Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.5 → 0.9

Time: 38.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 4.520106774411556088249689065409096453316 \cdot 10^{300}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r17781283 = x;
        double r17781284 = y;
        double r17781285 = r17781284 - r17781283;
        double r17781286 = z;
        double r17781287 = r17781285 * r17781286;
        double r17781288 = t;
        double r17781289 = r17781287 / r17781288;
        double r17781290 = r17781283 + r17781289;
        return r17781290;
}

↓

double f(double x, double y, double z, double t) {
        double r17781291 = x;
        double r17781292 = y;
        double r17781293 = r17781292 - r17781291;
        double r17781294 = z;
        double r17781295 = r17781293 * r17781294;
        double r17781296 = t;
        double r17781297 = r17781295 / r17781296;
        double r17781298 = r17781291 + r17781297;
        double r17781299 = -inf.0;
        bool r17781300 = r17781298 <= r17781299;
        double r17781301 = r17781293 / r17781296;
        double r17781302 = fma(r17781301, r17781294, r17781291);
        double r17781303 = 4.520106774411556e+300;
        bool r17781304 = r17781298 <= r17781303;
        double r17781305 = r17781304 ? r17781298 : r17781302;
        double r17781306 = r17781300 ? r17781302 : r17781305;
        return r17781306;
}

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	0.9

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

Derivation

1. Split input into 2 regimes

2. if (+ x (/ (* (- y x) z) t)) < -inf.0 or 4.520106774411556e+300 < (+ x (/ (* (- y x) z) t))

   1. Initial program 60.6

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

   2. Simplified 1.5

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

   if -inf.0 < (+ x (/ (* (- y x) z) t)) < 4.520106774411556e+300

   1. Initial program 0.8

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 4.520106774411556088249689065409096453316 \cdot 10^{300}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"

  :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)))