Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.6 → 2.2

Time: 23.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -5.562085220619439676639571842548833537494 \cdot 10^{-191} \lor \neg \left(x \le 9.895725275899004823919176625829480462873 \cdot 10^{-166}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - x\right) \cdot \frac{1}{t}, z, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r289184 = x;
        double r289185 = y;
        double r289186 = r289185 - r289184;
        double r289187 = z;
        double r289188 = r289186 * r289187;
        double r289189 = t;
        double r289190 = r289188 / r289189;
        double r289191 = r289184 + r289190;
        return r289191;
}

↓

double f(double x, double y, double z, double t) {
        double r289192 = x;
        double r289193 = -5.5620852206194397e-191;
        bool r289194 = r289192 <= r289193;
        double r289195 = 9.895725275899005e-166;
        bool r289196 = r289192 <= r289195;
        double r289197 = !r289196;
        bool r289198 = r289194 || r289197;
        double r289199 = y;
        double r289200 = r289199 - r289192;
        double r289201 = z;
        double r289202 = t;
        double r289203 = r289201 / r289202;
        double r289204 = r289200 * r289203;
        double r289205 = r289204 + r289192;
        double r289206 = 1.0;
        double r289207 = r289206 / r289202;
        double r289208 = r289200 * r289207;
        double r289209 = fma(r289208, r289201, r289192);
        double r289210 = r289198 ? r289205 : r289209;
        return r289210;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.6
Target	2.1
Herbie	2.2

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

Derivation

1. Split input into 2 regimes

2. if x < -5.5620852206194397e-191 or 9.895725275899005e-166 < x

   1. Initial program 6.9

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

   2. Simplified 6.9

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

   4. Applied div-inv 6.9

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

   6. Applied fma-udef 6.9

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

   7. Simplified 1.2

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

   if -5.5620852206194397e-191 < x < 9.895725275899005e-166

   1. Initial program 5.6

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

   2. Simplified 5.3

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

   4. Applied div-inv 5.4

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

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.562085220619439676639571842548833537494 \cdot 10^{-191} \lor \neg \left(x \le 9.895725275899004823919176625829480462873 \cdot 10^{-166}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - x\right) \cdot \frac{1}{t}, z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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)))