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

Average Error: 3.7 → 0.5

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.982808370584645728390571403324870832662 \cdot 10^{-59} \lor \neg \left(z \le 1.637385439772020732742719657482536703395 \cdot 10^{-52}\right):\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(1, x, \mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \frac{1}{z} \cdot \frac{\frac{t}{3}}{y}\right) - \frac{\frac{y}{3}}{z}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r696190 = x;
        double r696191 = y;
        double r696192 = z;
        double r696193 = 3.0;
        double r696194 = r696192 * r696193;
        double r696195 = r696191 / r696194;
        double r696196 = r696190 - r696195;
        double r696197 = t;
        double r696198 = r696194 * r696191;
        double r696199 = r696197 / r696198;
        double r696200 = r696196 + r696199;
        return r696200;
}

↓

double f(double x, double y, double z, double t) {
        double r696201 = z;
        double r696202 = -2.9828083705846457e-59;
        bool r696203 = r696201 <= r696202;
        double r696204 = 1.6373854397720207e-52;
        bool r696205 = r696201 <= r696204;
        double r696206 = !r696205;
        bool r696207 = r696203 || r696206;
        double r696208 = x;
        double r696209 = y;
        double r696210 = r696209 / r696201;
        double r696211 = 3.0;
        double r696212 = r696210 / r696211;
        double r696213 = r696208 - r696212;
        double r696214 = t;
        double r696215 = r696201 * r696211;
        double r696216 = r696215 * r696209;
        double r696217 = r696214 / r696216;
        double r696218 = r696213 + r696217;
        double r696219 = 1.0;
        double r696220 = r696219 / r696201;
        double r696221 = r696209 / r696211;
        double r696222 = -r696221;
        double r696223 = r696222 + r696221;
        double r696224 = r696214 / r696211;
        double r696225 = r696224 / r696209;
        double r696226 = r696220 * r696225;
        double r696227 = fma(r696220, r696223, r696226);
        double r696228 = r696221 / r696201;
        double r696229 = r696227 - r696228;
        double r696230 = fma(r696219, r696208, r696229);
        double r696231 = r696219 * r696230;
        double r696232 = r696207 ? r696218 : r696231;
        return r696232;
}

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 < -2.9828083705846457e-59 or 1.6373854397720207e-52 < 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-/r* 0.5

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

   if -2.9828083705846457e-59 < z < 1.6373854397720207e-52

   1. Initial program 13.7

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

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

   6. Applied times-frac 3.6

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

   7. Applied *-un-lft-identity 3.6

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

   8. Applied prod-diff 3.6

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

   9. Applied associate-+l+ 3.6

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

   10. Simplified 3.6

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

   12. Applied *-un-lft-identity 3.6

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

   13. Applied *-un-lft-identity 3.6

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

   14. Applied distribute-lft-out 3.6

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

   15. Simplified 3.6

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

   17. Applied *-un-lft-identity 3.6

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

   18. Applied *-un-lft-identity 3.6

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

   19. Applied times-frac 3.6

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

   20. Applied times-frac 0.3

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

   21. Simplified 0.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.982808370584645728390571403324870832662 \cdot 10^{-59} \lor \neg \left(z \le 1.637385439772020732742719657482536703395 \cdot 10^{-52}\right):\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(1, x, \mathsf{fma}\left(\frac{1}{z}, \left(-\frac{y}{3}\right) + \frac{y}{3}, \frac{1}{z} \cdot \frac{\frac{t}{3}}{y}\right) - \frac{\frac{y}{3}}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))))