Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.5 → 2.1

Time: 2.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.5028999843123688 \cdot 10^{-201} \lor \neg \left(x \le 5.86993056876950949 \cdot 10^{-45}\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{t}{\left(y - x\right) \cdot z}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r564508 = x;
        double r564509 = y;
        double r564510 = r564509 - r564508;
        double r564511 = z;
        double r564512 = r564510 * r564511;
        double r564513 = t;
        double r564514 = r564512 / r564513;
        double r564515 = r564508 + r564514;
        return r564515;
}

↓

double f(double x, double y, double z, double t) {
        double r564516 = x;
        double r564517 = -1.5028999843123688e-201;
        bool r564518 = r564516 <= r564517;
        double r564519 = 5.86993056876951e-45;
        bool r564520 = r564516 <= r564519;
        double r564521 = !r564520;
        bool r564522 = r564518 || r564521;
        double r564523 = y;
        double r564524 = r564523 - r564516;
        double r564525 = z;
        double r564526 = t;
        double r564527 = r564525 / r564526;
        double r564528 = r564524 * r564527;
        double r564529 = r564516 + r564528;
        double r564530 = 1.0;
        double r564531 = r564524 * r564525;
        double r564532 = r564526 / r564531;
        double r564533 = r564530 / r564532;
        double r564534 = r564516 + r564533;
        double r564535 = r564522 ? r564529 : r564534;
        return r564535;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.5
Target	1.8
Herbie	2.1

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

Derivation

1. Split input into 2 regimes

2. if x < -1.5028999843123688e-201 or 5.86993056876951e-45 < x

   1. Initial program 7.4

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

   3. Applied *-un-lft-identity 7.4

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

   4. Applied times-frac 0.9

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

   5. Simplified 0.9

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

   if -1.5028999843123688e-201 < x < 5.86993056876951e-45

   1. Initial program 4.5

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

   3. Applied clear-num 4.6

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.5028999843123688 \cdot 10^{-201} \lor \neg \left(x \le 5.86993056876950949 \cdot 10^{-45}\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{t}{\left(y - x\right) \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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)))