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

Average Error: 3.4 → 1.7

Time: 5.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -6.40431597217532072 \cdot 10^{-100}:\\ \;\;\;\;\left(x - \frac{1}{z} \cdot \frac{y}{3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r668091 = x;
        double r668092 = y;
        double r668093 = z;
        double r668094 = 3.0;
        double r668095 = r668093 * r668094;
        double r668096 = r668092 / r668095;
        double r668097 = r668091 - r668096;
        double r668098 = t;
        double r668099 = r668095 * r668092;
        double r668100 = r668098 / r668099;
        double r668101 = r668097 + r668100;
        return r668101;
}

↓

double f(double x, double y, double z, double t) {
        double r668102 = z;
        double r668103 = -6.404315972175321e-100;
        bool r668104 = r668102 <= r668103;
        double r668105 = x;
        double r668106 = 1.0;
        double r668107 = r668106 / r668102;
        double r668108 = y;
        double r668109 = 3.0;
        double r668110 = r668108 / r668109;
        double r668111 = r668107 * r668110;
        double r668112 = r668105 - r668111;
        double r668113 = t;
        double r668114 = r668102 * r668109;
        double r668115 = r668114 * r668108;
        double r668116 = r668113 / r668115;
        double r668117 = r668112 + r668116;
        double r668118 = r668108 / r668102;
        double r668119 = r668118 / r668109;
        double r668120 = r668105 - r668119;
        double r668121 = r668113 / r668114;
        double r668122 = r668121 / r668108;
        double r668123 = r668120 + r668122;
        double r668124 = r668104 ? r668117 : r668123;
        return r668124;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	3.4
Target	1.8
Herbie	1.7

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

Derivation

1. Split input into 2 regimes

2. if z < -6.404315972175321e-100

   1. Initial program 0.9

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

   3. Applied *-un-lft-identity 0.9

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

   4. Applied times-frac 1.0

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

   if -6.404315972175321e-100 < z

   1. Initial program 5.3

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

   3. Applied associate-/r* 2.1

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

   5. Applied associate-/r* 2.2

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

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.40431597217532072 \cdot 10^{-100}:\\ \;\;\;\;\left(x - \frac{1}{z} \cdot \frac{y}{3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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))))