Average Error: 11.3 → 2.4
Time: 15.1s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.414144362639176854803385722786278913018 \cdot 10^{-314} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 1.431802419250923180709933384670076216432 \cdot 10^{298}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\
\;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.414144362639176854803385722786278913018 \cdot 10^{-314} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 1.431802419250923180709933384670076216432 \cdot 10^{298}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r97693 = a1;
        double r97694 = a2;
        double r97695 = r97693 * r97694;
        double r97696 = b1;
        double r97697 = b2;
        double r97698 = r97696 * r97697;
        double r97699 = r97695 / r97698;
        return r97699;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r97700 = a1;
        double r97701 = a2;
        double r97702 = r97700 * r97701;
        double r97703 = b1;
        double r97704 = b2;
        double r97705 = r97703 * r97704;
        double r97706 = r97702 / r97705;
        double r97707 = -inf.0;
        bool r97708 = r97706 <= r97707;
        double r97709 = r97701 / r97704;
        double r97710 = r97709 / r97703;
        double r97711 = r97700 * r97710;
        double r97712 = -1.4141443626392e-314;
        bool r97713 = r97706 <= r97712;
        double r97714 = 0.0;
        bool r97715 = r97706 <= r97714;
        double r97716 = !r97715;
        double r97717 = 1.4318024192509232e+298;
        bool r97718 = r97706 <= r97717;
        bool r97719 = r97716 && r97718;
        bool r97720 = r97713 || r97719;
        double r97721 = r97700 / r97703;
        double r97722 = r97721 * r97709;
        double r97723 = r97720 ? r97706 : r97722;
        double r97724 = r97708 ? r97711 : r97723;
        return r97724;
}

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
Target11.2
Herbie2.4
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* a1 a2) (* b1 b2)) < -inf.0

    1. Initial program 64.0

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

    3. Applied times-frac11.2

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

    5. Applied div-inv11.3

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

    6. Applied associate-*l*19.0

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

    7. Simplified18.9

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

    if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -1.4141443626392e-314 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 1.4318024192509232e+298

    1. Initial program 0.9

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

    if -1.4141443626392e-314 < (/ (* a1 a2) (* b1 b2)) < 0.0 or 1.4318024192509232e+298 < (/ (* a1 a2) (* b1 b2))

    1. Initial program 22.7

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

    3. Applied times-frac3.2

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.414144362639176854803385722786278913018 \cdot 10^{-314} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 1.431802419250923180709933384670076216432 \cdot 10^{298}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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