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

Average Error: 14.9 → 6.5

Time: 7.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r93473 = x;
        double r93474 = y;
        double r93475 = z;
        double r93476 = r93474 / r93475;
        double r93477 = t;
        double r93478 = r93476 * r93477;
        double r93479 = r93478 / r93477;
        double r93480 = r93473 * r93479;
        return r93480;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r93481 = x;
        double r93482 = y;
        double r93483 = r93481 * r93482;
        double r93484 = z;
        double r93485 = r93483 / r93484;
        return r93485;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -1.19966857532151e+150 or 2.848942305436544e+181 < (/ y z)

   1. Initial program 36.7

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

   2. Simplified 20.1

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

   4. Applied add-cube-cbrt 20.9

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

   5. Applied *-un-lft-identity 20.9

      \[\leadsto x \cdot \frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

   6. Applied times-frac 20.9

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\right)}\]

   7. Applied associate-*r* 6.5

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{y}{\sqrt[3]{z}}}\]

   8. Simplified 6.5

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

   10. Applied *-un-lft-identity 6.5

       \[\leadsto \color{blue}{\left(1 \cdot \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)} \cdot \frac{y}{\sqrt[3]{z}}\]

   11. Applied associate-*l* 6.5

       \[\leadsto \color{blue}{1 \cdot \left(\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\right)}\]

   12. Simplified 2.3

       \[\leadsto 1 \cdot \color{blue}{\frac{x \cdot y}{z}}\]

   if -1.19966857532151e+150 < (/ y z) < -2.898468428317233e-278 or 0.0 < (/ y z) < 2.848942305436544e+181

   1. Initial program 9.9

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

   2. Simplified 2.2

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

   if -2.898468428317233e-278 < (/ y z) < 0.0

   1. Initial program 20.1

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

   2. Simplified 16.7

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

   4. Applied div-inv 16.7

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

   5. Applied associate-*r* 0.1

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.5

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

Reproduce

herbie shell --seed 2019308 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  :precision binary64
  (* x (/ (* (/ y z) t) t)))