Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.5 → 1.3

Time: 21.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 2.589721366200231900739645404502092751865 \cdot 10^{279}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r70380729 = x;
        double r70380730 = y;
        double r70380731 = r70380730 - r70380729;
        double r70380732 = z;
        double r70380733 = r70380731 * r70380732;
        double r70380734 = t;
        double r70380735 = r70380733 / r70380734;
        double r70380736 = r70380729 + r70380735;
        return r70380736;
}

↓

double f(double x, double y, double z, double t) {
        double r70380737 = x;
        double r70380738 = y;
        double r70380739 = r70380738 - r70380737;
        double r70380740 = z;
        double r70380741 = r70380739 * r70380740;
        double r70380742 = t;
        double r70380743 = r70380741 / r70380742;
        double r70380744 = r70380737 + r70380743;
        double r70380745 = -inf.0;
        bool r70380746 = r70380744 <= r70380745;
        double r70380747 = r70380739 / r70380742;
        double r70380748 = fma(r70380747, r70380740, r70380737);
        double r70380749 = 2.589721366200232e+279;
        bool r70380750 = r70380744 <= r70380749;
        double r70380751 = r70380750 ? r70380744 : r70380748;
        double r70380752 = r70380746 ? r70380748 : r70380751;
        return r70380752;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.5
Target	2.3
Herbie	1.3

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

Derivation

1. Split input into 2 regimes

2. if (+ x (/ (* (- y x) z) t)) < -inf.0 or 2.589721366200232e+279 < (+ x (/ (* (- y x) z) t))

   1. Initial program 48.0

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

   2. Simplified 5.4

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

   if -inf.0 < (+ x (/ (* (- y x) z) t)) < 2.589721366200232e+279

   1. Initial program 0.8

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 2.589721366200231900739645404502092751865 \cdot 10^{279}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +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)))