Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A

Average Error: 10.8 → 1.6

Time: 20.8s

Precision: 64

Math

\[\frac{x - y \cdot z}{t - a \cdot z}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.665105640857500107938904410076474960078 \cdot 10^{-4} \lor \neg \left(z \le 56659057657764167215480832\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{x}{t - a \cdot z}} \cdot \sqrt[3]{\frac{x}{t - a \cdot z}}\right) \cdot \sqrt[3]{\frac{x}{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \left(y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r584209 = x;
        double r584210 = y;
        double r584211 = z;
        double r584212 = r584210 * r584211;
        double r584213 = r584209 - r584212;
        double r584214 = t;
        double r584215 = a;
        double r584216 = r584215 * r584211;
        double r584217 = r584214 - r584216;
        double r584218 = r584213 / r584217;
        return r584218;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r584219 = z;
        double r584220 = -0.00046651056408575;
        bool r584221 = r584219 <= r584220;
        double r584222 = 5.665905765776417e+25;
        bool r584223 = r584219 <= r584222;
        double r584224 = !r584223;
        bool r584225 = r584221 || r584224;
        double r584226 = x;
        double r584227 = t;
        double r584228 = a;
        double r584229 = r584228 * r584219;
        double r584230 = r584227 - r584229;
        double r584231 = r584226 / r584230;
        double r584232 = cbrt(r584231);
        double r584233 = r584232 * r584232;
        double r584234 = r584233 * r584232;
        double r584235 = y;
        double r584236 = r584227 / r584219;
        double r584237 = r584236 - r584228;
        double r584238 = r584235 / r584237;
        double r584239 = r584234 - r584238;
        double r584240 = r584235 * r584219;
        double r584241 = 1.0;
        double r584242 = r584241 / r584230;
        double r584243 = r584240 * r584242;
        double r584244 = r584231 - r584243;
        double r584245 = r584225 ? r584239 : r584244;
        return r584245;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.8
Target	1.6
Herbie	1.6

\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -0.00046651056408575 or 5.665905765776417e+25 < z

   1. Initial program 22.0

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
   2. Using strategy rm

   3. Applied div-sub 22.0

      \[\leadsto \color{blue}{\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}}\]

   4. Simplified 13.9

      \[\leadsto \frac{x}{t - a \cdot z} - \color{blue}{y \cdot \frac{z}{t - a \cdot z}}\]
   5. Using strategy rm

   6. Applied pow1 13.9

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

   7. Applied pow1 13.9

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

   8. Applied pow-prod-down 13.9

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

   9. Simplified 2.6

      \[\leadsto \frac{x}{t - a \cdot z} - {\color{blue}{\left(\frac{y}{\frac{t}{z} - a}\right)}}^{1}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 2.9

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

   if -0.00046651056408575 < z < 5.665905765776417e+25

   1. Initial program 0.3

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
   2. Using strategy rm

   3. Applied div-sub 0.3

      \[\leadsto \color{blue}{\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}}\]

   4. Simplified 2.8

      \[\leadsto \frac{x}{t - a \cdot z} - \color{blue}{y \cdot \frac{z}{t - a \cdot z}}\]
   5. Using strategy rm

   6. Applied div-inv 2.9

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

   7. Applied associate-*r* 0.3

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.665105640857500107938904410076474960078 \cdot 10^{-4} \lor \neg \left(z \le 56659057657764167215480832\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{x}{t - a \cdot z}} \cdot \sqrt[3]{\frac{x}{t - a \cdot z}}\right) \cdot \sqrt[3]{\frac{x}{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \left(y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.51395223729782958e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))