Average Error: 11.3 → 5.0
Time: 3.0s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -6.08712058696868874 \cdot 10^{188}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -3.0178565167744215 \cdot 10^{-307}:\\ \;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{\frac{1}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.3291942180717 \cdot 10^{-313}:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{1} \cdot \left(\frac{\sqrt[3]{a1}}{b1} \cdot \frac{a2}{b2}\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 4.48816418979761226 \cdot 10^{154}:\\ \;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{\frac{1}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;a1 \cdot a2 \le -6.08712058696868874 \cdot 10^{188}:\\
\;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\

\mathbf{elif}\;a1 \cdot a2 \le -3.0178565167744215 \cdot 10^{-307}:\\
\;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{\frac{1}{b2}}{b1}\\

\mathbf{elif}\;a1 \cdot a2 \le 5.3291942180717 \cdot 10^{-313}:\\
\;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{1} \cdot \left(\frac{\sqrt[3]{a1}}{b1} \cdot \frac{a2}{b2}\right)\\

\mathbf{elif}\;a1 \cdot a2 \le 4.48816418979761226 \cdot 10^{154}:\\
\;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{\frac{1}{b2}}{b1}\\

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

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r124282 = a1;
        double r124283 = a2;
        double r124284 = r124282 * r124283;
        double r124285 = b1;
        double r124286 = b2;
        double r124287 = r124285 * r124286;
        double r124288 = r124284 / r124287;
        return r124288;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r124289 = a1;
        double r124290 = a2;
        double r124291 = r124289 * r124290;
        double r124292 = -6.087120586968689e+188;
        bool r124293 = r124291 <= r124292;
        double r124294 = b2;
        double r124295 = r124290 / r124294;
        double r124296 = b1;
        double r124297 = r124295 / r124296;
        double r124298 = r124289 * r124297;
        double r124299 = -3.0178565167744215e-307;
        bool r124300 = r124291 <= r124299;
        double r124301 = r124290 * r124289;
        double r124302 = 1.0;
        double r124303 = r124302 / r124294;
        double r124304 = r124303 / r124296;
        double r124305 = r124301 * r124304;
        double r124306 = 5.3291942180717e-313;
        bool r124307 = r124291 <= r124306;
        double r124308 = cbrt(r124289);
        double r124309 = r124308 * r124308;
        double r124310 = r124309 / r124302;
        double r124311 = r124308 / r124296;
        double r124312 = r124311 * r124295;
        double r124313 = r124310 * r124312;
        double r124314 = 4.488164189797612e+154;
        bool r124315 = r124291 <= r124314;
        double r124316 = r124315 ? r124305 : r124298;
        double r124317 = r124307 ? r124313 : r124316;
        double r124318 = r124300 ? r124305 : r124317;
        double r124319 = r124293 ? r124298 : r124318;
        return r124319;
}

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
Target11.7
Herbie5.0
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a1 a2) < -6.087120586968689e+188 or 4.488164189797612e+154 < (* a1 a2)

    1. Initial program 31.5

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
    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-inv11.2

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

    6. Applied associate-*l*9.8

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

    7. Simplified9.8

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

    if -6.087120586968689e+188 < (* a1 a2) < -3.0178565167744215e-307 or 5.3291942180717e-313 < (* a1 a2) < 4.488164189797612e+154

    1. Initial program 4.5

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
    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 div-inv13.9

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

    6. Applied associate-*l*13.6

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

    7. Simplified13.5

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

    9. Applied *-un-lft-identity13.5

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

    10. Applied div-inv13.6

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

    11. Applied times-frac10.3

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

    12. Applied associate-*r*4.6

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

    13. Simplified4.6

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

    if -3.0178565167744215e-307 < (* a1 a2) < 5.3291942180717e-313

    1. Initial program 21.6

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

    3. Applied times-frac2.9

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

    5. Applied *-un-lft-identity2.9

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

    6. Applied add-cube-cbrt3.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \sqrt[3]{a1}}}{1 \cdot b1} \cdot \frac{a2}{b2}\]

    7. Applied times-frac3.2

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{1} \cdot \frac{\sqrt[3]{a1}}{b1}\right)} \cdot \frac{a2}{b2}\]

    8. Applied associate-*l*2.4

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

  4. Final simplification5.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -6.08712058696868874 \cdot 10^{188}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -3.0178565167744215 \cdot 10^{-307}:\\ \;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{\frac{1}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.3291942180717 \cdot 10^{-313}:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{1} \cdot \left(\frac{\sqrt[3]{a1}}{b1} \cdot \frac{a2}{b2}\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 4.48816418979761226 \cdot 10^{154}:\\ \;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{\frac{1}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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