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

Average Error: 7.8 → 1.0

Time: 4.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 3.3253015755175787 \cdot 10^{290}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x \cdot y\right) \cdot \frac{1}{a}\right) - \frac{4.5 \cdot \left(t \cdot z\right)}{a}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r687231 = x;
        double r687232 = y;
        double r687233 = r687231 * r687232;
        double r687234 = z;
        double r687235 = 9.0;
        double r687236 = r687234 * r687235;
        double r687237 = t;
        double r687238 = r687236 * r687237;
        double r687239 = r687233 - r687238;
        double r687240 = a;
        double r687241 = 2.0;
        double r687242 = r687240 * r687241;
        double r687243 = r687239 / r687242;
        return r687243;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r687244 = x;
        double r687245 = y;
        double r687246 = r687244 * r687245;
        double r687247 = z;
        double r687248 = 9.0;
        double r687249 = r687247 * r687248;
        double r687250 = t;
        double r687251 = r687249 * r687250;
        double r687252 = r687246 - r687251;
        double r687253 = -inf.0;
        bool r687254 = r687252 <= r687253;
        double r687255 = 3.325301575517579e+290;
        bool r687256 = r687252 <= r687255;
        double r687257 = !r687256;
        bool r687258 = r687254 || r687257;
        double r687259 = 0.5;
        double r687260 = a;
        double r687261 = cbrt(r687260);
        double r687262 = r687261 * r687261;
        double r687263 = r687244 / r687262;
        double r687264 = r687245 / r687261;
        double r687265 = r687263 * r687264;
        double r687266 = r687259 * r687265;
        double r687267 = 4.5;
        double r687268 = r687260 / r687247;
        double r687269 = r687250 / r687268;
        double r687270 = r687267 * r687269;
        double r687271 = r687266 - r687270;
        double r687272 = 1.0;
        double r687273 = r687272 / r687260;
        double r687274 = r687246 * r687273;
        double r687275 = r687259 * r687274;
        double r687276 = r687250 * r687247;
        double r687277 = r687267 * r687276;
        double r687278 = r687277 / r687260;
        double r687279 = r687275 - r687278;
        double r687280 = r687258 ? r687271 : r687279;
        return r687280;
}

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	5.5
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if (- (* x y) (* (* z 9.0) t)) < -inf.0 or 3.325301575517579e+290 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 60.0

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

   2. Taylor expanded around 0 59.2

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

   4. Applied add-cube-cbrt 59.2

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}} - 4.5 \cdot \frac{t \cdot z}{a}\]

   5. Applied times-frac 33.3

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right)} - 4.5 \cdot \frac{t \cdot z}{a}\]
   6. Using strategy rm

   7. Applied associate-/l* 1.1

      \[\leadsto 0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - 4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}}\]

   if -inf.0 < (- (* x y) (* (* z 9.0) t)) < 3.325301575517579e+290

   1. Initial program 0.9

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

   2. Taylor expanded around 0 0.9

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

   4. Applied associate-*r/ 0.9

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

   6. Applied div-inv 1.0

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(x \cdot y\right) \cdot \frac{1}{a}\right)} - \frac{4.5 \cdot \left(t \cdot z\right)}{a}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 3.3253015755175787 \cdot 10^{290}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x \cdot y\right) \cdot \frac{1}{a}\right) - \frac{4.5 \cdot \left(t \cdot z\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :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 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9) t)) (* a 2)))