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

Average Error: 14.6 → 2.9

Time: 2.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.68209557109898762 \cdot 10^{256} \lor \neg \left(\frac{y}{z} \le -8.7578361305297263 \cdot 10^{-224}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r722673 = x;
        double r722674 = y;
        double r722675 = z;
        double r722676 = r722674 / r722675;
        double r722677 = t;
        double r722678 = r722676 * r722677;
        double r722679 = r722678 / r722677;
        double r722680 = r722673 * r722679;
        return r722680;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r722681 = y;
        double r722682 = z;
        double r722683 = r722681 / r722682;
        double r722684 = -1.6820955710989876e+256;
        bool r722685 = r722683 <= r722684;
        double r722686 = -8.757836130529726e-224;
        bool r722687 = r722683 <= r722686;
        double r722688 = !r722687;
        bool r722689 = r722685 || r722688;
        double r722690 = x;
        double r722691 = r722690 * r722681;
        double r722692 = 1.0;
        double r722693 = r722692 / r722682;
        double r722694 = r722691 * r722693;
        double r722695 = r722682 / r722681;
        double r722696 = r722690 / r722695;
        double r722697 = r722689 ? r722694 : r722696;
        return r722697;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.6
Target	1.7
Herbie	2.9

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

Derivation

1. Split input into 2 regimes

2. if (/ y z) < -1.6820955710989876e+256 or -8.757836130529726e-224 < (/ y z)

   1. Initial program 17.3

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

   2. Simplified 9.1

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

   4. Applied div-inv 9.2

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

   5. Applied associate-*r* 4.3

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

   if -1.6820955710989876e+256 < (/ y z) < -8.757836130529726e-224

   1. Initial program 9.3

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

   2. Simplified 0.2

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

   4. Applied associate-*r/ 8.9

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

   6. Applied associate-/l* 0.2

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.68209557109898762 \cdot 10^{256} \lor \neg \left(\frac{y}{z} \le -8.7578361305297263 \cdot 10^{-224}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(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)))