Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.4 → 2.1

Time: 17.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -7.797770989917811 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x - \frac{z}{t} \cdot x\right)\\ \mathbf{elif}\;y \le -1.8127477951727028 \cdot 10^{-264}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x - \frac{z}{t} \cdot x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r19651327 = x;
        double r19651328 = y;
        double r19651329 = r19651328 - r19651327;
        double r19651330 = z;
        double r19651331 = r19651329 * r19651330;
        double r19651332 = t;
        double r19651333 = r19651331 / r19651332;
        double r19651334 = r19651327 + r19651333;
        return r19651334;
}

↓

double f(double x, double y, double z, double t) {
        double r19651335 = y;
        double r19651336 = -7.797770989917811e-101;
        bool r19651337 = r19651335 <= r19651336;
        double r19651338 = z;
        double r19651339 = t;
        double r19651340 = r19651338 / r19651339;
        double r19651341 = x;
        double r19651342 = r19651340 * r19651341;
        double r19651343 = r19651341 - r19651342;
        double r19651344 = fma(r19651340, r19651335, r19651343);
        double r19651345 = -1.8127477951727028e-264;
        bool r19651346 = r19651335 <= r19651345;
        double r19651347 = r19651335 - r19651341;
        double r19651348 = r19651347 * r19651338;
        double r19651349 = r19651348 / r19651339;
        double r19651350 = r19651341 + r19651349;
        double r19651351 = r19651346 ? r19651350 : r19651344;
        double r19651352 = r19651337 ? r19651344 : r19651351;
        return r19651352;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.4
Target	1.8
Herbie	2.1

\[\begin{array}{l} \mathbf{if}\;x \lt -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -7.797770989917811e-101 or -1.8127477951727028e-264 < y

   1. Initial program 6.6

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

   2. Simplified 6.2

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

   3. Taylor expanded around 0 6.6

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

   4. Simplified 4.9

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

   6. Applied div-inv 4.9

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

   7. Applied associate-*l* 1.7

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

   8. Simplified 1.7

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

   if -7.797770989917811e-101 < y < -1.8127477951727028e-264

   1. Initial program 4.9

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.797770989917811 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x - \frac{z}{t} \cdot x\right)\\ \mathbf{elif}\;y \le -1.8127477951727028 \cdot 10^{-264}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x - \frac{z}{t} \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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)))