Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, H

Average Error: 3.5 → 1.0

Time: 4.7s

Precision: 64

Math

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.23853620324777987 \cdot 10^{141}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{3} \cdot \frac{\frac{1}{y}}{z}\\ \mathbf{elif}\;t \le 8.57969581197539482 \cdot 10^{48}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + 1 \cdot \frac{\frac{t}{z \cdot y}}{3}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r805809 = x;
        double r805810 = y;
        double r805811 = z;
        double r805812 = 3.0;
        double r805813 = r805811 * r805812;
        double r805814 = r805810 / r805813;
        double r805815 = r805809 - r805814;
        double r805816 = t;
        double r805817 = r805813 * r805810;
        double r805818 = r805816 / r805817;
        double r805819 = r805815 + r805818;
        return r805819;
}

↓

double f(double x, double y, double z, double t) {
        double r805820 = t;
        double r805821 = -1.2385362032477799e+141;
        bool r805822 = r805820 <= r805821;
        double r805823 = x;
        double r805824 = y;
        double r805825 = z;
        double r805826 = 3.0;
        double r805827 = r805825 * r805826;
        double r805828 = r805824 / r805827;
        double r805829 = r805823 - r805828;
        double r805830 = r805820 / r805826;
        double r805831 = 1.0;
        double r805832 = r805831 / r805824;
        double r805833 = r805832 / r805825;
        double r805834 = r805830 * r805833;
        double r805835 = r805829 + r805834;
        double r805836 = 8.579695811975395e+48;
        bool r805837 = r805820 <= r805836;
        double r805838 = r805820 / r805827;
        double r805839 = r805838 / r805824;
        double r805840 = r805829 + r805839;
        double r805841 = r805824 / r805825;
        double r805842 = r805841 / r805826;
        double r805843 = r805823 - r805842;
        double r805844 = r805825 * r805824;
        double r805845 = r805820 / r805844;
        double r805846 = r805845 / r805826;
        double r805847 = r805831 * r805846;
        double r805848 = r805843 + r805847;
        double r805849 = r805837 ? r805840 : r805848;
        double r805850 = r805822 ? r805835 : r805849;
        return r805850;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	3.5
Target	1.9
Herbie	1.0

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\]

Derivation

1. Split input into 3 regimes

2. if t < -1.2385362032477799e+141

   1. Initial program 1.1

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
   2. Using strategy rm

   3. Applied pow1 1.1

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

   4. Applied pow1 1.1

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

   5. Applied pow1 1.1

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

   6. Applied pow-prod-down 1.1

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

   7. Applied pow-prod-down 1.1

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

   8. Simplified 1.0

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{{\color{blue}{\left(3 \cdot \left(z \cdot y\right)\right)}}^{1}}\]
   9. Using strategy rm

   10. Applied clear-num 1.1

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

   11. Simplified 1.0

       \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{1}{\color{blue}{\frac{3}{\frac{t}{z \cdot y}}}}\]
   12. Using strategy rm

   13. Applied associate-/r/ 1.1

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

   14. Applied add-cube-cbrt 1.1

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

   15. Applied times-frac 1.1

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

   16. Simplified 1.0

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

   17. Simplified 1.0

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

   if -1.2385362032477799e+141 < t < 8.579695811975395e+48

   1. Initial program 4.6

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
   2. Using strategy rm

   3. Applied associate-/r* 1.0

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

   if 8.579695811975395e+48 < t

   1. Initial program 0.6

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
   2. Using strategy rm

   3. Applied pow1 0.6

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

   4. Applied pow1 0.6

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

   5. Applied pow1 0.6

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

   6. Applied pow-prod-down 0.6

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

   7. Applied pow-prod-down 0.6

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

   8. Simplified 0.6

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{{\color{blue}{\left(3 \cdot \left(z \cdot y\right)\right)}}^{1}}\]
   9. Using strategy rm

   10. Applied clear-num 0.6

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

   11. Simplified 0.8

       \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{1}{\color{blue}{\frac{3}{\frac{t}{z \cdot y}}}}\]
   12. Using strategy rm

   13. Applied *-un-lft-identity 0.8

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

   14. Applied *-un-lft-identity 0.8

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

   15. Applied times-frac 0.8

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

   16. Applied add-cube-cbrt 0.8

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

   17. Applied times-frac 0.8

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

   18. Simplified 0.8

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

   19. Simplified 0.7

       \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + 1 \cdot \color{blue}{\frac{\frac{t}{z \cdot y}}{3}}\]
   20. Using strategy rm

   21. Applied associate-/r* 0.8

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.23853620324777987 \cdot 10^{141}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{3} \cdot \frac{\frac{1}{y}}{z}\\ \mathbf{elif}\;t \le 8.57969581197539482 \cdot 10^{48}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + 1 \cdot \frac{\frac{t}{z \cdot y}}{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3))) (/ (/ t (* z 3)) y))

  (+ (- x (/ y (* z 3))) (/ t (* (* z 3) y))))