Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.6 → 2.2

Time: 23.5s

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 r341733 = x;
        double r341734 = y;
        double r341735 = r341734 - r341733;
        double r341736 = z;
        double r341737 = r341735 * r341736;
        double r341738 = t;
        double r341739 = r341737 / r341738;
        double r341740 = r341733 + r341739;
        return r341740;
}

↓

double f(double x, double y, double z, double t) {
        double r341741 = x;
        double r341742 = -5.5620852206194397e-191;
        bool r341743 = r341741 <= r341742;
        double r341744 = 9.895725275899005e-166;
        bool r341745 = r341741 <= r341744;
        double r341746 = !r341745;
        bool r341747 = r341743 || r341746;
        double r341748 = y;
        double r341749 = r341748 - r341741;
        double r341750 = z;
        double r341751 = t;
        double r341752 = r341750 / r341751;
        double r341753 = r341749 * r341752;
        double r341754 = r341753 + r341741;
        double r341755 = 1.0;
        double r341756 = r341755 / r341751;
        double r341757 = r341749 * r341756;
        double r341758 = fma(r341757, r341750, r341741);
        double r341759 = r341747 ? r341754 : r341758;
        return r341759;
}

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)))