Average Error: 11.4 → 5.4
Time: 4.0s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -2.12735229765193038 \cdot 10^{182}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le -1.0024754663520561 \cdot 10^{-263}:\\ \;\;\;\;\frac{\left(a1 \cdot a2\right) \cdot \frac{1}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.48183825434908302 \cdot 10^{-297} \lor \neg \left(a1 \cdot a2 \le 2.32454057704183092 \cdot 10^{123}\right):\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;a1 \cdot a2 \le -2.12735229765193038 \cdot 10^{182}:\\
\;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\

\mathbf{elif}\;a1 \cdot a2 \le -1.0024754663520561 \cdot 10^{-263}:\\
\;\;\;\;\frac{\left(a1 \cdot a2\right) \cdot \frac{1}{b2}}{b1}\\

\mathbf{elif}\;a1 \cdot a2 \le 5.48183825434908302 \cdot 10^{-297} \lor \neg \left(a1 \cdot a2 \le 2.32454057704183092 \cdot 10^{123}\right):\\
\;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r233283 = a1;
        double r233284 = a2;
        double r233285 = r233283 * r233284;
        double r233286 = b1;
        double r233287 = b2;
        double r233288 = r233286 * r233287;
        double r233289 = r233285 / r233288;
        return r233289;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r233290 = a1;
        double r233291 = a2;
        double r233292 = r233290 * r233291;
        double r233293 = -2.1273522976519304e+182;
        bool r233294 = r233292 <= r233293;
        double r233295 = b1;
        double r233296 = b2;
        double r233297 = r233296 / r233291;
        double r233298 = r233295 * r233297;
        double r233299 = r233290 / r233298;
        double r233300 = -1.002475466352056e-263;
        bool r233301 = r233292 <= r233300;
        double r233302 = 1.0;
        double r233303 = r233302 / r233296;
        double r233304 = r233292 * r233303;
        double r233305 = r233304 / r233295;
        double r233306 = 5.481838254349083e-297;
        bool r233307 = r233292 <= r233306;
        double r233308 = 2.324540577041831e+123;
        bool r233309 = r233292 <= r233308;
        double r233310 = !r233309;
        bool r233311 = r233307 || r233310;
        double r233312 = r233292 / r233295;
        double r233313 = r233312 / r233296;
        double r233314 = r233311 ? r233299 : r233313;
        double r233315 = r233301 ? r233305 : r233314;
        double r233316 = r233294 ? r233299 : r233315;
        return r233316;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.4
Target11.0
Herbie5.4
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a1 a2) < -2.1273522976519304e+182 or -1.002475466352056e-263 < (* a1 a2) < 5.481838254349083e-297 or 2.324540577041831e+123 < (* a1 a2)

    1. Initial program 22.5

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

    3. Applied associate-/l*11.6

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

    4. Simplified6.9

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

    if -2.1273522976519304e+182 < (* a1 a2) < -1.002475466352056e-263

    1. Initial program 4.8

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

    3. Applied times-frac13.9

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

    5. Applied associate-*l/11.0

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

    7. Applied div-inv11.0

      \[\leadsto \frac{a1 \cdot \color{blue}{\left(a2 \cdot \frac{1}{b2}\right)}}{b1}\]

    8. Applied associate-*r*4.6

      \[\leadsto \frac{\color{blue}{\left(a1 \cdot a2\right) \cdot \frac{1}{b2}}}{b1}\]

    if 5.481838254349083e-297 < (* a1 a2) < 2.324540577041831e+123

    1. Initial program 4.8

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

    3. Applied associate-/r*4.4

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

  4. Final simplification5.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -2.12735229765193038 \cdot 10^{182}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le -1.0024754663520561 \cdot 10^{-263}:\\ \;\;\;\;\frac{\left(a1 \cdot a2\right) \cdot \frac{1}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.48183825434908302 \cdot 10^{-297} \lor \neg \left(a1 \cdot a2 \le 2.32454057704183092 \cdot 10^{123}\right):\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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