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

Average Error: 3.5 → 1.0

Time: 9.3s

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 r4348 = x;
        double r4349 = y;
        double r4350 = z;
        double r4351 = 3.0;
        double r4352 = r4350 * r4351;
        double r4353 = r4349 / r4352;
        double r4354 = r4348 - r4353;
        double r4355 = t;
        double r4356 = r4352 * r4349;
        double r4357 = r4355 / r4356;
        double r4358 = r4354 + r4357;
        return r4358;
}

↓

double f(double x, double y, double z, double t) {
        double r4359 = t;
        double r4360 = -1.2385362032477799e+141;
        bool r4361 = r4359 <= r4360;
        double r4362 = x;
        double r4363 = y;
        double r4364 = z;
        double r4365 = 3.0;
        double r4366 = r4364 * r4365;
        double r4367 = r4363 / r4366;
        double r4368 = r4362 - r4367;
        double r4369 = r4359 / r4365;
        double r4370 = 1.0;
        double r4371 = r4370 / r4363;
        double r4372 = r4371 / r4364;
        double r4373 = r4369 * r4372;
        double r4374 = r4368 + r4373;
        double r4375 = 8.579695811975395e+48;
        bool r4376 = r4359 <= r4375;
        double r4377 = r4359 / r4366;
        double r4378 = r4377 / r4363;
        double r4379 = r4368 + r4378;
        double r4380 = r4363 / r4364;
        double r4381 = r4380 / r4365;
        double r4382 = r4362 - r4381;
        double r4383 = r4364 * r4363;
        double r4384 = r4359 / r4383;
        double r4385 = r4384 / r4365;
        double r4386 = r4370 * r4385;
        double r4387 = r4382 + r4386;
        double r4388 = r4376 ? r4379 : r4387;
        double r4389 = r4361 ? r4374 : r4388;
        return r4389;
}

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))))