Average Error: 11.3 → 5.5
Time: 6.4s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.512192312330299 \cdot 10^{168}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \le -1.78311602156644772 \cdot 10^{-257}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 5.89224416336 \cdot 10^{-314}:\\ \;\;\;\;\frac{1}{\frac{\frac{b2}{a2}}{a1} \cdot b1}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.7927717050604466 \cdot 10^{257}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;b1 \cdot b2 \le -1.512192312330299 \cdot 10^{168}:\\
\;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\

\mathbf{elif}\;b1 \cdot b2 \le -1.78311602156644772 \cdot 10^{-257}:\\
\;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\

\mathbf{elif}\;b1 \cdot b2 \le 5.89224416336 \cdot 10^{-314}:\\
\;\;\;\;\frac{1}{\frac{\frac{b2}{a2}}{a1} \cdot b1}\\

\mathbf{elif}\;b1 \cdot b2 \le 1.7927717050604466 \cdot 10^{257}:\\
\;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r116244 = a1;
        double r116245 = a2;
        double r116246 = r116244 * r116245;
        double r116247 = b1;
        double r116248 = b2;
        double r116249 = r116247 * r116248;
        double r116250 = r116246 / r116249;
        return r116250;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r116251 = b1;
        double r116252 = b2;
        double r116253 = r116251 * r116252;
        double r116254 = -1.512192312330299e+168;
        bool r116255 = r116253 <= r116254;
        double r116256 = a1;
        double r116257 = a2;
        double r116258 = r116257 / r116252;
        double r116259 = r116256 * r116258;
        double r116260 = r116259 / r116251;
        double r116261 = -1.7831160215664477e-257;
        bool r116262 = r116253 <= r116261;
        double r116263 = r116256 * r116257;
        double r116264 = 1.0;
        double r116265 = r116264 / r116253;
        double r116266 = r116263 * r116265;
        double r116267 = 5.8922441633555e-314;
        bool r116268 = r116253 <= r116267;
        double r116269 = r116252 / r116257;
        double r116270 = r116269 / r116256;
        double r116271 = r116270 * r116251;
        double r116272 = r116264 / r116271;
        double r116273 = 1.7927717050604466e+257;
        bool r116274 = r116253 <= r116273;
        double r116275 = r116274 ? r116266 : r116260;
        double r116276 = r116268 ? r116272 : r116275;
        double r116277 = r116262 ? r116266 : r116276;
        double r116278 = r116255 ? r116260 : r116277;
        return r116278;
}

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

Original11.3
Target11.3
Herbie5.5
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* b1 b2) < -1.512192312330299e+168 or 1.7927717050604466e+257 < (* b1 b2)

    1. Initial program 15.6

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

    3. Applied associate-/r*7.8

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

    5. Applied clear-num8.4

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

    7. Applied associate-/r/8.6

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

    8. Applied associate-/r*8.0

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

    9. Simplified4.6

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

    if -1.512192312330299e+168 < (* b1 b2) < -1.7831160215664477e-257 or 5.8922441633555e-314 < (* b1 b2) < 1.7927717050604466e+257

    1. Initial program 5.4

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

    3. Applied div-inv5.5

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

    if -1.7831160215664477e-257 < (* b1 b2) < 5.8922441633555e-314

    1. Initial program 50.6

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

    3. Applied associate-/r*20.2

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

    5. Applied clear-num20.3

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

    7. Applied associate-/r/20.3

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

    8. Simplified7.9

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{b2}{a2}}{a1}} \cdot b1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.512192312330299 \cdot 10^{168}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \le -1.78311602156644772 \cdot 10^{-257}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 5.89224416336 \cdot 10^{-314}:\\ \;\;\;\;\frac{1}{\frac{\frac{b2}{a2}}{a1} \cdot b1}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.7927717050604466 \cdot 10^{257}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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