Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.7 → 1.5

Time: 3.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -392971998.98787081 \lor \neg \left(t \le 1.332006755273701 \cdot 10^{-307}\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{t}{\left(y - x\right) \cdot z}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r456169 = x;
        double r456170 = y;
        double r456171 = r456170 - r456169;
        double r456172 = z;
        double r456173 = r456171 * r456172;
        double r456174 = t;
        double r456175 = r456173 / r456174;
        double r456176 = r456169 + r456175;
        return r456176;
}

↓

double f(double x, double y, double z, double t) {
        double r456177 = t;
        double r456178 = -392971998.9878708;
        bool r456179 = r456177 <= r456178;
        double r456180 = 1.3320067552737007e-307;
        bool r456181 = r456177 <= r456180;
        double r456182 = !r456181;
        bool r456183 = r456179 || r456182;
        double r456184 = x;
        double r456185 = y;
        double r456186 = r456185 - r456184;
        double r456187 = z;
        double r456188 = r456187 / r456177;
        double r456189 = r456186 * r456188;
        double r456190 = r456184 + r456189;
        double r456191 = 1.0;
        double r456192 = r456186 * r456187;
        double r456193 = r456177 / r456192;
        double r456194 = r456191 / r456193;
        double r456195 = r456184 + r456194;
        double r456196 = r456183 ? r456190 : r456195;
        return r456196;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.7
Target	2.0
Herbie	1.5

\[\begin{array}{l} \mathbf{if}\;x \lt -9.0255111955330046 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.2750321637007147 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < -392971998.9878708 or 1.3320067552737007e-307 < t

   1. Initial program 8.0

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

   3. Applied *-un-lft-identity 8.0

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

   4. Applied times-frac 1.5

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

   5. Simplified 1.5

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

   if -392971998.9878708 < t < 1.3320067552737007e-307

   1. Initial program 1.5

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

   3. Applied clear-num 1.6

      \[\leadsto x + \color{blue}{\frac{1}{\frac{t}{\left(y - x\right) \cdot z}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -392971998.98787081 \lor \neg \left(t \le 1.332006755273701 \cdot 10^{-307}\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{t}{\left(y - x\right) \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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)))