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

Average Error: 14.7 → 1.8

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -4.55389167634140265 \cdot 10^{-149}:\\ \;\;\;\;1 \cdot \frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 1.4704572909860111 \cdot 10^{-239}:\\ \;\;\;\;1 \cdot \frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 7.35260291402330968 \cdot 10^{277}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{1}{\frac{z}{x \cdot y}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r114973 = x;
        double r114974 = y;
        double r114975 = z;
        double r114976 = r114974 / r114975;
        double r114977 = t;
        double r114978 = r114976 * r114977;
        double r114979 = r114978 / r114977;
        double r114980 = r114973 * r114979;
        return r114980;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r114981 = y;
        double r114982 = z;
        double r114983 = r114981 / r114982;
        double r114984 = -4.553891676341403e-149;
        bool r114985 = r114983 <= r114984;
        double r114986 = 1.0;
        double r114987 = x;
        double r114988 = r114982 / r114981;
        double r114989 = r114987 / r114988;
        double r114990 = r114986 * r114989;
        double r114991 = 1.470457290986011e-239;
        bool r114992 = r114983 <= r114991;
        double r114993 = r114987 * r114981;
        double r114994 = r114993 / r114982;
        double r114995 = r114986 * r114994;
        double r114996 = 7.35260291402331e+277;
        bool r114997 = r114983 <= r114996;
        double r114998 = r114987 * r114983;
        double r114999 = r114982 / r114993;
        double r115000 = r114986 / r114999;
        double r115001 = r114986 * r115000;
        double r115002 = r114997 ? r114998 : r115001;
        double r115003 = r114992 ? r114995 : r115002;
        double r115004 = r114985 ? r114990 : r115003;
        return r115004;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Split input into 4 regimes

2. if (/ y z) < -4.553891676341403e-149

   1. Initial program 13.5

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

   2. Simplified 5.0

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

   4. Applied add-cube-cbrt 6.0

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

   5. Applied *-un-lft-identity 6.0

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

   6. Applied times-frac 6.0

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

   7. Applied associate-*r* 7.4

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

   8. Simplified 7.4

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

   10. Applied *-un-lft-identity 7.4

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

   11. Applied associate-*l* 7.4

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

   12. Simplified 9.7

       \[\leadsto 1 \cdot \color{blue}{\frac{x \cdot y}{z}}\]
   13. Using strategy rm

   14. Applied associate-/l* 4.4

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

   if -4.553891676341403e-149 < (/ y z) < 1.470457290986011e-239

   1. Initial program 17.2

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

   2. Simplified 10.4

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

   4. Applied add-cube-cbrt 10.7

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

   5. Applied *-un-lft-identity 10.7

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

   6. Applied times-frac 10.7

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

   7. Applied associate-*r* 2.6

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

   8. Simplified 2.6

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

   10. Applied *-un-lft-identity 2.6

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

   11. Applied associate-*l* 2.6

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

   12. Simplified 0.9

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

   if 1.470457290986011e-239 < (/ y z) < 7.35260291402331e+277

   1. Initial program 9.8

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

   2. Simplified 0.2

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

   if 7.35260291402331e+277 < (/ y z)

   1. Initial program 55.4

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

   2. Simplified 49.4

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

   4. Applied add-cube-cbrt 49.7

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

   5. Applied *-un-lft-identity 49.7

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

   6. Applied times-frac 49.7

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

   7. Applied associate-*r* 12.6

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

   8. Simplified 12.5

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

   10. Applied *-un-lft-identity 12.5

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

   11. Applied associate-*l* 12.5

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

   12. Simplified 0.3

       \[\leadsto 1 \cdot \color{blue}{\frac{x \cdot y}{z}}\]
   13. Using strategy rm

   14. Applied clear-num 0.4

       \[\leadsto 1 \cdot \color{blue}{\frac{1}{\frac{z}{x \cdot y}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -4.55389167634140265 \cdot 10^{-149}:\\ \;\;\;\;1 \cdot \frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 1.4704572909860111 \cdot 10^{-239}:\\ \;\;\;\;1 \cdot \frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 7.35260291402330968 \cdot 10^{277}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{1}{\frac{z}{x \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(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)))