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

Average Error: 3.5 → 0.4

Time: 9.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \le -3.25930114308438566002761827130284834267 \cdot 10^{51} \lor \neg \left(z \cdot 3 \le 4698335802131639211560781442472125558424000\right):\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \frac{t}{z \cdot y}, x\right) - 0.3333333333333333148296162562473909929395 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r647378 = x;
        double r647379 = y;
        double r647380 = z;
        double r647381 = 3.0;
        double r647382 = r647380 * r647381;
        double r647383 = r647379 / r647382;
        double r647384 = r647378 - r647383;
        double r647385 = t;
        double r647386 = r647382 * r647379;
        double r647387 = r647385 / r647386;
        double r647388 = r647384 + r647387;
        return r647388;
}

↓

double f(double x, double y, double z, double t) {
        double r647389 = z;
        double r647390 = 3.0;
        double r647391 = r647389 * r647390;
        double r647392 = -3.2593011430843857e+51;
        bool r647393 = r647391 <= r647392;
        double r647394 = 4.698335802131639e+42;
        bool r647395 = r647391 <= r647394;
        double r647396 = !r647395;
        bool r647397 = r647393 || r647396;
        double r647398 = 0.3333333333333333;
        double r647399 = t;
        double r647400 = y;
        double r647401 = r647389 * r647400;
        double r647402 = r647399 / r647401;
        double r647403 = x;
        double r647404 = fma(r647398, r647402, r647403);
        double r647405 = r647400 / r647389;
        double r647406 = r647398 * r647405;
        double r647407 = r647404 - r647406;
        double r647408 = r647400 / r647391;
        double r647409 = r647403 - r647408;
        double r647410 = 1.0;
        double r647411 = r647410 / r647389;
        double r647412 = r647399 / r647390;
        double r647413 = r647400 / r647412;
        double r647414 = r647411 / r647413;
        double r647415 = r647409 + r647414;
        double r647416 = r647397 ? r647407 : r647415;
        return r647416;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	3.5
Target	1.8
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (* z 3.0) < -3.2593011430843857e+51 or 4.698335802131639e+42 < (* 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

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

   3. Simplified 0.4

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

   if -3.2593011430843857e+51 < (* z 3.0) < 4.698335802131639e+42

   1. Initial program 7.6

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

   3. Applied associate-/r* 2.6

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

   5. Applied *-un-lft-identity 2.6

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

   6. Applied times-frac 2.6

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

   7. Applied associate-/l* 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \le -3.25930114308438566002761827130284834267 \cdot 10^{51} \lor \neg \left(z \cdot 3 \le 4698335802131639211560781442472125558424000\right):\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \frac{t}{z \cdot y}, x\right) - 0.3333333333333333148296162562473909929395 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +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))))