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

Average Error: 3.6 → 0.6

Time: 41.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \le -7.959906075858907547200374344342532215009 \cdot 10^{-64}:\\ \;\;\;\;\left(\frac{1}{\frac{y}{\frac{t}{z \cdot 3}}} - \frac{\frac{y}{3}}{z}\right) + x\\ \mathbf{elif}\;z \cdot 3 \le 4741144259583525519360:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{1}{z \cdot 3} \cdot \frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.3333333333333333148296162562473909929395 \cdot \left(\frac{t}{y \cdot z} - \frac{y}{z}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r32856529 = x;
        double r32856530 = y;
        double r32856531 = z;
        double r32856532 = 3.0;
        double r32856533 = r32856531 * r32856532;
        double r32856534 = r32856530 / r32856533;
        double r32856535 = r32856529 - r32856534;
        double r32856536 = t;
        double r32856537 = r32856533 * r32856530;
        double r32856538 = r32856536 / r32856537;
        double r32856539 = r32856535 + r32856538;
        return r32856539;
}

↓

double f(double x, double y, double z, double t) {
        double r32856540 = z;
        double r32856541 = 3.0;
        double r32856542 = r32856540 * r32856541;
        double r32856543 = -7.959906075858908e-64;
        bool r32856544 = r32856542 <= r32856543;
        double r32856545 = 1.0;
        double r32856546 = y;
        double r32856547 = t;
        double r32856548 = r32856547 / r32856542;
        double r32856549 = r32856546 / r32856548;
        double r32856550 = r32856545 / r32856549;
        double r32856551 = r32856546 / r32856541;
        double r32856552 = r32856551 / r32856540;
        double r32856553 = r32856550 - r32856552;
        double r32856554 = x;
        double r32856555 = r32856553 + r32856554;
        double r32856556 = 4.741144259583526e+21;
        bool r32856557 = r32856542 <= r32856556;
        double r32856558 = r32856546 / r32856542;
        double r32856559 = r32856554 - r32856558;
        double r32856560 = r32856545 / r32856542;
        double r32856561 = r32856547 / r32856546;
        double r32856562 = r32856560 * r32856561;
        double r32856563 = r32856559 + r32856562;
        double r32856564 = 0.3333333333333333;
        double r32856565 = r32856546 * r32856540;
        double r32856566 = r32856547 / r32856565;
        double r32856567 = r32856546 / r32856540;
        double r32856568 = r32856566 - r32856567;
        double r32856569 = r32856564 * r32856568;
        double r32856570 = r32856554 + r32856569;
        double r32856571 = r32856557 ? r32856563 : r32856570;
        double r32856572 = r32856544 ? r32856555 : r32856571;
        return r32856572;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	3.6
Target	1.8
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if (* z 3.0) < -7.959906075858908e-64

   1. Initial program 0.4

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

   3. Applied sub-neg 0.4

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

   4. Applied associate-+l+ 0.4

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

   5. Simplified 1.1

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

   7. Applied associate-/r* 1.1

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

   9. Applied clear-num 1.1

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

   if -7.959906075858908e-64 < (* z 3.0) < 4.741144259583526e+21

   1. Initial program 11.2

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

   3. Applied *-un-lft-identity 11.2

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

   4. Applied times-frac 0.3

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

   if 4.741144259583526e+21 < (* z 3.0)

   1. Initial program 0.4

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

   2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\left(x + 0.3333333333333333148296162562473909929395 \cdot \frac{t}{z \cdot y}\right) - 0.3333333333333333148296162562473909929395 \cdot \frac{y}{z}}\]

   3. Simplified 0.4

      \[\leadsto \color{blue}{0.3333333333333333148296162562473909929395 \cdot \left(\frac{t}{z \cdot y} - \frac{y}{z}\right) + x}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \le -7.959906075858907547200374344342532215009 \cdot 10^{-64}:\\ \;\;\;\;\left(\frac{1}{\frac{y}{\frac{t}{z \cdot 3}}} - \frac{\frac{y}{3}}{z}\right) + x\\ \mathbf{elif}\;z \cdot 3 \le 4741144259583525519360:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{1}{z \cdot 3} \cdot \frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.3333333333333333148296162562473909929395 \cdot \left(\frac{t}{y \cdot z} - \frac{y}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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))))