Average Error: 11.4 → 4.9
Time: 11.4s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.044587333497852823924878268479973587724 \cdot 10^{-294}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{1}{\frac{b1}{a1} \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 3.005399214482244746567528503375041565374 \cdot 10^{300}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{b1}}{\frac{b2}{a1}} \cdot a2\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.044587333497852823924878268479973587724 \cdot 10^{-294}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\
\;\;\;\;\frac{1}{\frac{b1}{a1} \cdot \frac{b2}{a2}}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 3.005399214482244746567528503375041565374 \cdot 10^{300}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r119991 = a1;
        double r119992 = a2;
        double r119993 = r119991 * r119992;
        double r119994 = b1;
        double r119995 = b2;
        double r119996 = r119994 * r119995;
        double r119997 = r119993 / r119996;
        return r119997;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r119998 = a1;
        double r119999 = a2;
        double r120000 = r119998 * r119999;
        double r120001 = b1;
        double r120002 = b2;
        double r120003 = r120001 * r120002;
        double r120004 = r120000 / r120003;
        double r120005 = -1.0445873334978528e-294;
        bool r120006 = r120004 <= r120005;
        double r120007 = 0.0;
        bool r120008 = r120004 <= r120007;
        double r120009 = 1.0;
        double r120010 = r120001 / r119998;
        double r120011 = r120002 / r119999;
        double r120012 = r120010 * r120011;
        double r120013 = r120009 / r120012;
        double r120014 = 3.005399214482245e+300;
        bool r120015 = r120004 <= r120014;
        double r120016 = r120009 / r120001;
        double r120017 = r120002 / r119998;
        double r120018 = r120016 / r120017;
        double r120019 = r120018 * r119999;
        double r120020 = r120015 ? r120004 : r120019;
        double r120021 = r120008 ? r120013 : r120020;
        double r120022 = r120006 ? r120004 : r120021;
        return r120022;
}

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
Target11.3
Herbie4.9
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* a1 a2) (* b1 b2)) < -1.0445873334978528e-294 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 3.005399214482245e+300

    1. Initial program 4.5

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

    if -1.0445873334978528e-294 < (/ (* a1 a2) (* b1 b2)) < 0.0

    1. Initial program 14.0

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

    3. Applied clear-num14.4

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

    4. Simplified3.5

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

    6. Applied pow13.5

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

    7. Applied pow13.5

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

    8. Applied pow-prod-down3.5

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

    9. Simplified3.8

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

    if 3.005399214482245e+300 < (/ (* a1 a2) (* b1 b2))

    1. Initial program 61.7

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

    3. Applied clear-num61.7

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

    4. Simplified6.4

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

    6. Applied associate-*r/13.4

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

    7. Applied associate-/r/13.8

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

    8. Simplified13.6

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

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.044587333497852823924878268479973587724 \cdot 10^{-294}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{1}{\frac{b1}{a1} \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 3.005399214482244746567528503375041565374 \cdot 10^{300}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{b1}}{\frac{b2}{a1}} \cdot a2\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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