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

Average Error: 15.0 → 6.3

Time: 13.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r77922 = x;
        double r77923 = y;
        double r77924 = z;
        double r77925 = r77923 / r77924;
        double r77926 = t;
        double r77927 = r77925 * r77926;
        double r77928 = r77927 / r77926;
        double r77929 = r77922 * r77928;
        return r77929;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r77930 = x;
        double r77931 = y;
        double r77932 = z;
        double r77933 = r77931 / r77932;
        double r77934 = r77930 * r77933;
        return r77934;
}

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) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   4. Applied div-inv 64.0

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

   5. Applied associate-*r* 0.3

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

   if -inf.0 < (/ y z) < -4.471661983018054e-179 or 5.2143480045202775e-160 < (/ y z) < 8.142281339311981e+291

   1. Initial program 10.1

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

   2. Simplified 0.2

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

   if -4.471661983018054e-179 < (/ y z) < 5.2143480045202775e-160 or 8.142281339311981e+291 < (/ y z)

   1. Initial program 20.5

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

   2. Simplified 13.3

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

   4. Applied add-cube-cbrt 13.6

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

   5. Applied *-un-lft-identity 13.6

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

   6. Applied times-frac 13.6

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

   7. Applied associate-*r* 3.6

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

   8. Simplified 3.6

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

   10. Applied associate-*r/ 1.1

       \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot y}{\sqrt[3]{z}}}\]

   11. Simplified 3.6

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

   13. Applied frac-times 1.4

       \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}}{\sqrt[3]{z}}\]

   14. Applied associate-/l/ 1.4

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

   15. Simplified 0.8

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

4. Final simplification 6.3

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

Reproduce

herbie shell --seed 2019298 
(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)))