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

Average Error: 14.8 → 0.8

Time: 3.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -3.06893781831439348 \cdot 10^{259}:\\ \;\;\;\;{\left(\left(x \cdot y\right) \cdot \frac{1}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le -9.6741057542884294 \cdot 10^{-296}:\\ \;\;\;\;{\left(\frac{x}{\frac{z}{y}}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 3.1054900882765682 \cdot 10^{-293}:\\ \;\;\;\;{\left(\left(x \cdot y\right) \cdot \frac{1}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 3.9706404122678694 \cdot 10^{93}:\\ \;\;\;\;{\left(\frac{x}{\frac{z}{y}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\frac{z}{x \cdot y}}\right)}^{1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r490086 = x;
        double r490087 = y;
        double r490088 = z;
        double r490089 = r490087 / r490088;
        double r490090 = t;
        double r490091 = r490089 * r490090;
        double r490092 = r490091 / r490090;
        double r490093 = r490086 * r490092;
        return r490093;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r490094 = y;
        double r490095 = z;
        double r490096 = r490094 / r490095;
        double r490097 = -3.0689378183143935e+259;
        bool r490098 = r490096 <= r490097;
        double r490099 = x;
        double r490100 = r490099 * r490094;
        double r490101 = 1.0;
        double r490102 = r490101 / r490095;
        double r490103 = r490100 * r490102;
        double r490104 = pow(r490103, r490101);
        double r490105 = -9.674105754288429e-296;
        bool r490106 = r490096 <= r490105;
        double r490107 = r490095 / r490094;
        double r490108 = r490099 / r490107;
        double r490109 = pow(r490108, r490101);
        double r490110 = 3.105490088276568e-293;
        bool r490111 = r490096 <= r490110;
        double r490112 = 3.9706404122678694e+93;
        bool r490113 = r490096 <= r490112;
        double r490114 = r490095 / r490100;
        double r490115 = r490101 / r490114;
        double r490116 = pow(r490115, r490101);
        double r490117 = r490113 ? r490109 : r490116;
        double r490118 = r490111 ? r490104 : r490117;
        double r490119 = r490106 ? r490109 : r490118;
        double r490120 = r490098 ? r490104 : r490119;
        return r490120;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.8
Target	1.6
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -3.0689378183143935e+259 or -9.674105754288429e-296 < (/ y z) < 3.105490088276568e-293

   1. Initial program 25.8

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

   2. Simplified 22.1

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

   4. Applied add-cube-cbrt 22.3

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

   5. Applied *-un-lft-identity 22.3

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

   6. Applied times-frac 22.2

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

   7. Applied associate-*r* 4.4

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

   8. Simplified 4.4

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

   10. Applied pow1 4.4

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

   11. Applied pow1 4.4

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

   12. Applied pow-prod-down 4.4

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

   13. Simplified 0.1

       \[\leadsto {\color{blue}{\left(\frac{x \cdot y}{z}\right)}}^{1}\]
   14. Using strategy rm

   15. Applied div-inv 0.2

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

   if -3.0689378183143935e+259 < (/ y z) < -9.674105754288429e-296 or 3.105490088276568e-293 < (/ y z) < 3.9706404122678694e+93

   1. Initial program 9.4

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

   2. Simplified 0.2

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

   4. Applied add-cube-cbrt 1.2

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

   5. Applied *-un-lft-identity 1.2

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

   6. Applied times-frac 1.2

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

   7. Applied associate-*r* 5.9

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

   8. Simplified 5.9

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

   10. Applied pow1 5.9

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

   11. Applied pow1 5.9

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

   12. Applied pow-prod-down 5.9

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

   13. Simplified 8.1

       \[\leadsto {\color{blue}{\left(\frac{x \cdot y}{z}\right)}}^{1}\]
   14. Using strategy rm

   15. Applied associate-/l* 0.2

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

   if 3.9706404122678694e+93 < (/ y z)

   1. Initial program 27.5

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

   2. Simplified 13.7

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

   4. Applied add-cube-cbrt 14.6

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

   5. Applied *-un-lft-identity 14.6

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

   6. Applied times-frac 14.6

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

   7. Applied associate-*r* 4.6

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

   8. Simplified 4.5

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

   10. Applied pow1 4.5

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

   11. Applied pow1 4.5

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

   12. Applied pow-prod-down 4.5

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

   13. Simplified 4.8

       \[\leadsto {\color{blue}{\left(\frac{x \cdot y}{z}\right)}}^{1}\]
   14. Using strategy rm

   15. Applied clear-num 4.9

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -3.06893781831439348 \cdot 10^{259}:\\ \;\;\;\;{\left(\left(x \cdot y\right) \cdot \frac{1}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le -9.6741057542884294 \cdot 10^{-296}:\\ \;\;\;\;{\left(\frac{x}{\frac{z}{y}}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 3.1054900882765682 \cdot 10^{-293}:\\ \;\;\;\;{\left(\left(x \cdot y\right) \cdot \frac{1}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 3.9706404122678694 \cdot 10^{93}:\\ \;\;\;\;{\left(\frac{x}{\frac{z}{y}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\frac{z}{x \cdot y}}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(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)))