Quotient of products

Average Error: 11.5 → 2.3

Time: 4.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{a1 \cdot a2}{b1 \cdot b2}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -8.356672191202089856632868574789828922924 \cdot 10^{302}:\\ \;\;\;\;\frac{1}{\frac{b1}{a1} \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -6.627043998342215562621926848971746206689 \cdot 10^{-272}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 6.467713212022452866787113437472850598121 \cdot 10^{306}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b1}{a1} \cdot \frac{b2}{a2}}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r125284 = a1;
        double r125285 = a2;
        double r125286 = r125284 * r125285;
        double r125287 = b1;
        double r125288 = b2;
        double r125289 = r125287 * r125288;
        double r125290 = r125286 / r125289;
        return r125290;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r125291 = a1;
        double r125292 = a2;
        double r125293 = r125291 * r125292;
        double r125294 = b1;
        double r125295 = b2;
        double r125296 = r125294 * r125295;
        double r125297 = r125293 / r125296;
        double r125298 = -8.35667219120209e+302;
        bool r125299 = r125297 <= r125298;
        double r125300 = 1.0;
        double r125301 = r125294 / r125291;
        double r125302 = r125295 / r125292;
        double r125303 = r125301 * r125302;
        double r125304 = r125300 / r125303;
        double r125305 = -6.627043998342216e-272;
        bool r125306 = r125297 <= r125305;
        double r125307 = 0.0;
        bool r125308 = r125297 <= r125307;
        double r125309 = r125291 / r125294;
        double r125310 = r125292 / r125295;
        double r125311 = r125309 * r125310;
        double r125312 = 6.467713212022453e+306;
        bool r125313 = r125297 <= r125312;
        double r125314 = r125313 ? r125297 : r125304;
        double r125315 = r125308 ? r125311 : r125314;
        double r125316 = r125306 ? r125297 : r125315;
        double r125317 = r125299 ? r125304 : r125316;
        return r125317;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.5
Target	11.5
Herbie	2.3

\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -8.35667219120209e+302 or 6.467713212022453e+306 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 63.1

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
   2. Using strategy rm

   3. Applied clear-num 63.1

      \[\leadsto \color{blue}{\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}}\]
   4. Using strategy rm

   5. Applied times-frac 7.5

      \[\leadsto \frac{1}{\color{blue}{\frac{b1}{a1} \cdot \frac{b2}{a2}}}\]

   if -8.35667219120209e+302 < (/ (* a1 a2) (* b1 b2)) < -6.627043998342216e-272 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 6.467713212022453e+306

   1. Initial program 3.6

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
   2. Using strategy rm

   3. Applied clear-num 3.7

      \[\leadsto \color{blue}{\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}}\]

   4. Taylor expanded around 0 3.6

      \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}}\]

   if -6.627043998342216e-272 < (/ (* a1 a2) (* b1 b2)) < 0.0

   1. Initial program 12.3

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
   2. Using strategy rm

   3. Applied times-frac 4.3

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -8.356672191202089856632868574789828922924 \cdot 10^{302}:\\ \;\;\;\;\frac{1}{\frac{b1}{a1} \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -6.627043998342215562621926848971746206689 \cdot 10^{-272}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 6.467713212022452866787113437472850598121 \cdot 10^{306}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b1}{a1} \cdot \frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

  :herbie-target
  (* (/ a1 b1) (/ a2 b2))

  (/ (* a1 a2) (* b1 b2)))