Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.5 → 1.9

Time: 13.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8.197251181992450336177184911706312715011 \cdot 10^{-110} \lor \neg \left(x \le 1.163685728587467934538355368329212838924 \cdot 10^{-252}\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r379540 = x;
        double r379541 = y;
        double r379542 = r379541 - r379540;
        double r379543 = z;
        double r379544 = r379542 * r379543;
        double r379545 = t;
        double r379546 = r379544 / r379545;
        double r379547 = r379540 + r379546;
        return r379547;
}

↓

double f(double x, double y, double z, double t) {
        double r379548 = x;
        double r379549 = -8.19725118199245e-110;
        bool r379550 = r379548 <= r379549;
        double r379551 = 1.163685728587468e-252;
        bool r379552 = r379548 <= r379551;
        double r379553 = !r379552;
        bool r379554 = r379550 || r379553;
        double r379555 = y;
        double r379556 = r379555 - r379548;
        double r379557 = z;
        double r379558 = t;
        double r379559 = r379557 / r379558;
        double r379560 = r379556 * r379559;
        double r379561 = r379548 + r379560;
        double r379562 = r379556 * r379557;
        double r379563 = r379562 / r379558;
        double r379564 = r379563 + r379548;
        double r379565 = r379554 ? r379561 : r379564;
        return r379565;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.5
Target	2.0
Herbie	1.9

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

Derivation

1. Split input into 2 regimes

2. if x < -8.19725118199245e-110 or 1.163685728587468e-252 < x

   1. Initial program 7.0

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

   3. Applied *-un-lft-identity 7.0

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

   4. Applied times-frac 1.2

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

   5. Simplified 1.2

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

   if -8.19725118199245e-110 < x < 1.163685728587468e-252

   1. Initial program 4.5

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

   3. Applied *-un-lft-identity 4.5

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.197251181992450336177184911706312715011 \cdot 10^{-110} \lor \neg \left(x \le 1.163685728587467934538355368329212838924 \cdot 10^{-252}\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.0255111955330046e-135) (- x (* (/ z t) (- x y))) (if (< x 4.2750321637007147e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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