Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B

Average Error: 13.9 → 3.0

Time: 3.4s

Precision: 64

Math

\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -9.8899108197747985 \cdot 10^{-269} \lor \neg \left(\frac{y}{z} \le 1.8387934390842539 \cdot 10^{-189}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r524827 = x;
        double r524828 = y;
        double r524829 = z;
        double r524830 = r524828 / r524829;
        double r524831 = t;
        double r524832 = r524830 * r524831;
        double r524833 = r524832 / r524831;
        double r524834 = r524827 * r524833;
        return r524834;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r524835 = y;
        double r524836 = z;
        double r524837 = r524835 / r524836;
        double r524838 = -9.889910819774798e-269;
        bool r524839 = r524837 <= r524838;
        double r524840 = 1.838793439084254e-189;
        bool r524841 = r524837 <= r524840;
        double r524842 = !r524841;
        bool r524843 = r524839 || r524842;
        double r524844 = x;
        double r524845 = r524836 / r524835;
        double r524846 = r524844 / r524845;
        double r524847 = r524844 * r524835;
        double r524848 = r524847 / r524836;
        double r524849 = r524843 ? r524846 : r524848;
        return r524849;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	13.9
Target	1.8
Herbie	3.0

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \lt -1.20672205123045005 \cdot 10^{245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt -5.90752223693390633 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 5.65895442315341522 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 2.0087180502407133 \cdot 10^{217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ y z) < -9.889910819774798e-269 or 1.838793439084254e-189 < (/ y z)

   1. Initial program 13.2

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 4.1

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
   3. Using strategy rm

   4. Applied associate-*r/ 8.0

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}}\]
   5. Using strategy rm

   6. Applied associate-/l* 3.8

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]

   if -9.889910819774798e-269 < (/ y z) < 1.838793439084254e-189

   1. Initial program 15.9

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 10.8

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
   3. Using strategy rm

   4. Applied associate-*r/ 0.5

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -9.8899108197747985 \cdot 10^{-269} \lor \neg \left(\frac{y}{z} \le 1.8387934390842539 \cdot 10^{-189}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
  :precision binary64

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

  (* x (/ (* (/ y z) t) t)))