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

Average Error: 14.7 → 1.8

Time: 3.8s

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 r698070 = x;
        double r698071 = y;
        double r698072 = z;
        double r698073 = r698071 / r698072;
        double r698074 = t;
        double r698075 = r698073 * r698074;
        double r698076 = r698075 / r698074;
        double r698077 = r698070 * r698076;
        return r698077;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r698078 = y;
        double r698079 = z;
        double r698080 = r698078 / r698079;
        double r698081 = -4.553891676341403e-149;
        bool r698082 = r698080 <= r698081;
        double r698083 = 1.0;
        double r698084 = x;
        double r698085 = r698079 / r698078;
        double r698086 = r698084 / r698085;
        double r698087 = r698083 * r698086;
        double r698088 = 1.470457290986011e-239;
        bool r698089 = r698080 <= r698088;
        double r698090 = r698084 * r698078;
        double r698091 = r698090 / r698079;
        double r698092 = r698083 * r698091;
        double r698093 = 7.35260291402331e+277;
        bool r698094 = r698080 <= r698093;
        double r698095 = r698084 * r698080;
        double r698096 = r698079 / r698090;
        double r698097 = r698083 / r698096;
        double r698098 = r698083 * r698097;
        double r698099 = r698094 ? r698095 : r698098;
        double r698100 = r698089 ? r698092 : r698099;
        double r698101 = r698082 ? r698087 : r698100;
        return r698101;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.7
Target	1.6
Herbie	1.8

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

Derivation

1. Split input into 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 
(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)))