{-72}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

  (+ x (/ (* (- y x) z) t)))