Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.5 → 0.7

Time: 2.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 5.341209302805126244331680067677643806813 \cdot 10^{306}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r387372 = x;
        double r387373 = y;
        double r387374 = r387373 - r387372;
        double r387375 = z;
        double r387376 = r387374 * r387375;
        double r387377 = t;
        double r387378 = r387376 / r387377;
        double r387379 = r387372 + r387378;
        return r387379;
}

↓

double f(double x, double y, double z, double t) {
        double r387380 = x;
        double r387381 = y;
        double r387382 = r387381 - r387380;
        double r387383 = z;
        double r387384 = r387382 * r387383;
        double r387385 = t;
        double r387386 = r387384 / r387385;
        double r387387 = r387380 + r387386;
        double r387388 = -inf.0;
        bool r387389 = r387387 <= r387388;
        double r387390 = r387382 / r387385;
        double r387391 = fma(r387390, r387383, r387380);
        double r387392 = 5.341209302805126e+306;
        bool r387393 = r387387 <= r387392;
        double r387394 = r387385 / r387383;
        double r387395 = r387382 / r387394;
        double r387396 = r387380 + r387395;
        double r387397 = r387393 ? r387387 : r387396;
        double r387398 = r387389 ? r387391 : r387397;
        return r387398;
}

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	0.7

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Simplified 0.2

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

   if -inf.0 < (+ x (/ (* (- y x) z) t)) < 5.341209302805126e+306

   1. Initial program 0.7

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

   if 5.341209302805126e+306 < (+ x (/ (* (- y x) z) t))

   1. Initial program 62.1

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

   3. Applied associate-/l* 0.2

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 5.341209302805126244331680067677643806813 \cdot 10^{306}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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)))