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

Average Error: 3.7 → 0.5

Time: 21.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.225043477922090241184253274805260914534 \cdot 10^{-45}:\\ \;\;\;\;\frac{t}{z \cdot \left(y \cdot 3\right)} + \left(x - \frac{y}{3 \cdot z}\right)\\ \mathbf{elif}\;z \le 4.972548460705735309806705638542917117847 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{t}{3}}{y} + \left(x - \frac{y}{3 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z \cdot \left(y \cdot 3\right)} + \left(x - \frac{y}{3 \cdot z}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r35399565 = x;
        double r35399566 = y;
        double r35399567 = z;
        double r35399568 = 3.0;
        double r35399569 = r35399567 * r35399568;
        double r35399570 = r35399566 / r35399569;
        double r35399571 = r35399565 - r35399570;
        double r35399572 = t;
        double r35399573 = r35399569 * r35399566;
        double r35399574 = r35399572 / r35399573;
        double r35399575 = r35399571 + r35399574;
        return r35399575;
}

↓

double f(double x, double y, double z, double t) {
        double r35399576 = z;
        double r35399577 = -2.2250434779220902e-45;
        bool r35399578 = r35399576 <= r35399577;
        double r35399579 = t;
        double r35399580 = y;
        double r35399581 = 3.0;
        double r35399582 = r35399580 * r35399581;
        double r35399583 = r35399576 * r35399582;
        double r35399584 = r35399579 / r35399583;
        double r35399585 = x;
        double r35399586 = r35399581 * r35399576;
        double r35399587 = r35399580 / r35399586;
        double r35399588 = r35399585 - r35399587;
        double r35399589 = r35399584 + r35399588;
        double r35399590 = 4.972548460705735e-55;
        bool r35399591 = r35399576 <= r35399590;
        double r35399592 = 1.0;
        double r35399593 = r35399592 / r35399576;
        double r35399594 = r35399579 / r35399581;
        double r35399595 = r35399594 / r35399580;
        double r35399596 = r35399593 * r35399595;
        double r35399597 = r35399596 + r35399588;
        double r35399598 = r35399591 ? r35399597 : r35399589;
        double r35399599 = r35399578 ? r35399589 : r35399598;
        return r35399599;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	3.7
Target	1.6
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if z < -2.2250434779220902e-45 or 4.972548460705735e-55 < z

   1. Initial program 0.5

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

   3. Applied associate-*l* 0.5

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

   if -2.2250434779220902e-45 < z < 4.972548460705735e-55

   1. Initial program 13.5

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

   3. Applied associate-/r* 3.6

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

   5. Applied *-un-lft-identity 3.6

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

   6. Applied *-un-lft-identity 3.6

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

   7. Applied times-frac 3.6

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

   8. Applied times-frac 0.3

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

   9. Simplified 0.3

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{1}{z}} \cdot \frac{\frac{t}{3}}{y}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.225043477922090241184253274805260914534 \cdot 10^{-45}:\\ \;\;\;\;\frac{t}{z \cdot \left(y \cdot 3\right)} + \left(x - \frac{y}{3 \cdot z}\right)\\ \mathbf{elif}\;z \le 4.972548460705735309806705638542917117847 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{t}{3}}{y} + \left(x - \frac{y}{3 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z \cdot \left(y \cdot 3\right)} + \left(x - \frac{y}{3 \cdot z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"

  :herbie-target
  (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))