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

Average Error: 14.8 → 0.8

Time: 3.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.03192616381755189 \cdot 10^{-277}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 7.34626099988 \cdot 10^{-313}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 2.1777827215712954 \cdot 10^{98}:\\ \;\;\;\;{\left(\frac{x}{\frac{z}{y}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r531009 = x;
        double r531010 = y;
        double r531011 = z;
        double r531012 = r531010 / r531011;
        double r531013 = t;
        double r531014 = r531012 * r531013;
        double r531015 = r531014 / r531013;
        double r531016 = r531009 * r531015;
        return r531016;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r531017 = y;
        double r531018 = z;
        double r531019 = r531017 / r531018;
        double r531020 = -inf.0;
        bool r531021 = r531019 <= r531020;
        double r531022 = x;
        double r531023 = r531022 * r531017;
        double r531024 = 1.0;
        double r531025 = r531024 / r531018;
        double r531026 = r531023 * r531025;
        double r531027 = -1.0319261638175519e-277;
        bool r531028 = r531019 <= r531027;
        double r531029 = r531022 * r531019;
        double r531030 = 7.3462609998833e-313;
        bool r531031 = r531019 <= r531030;
        double r531032 = r531023 / r531018;
        double r531033 = pow(r531032, r531024);
        double r531034 = 2.1777827215712954e+98;
        bool r531035 = r531019 <= r531034;
        double r531036 = r531018 / r531017;
        double r531037 = r531022 / r531036;
        double r531038 = pow(r531037, r531024);
        double r531039 = r531035 ? r531038 : r531033;
        double r531040 = r531031 ? r531033 : r531039;
        double r531041 = r531028 ? r531029 : r531040;
        double r531042 = r531021 ? r531026 : r531041;
        return r531042;
}

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 4 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) < -1.0319261638175519e-277

   1. Initial program 11.1

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

   2. Simplified 0.2

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

   if -1.0319261638175519e-277 < (/ y z) < 7.3462609998833e-313 or 2.1777827215712954e+98 < (/ y z)

   1. Initial program 23.4

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

   2. Simplified 16.6

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

   4. Applied add-cube-cbrt 16.9

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

   5. Applied *-un-lft-identity 16.9

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

   6. Applied times-frac 17.0

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

   7. Applied associate-*r* 4.1

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

   8. Simplified 4.1

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

   10. Applied pow1 4.1

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

   11. Applied pow1 4.1

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

   12. Applied pow-prod-down 4.1

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

   13. Simplified 2.0

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

   if 7.3462609998833e-313 < (/ y z) < 2.1777827215712954e+98

   1. Initial program 8.2

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

   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* 6.1

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

   8. Simplified 6.1

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

   10. Applied pow1 6.1

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

   11. Applied pow1 6.1

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

   12. Applied pow-prod-down 6.1

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

   13. Simplified 8.1

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

   15. Applied associate-/l* 0.4

       \[\leadsto {\color{blue}{\left(\frac{x}{\frac{z}{y}}\right)}}^{1}\]
3. Recombined 4 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.03192616381755189 \cdot 10^{-277}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 7.34626099988 \cdot 10^{-313}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 2.1777827215712954 \cdot 10^{98}:\\ \;\;\;\;{\left(\frac{x}{\frac{z}{y}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(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)))