Average Error: 10.9 → 5.8
Time: 2.4s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.233204880322462 \cdot 10^{-264}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.307211939997339 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{b1}{a1}}{a2}}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 3.65318349007526422 \cdot 10^{92}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{b1}{a1}}{a2}}}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;b1 \cdot b2 = -\infty:\\
\;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\

\mathbf{elif}\;b1 \cdot b2 \le -4.233204880322462 \cdot 10^{-264}:\\
\;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\

\mathbf{elif}\;b1 \cdot b2 \le 1.307211939997339 \cdot 10^{-147}:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{b1}{a1}}{a2}}}{b2}\\

\mathbf{elif}\;b1 \cdot b2 \le 3.65318349007526422 \cdot 10^{92}:\\
\;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{b1}{a1}}{a2}}}{b2}\\

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r92088 = a1;
        double r92089 = a2;
        double r92090 = r92088 * r92089;
        double r92091 = b1;
        double r92092 = b2;
        double r92093 = r92091 * r92092;
        double r92094 = r92090 / r92093;
        return r92094;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r92095 = b1;
        double r92096 = b2;
        double r92097 = r92095 * r92096;
        double r92098 = -inf.0;
        bool r92099 = r92097 <= r92098;
        double r92100 = a1;
        double r92101 = r92100 / r92095;
        double r92102 = a2;
        double r92103 = r92102 / r92096;
        double r92104 = r92101 * r92103;
        double r92105 = -4.233204880322462e-264;
        bool r92106 = r92097 <= r92105;
        double r92107 = r92097 / r92102;
        double r92108 = r92100 / r92107;
        double r92109 = 1.3072119399973387e-147;
        bool r92110 = r92097 <= r92109;
        double r92111 = 1.0;
        double r92112 = r92095 / r92100;
        double r92113 = r92112 / r92102;
        double r92114 = r92111 / r92113;
        double r92115 = r92114 / r92096;
        double r92116 = 3.653183490075264e+92;
        bool r92117 = r92097 <= r92116;
        double r92118 = r92117 ? r92108 : r92115;
        double r92119 = r92110 ? r92115 : r92118;
        double r92120 = r92106 ? r92108 : r92119;
        double r92121 = r92099 ? r92104 : r92120;
        return r92121;
}

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

Original10.9
Target11.1
Herbie5.8
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* b1 b2) < -inf.0

    1. Initial program 20.9

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

    3. Applied times-frac2.0

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

    if -inf.0 < (* b1 b2) < -4.233204880322462e-264 or 1.3072119399973387e-147 < (* b1 b2) < 3.653183490075264e+92

    1. Initial program 4.6

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

    3. Applied associate-/l*4.9

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

    if -4.233204880322462e-264 < (* b1 b2) < 1.3072119399973387e-147 or 3.653183490075264e+92 < (* b1 b2)

    1. Initial program 18.8

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

    3. Applied associate-/r*11.6

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

    5. Applied clear-num11.8

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

    7. Applied associate-/r*8.3

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

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -4.233204880322462 \cdot 10^{-264}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.307211939997339 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{b1}{a1}}{a2}}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 3.65318349007526422 \cdot 10^{92}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{b1}{a1}}{a2}}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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