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

Average Error: 7.8 → 1.5

Time: 31.7s

Precision: 64

Math

\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -5.8428229986270315 \cdot 10^{139} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 4.93643854715891061 \cdot 10^{207}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r475221 = x;
        double r475222 = y;
        double r475223 = r475221 * r475222;
        double r475224 = z;
        double r475225 = 9.0;
        double r475226 = r475224 * r475225;
        double r475227 = t;
        double r475228 = r475226 * r475227;
        double r475229 = r475223 - r475228;
        double r475230 = a;
        double r475231 = 2.0;
        double r475232 = r475230 * r475231;
        double r475233 = r475229 / r475232;
        return r475233;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r475234 = x;
        double r475235 = y;
        double r475236 = r475234 * r475235;
        double r475237 = z;
        double r475238 = 9.0;
        double r475239 = r475237 * r475238;
        double r475240 = t;
        double r475241 = r475239 * r475240;
        double r475242 = r475236 - r475241;
        double r475243 = -5.842822998627031e+139;
        bool r475244 = r475242 <= r475243;
        double r475245 = 4.9364385471589106e+207;
        bool r475246 = r475242 <= r475245;
        double r475247 = !r475246;
        bool r475248 = r475244 || r475247;
        double r475249 = 0.5;
        double r475250 = a;
        double r475251 = r475235 / r475250;
        double r475252 = r475234 * r475251;
        double r475253 = r475249 * r475252;
        double r475254 = 4.5;
        double r475255 = r475254 * r475240;
        double r475256 = r475237 / r475250;
        double r475257 = r475255 * r475256;
        double r475258 = r475253 - r475257;
        double r475259 = 1.0;
        double r475260 = 2.0;
        double r475261 = r475250 * r475260;
        double r475262 = r475259 / r475261;
        double r475263 = r475242 * r475262;
        double r475264 = r475248 ? r475258 : r475263;
        return r475264;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.8
Target	5.4
Herbie	1.5

\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (* x y) (* (* z 9.0) t)) < -5.842822998627031e+139 or 4.9364385471589106e+207 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 24.2

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 23.9

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 23.9

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{\color{blue}{1 \cdot a}}\]

   5. Applied times-frac 13.9

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\left(\frac{t}{1} \cdot \frac{z}{a}\right)}\]

   6. Applied associate-*r* 14.0

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(4.5 \cdot \frac{t}{1}\right) \cdot \frac{z}{a}}\]

   7. Simplified 14.0

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(4.5 \cdot t\right)} \cdot \frac{z}{a}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 14.0

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{\color{blue}{1 \cdot a}} - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\]

   10. Applied times-frac 2.9

       \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{y}{a}\right)} - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\]

   11. Simplified 2.9

       \[\leadsto 0.5 \cdot \left(\color{blue}{x} \cdot \frac{y}{a}\right) - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\]

   if -5.842822998627031e+139 < (- (* x y) (* (* z 9.0) t)) < 4.9364385471589106e+207

   1. Initial program 0.9

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
   2. Using strategy rm

   3. Applied div-inv 1.0

      \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -5.8428229986270315 \cdot 10^{139} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 4.93643854715891061 \cdot 10^{207}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))