Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.6 → 1.9

Time: 2.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.15003287352819712 \cdot 10^{-238} \lor \neg \left(x \le 1.0116038839391772 \cdot 10^{-300}\right):\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r476280 = x;
        double r476281 = y;
        double r476282 = r476281 - r476280;
        double r476283 = z;
        double r476284 = r476282 * r476283;
        double r476285 = t;
        double r476286 = r476284 / r476285;
        double r476287 = r476280 + r476286;
        return r476287;
}

↓

double f(double x, double y, double z, double t) {
        double r476288 = x;
        double r476289 = -1.1500328735281971e-238;
        bool r476290 = r476288 <= r476289;
        double r476291 = 1.0116038839391772e-300;
        bool r476292 = r476288 <= r476291;
        double r476293 = !r476292;
        bool r476294 = r476290 || r476293;
        double r476295 = z;
        double r476296 = t;
        double r476297 = r476295 / r476296;
        double r476298 = y;
        double r476299 = r476298 - r476288;
        double r476300 = r476297 * r476299;
        double r476301 = r476300 + r476288;
        double r476302 = r476299 / r476296;
        double r476303 = fma(r476302, r476295, r476288);
        double r476304 = r476294 ? r476301 : r476303;
        return r476304;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.6
Target	1.9
Herbie	1.9

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

Derivation

1. Split input into 2 regimes

2. if x < -1.1500328735281971e-238 or 1.0116038839391772e-300 < x

   1. Initial program 6.7

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

   2. Simplified 6.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)}\]
   3. Using strategy rm

   4. Applied div-inv 6.5

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

   5. Taylor expanded around 0 6.7

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

   6. Simplified 5.6

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

   7. Taylor expanded around 0 6.7

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

   8. Simplified 1.7

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

   if -1.1500328735281971e-238 < x < 1.0116038839391772e-300

   1. Initial program 5.6

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

   2. Simplified 4.6

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.15003287352819712 \cdot 10^{-238} \lor \neg \left(x \le 1.0116038839391772 \cdot 10^{-300}\right):\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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)))