Quotient of products

Average Error: 11.7 → 3.2

Time: 3.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.72595112384748723 \cdot 10^{296}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.5493029940074664 \cdot 10^{-307}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 8.09671095346136017 \cdot 10^{300}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r156775 = a1;
        double r156776 = a2;
        double r156777 = r156775 * r156776;
        double r156778 = b1;
        double r156779 = b2;
        double r156780 = r156778 * r156779;
        double r156781 = r156777 / r156780;
        return r156781;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r156782 = a1;
        double r156783 = a2;
        double r156784 = r156782 * r156783;
        double r156785 = b1;
        double r156786 = b2;
        double r156787 = r156785 * r156786;
        double r156788 = r156784 / r156787;
        double r156789 = -1.7259511238474872e+296;
        bool r156790 = r156788 <= r156789;
        double r156791 = r156783 / r156786;
        double r156792 = r156791 / r156785;
        double r156793 = r156782 * r156792;
        double r156794 = -3.5493029940074664e-307;
        bool r156795 = r156788 <= r156794;
        double r156796 = 0.0;
        bool r156797 = r156788 <= r156796;
        double r156798 = r156782 / r156785;
        double r156799 = cbrt(r156786);
        double r156800 = r156799 * r156799;
        double r156801 = r156798 / r156800;
        double r156802 = r156783 / r156799;
        double r156803 = r156801 * r156802;
        double r156804 = 8.09671095346136e+300;
        bool r156805 = r156788 <= r156804;
        double r156806 = r156782 * r156791;
        double r156807 = r156806 / r156785;
        double r156808 = r156805 ? r156788 : r156807;
        double r156809 = r156797 ? r156803 : r156808;
        double r156810 = r156795 ? r156788 : r156809;
        double r156811 = r156790 ? r156793 : r156810;
        return r156811;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.7
Target	10.7
Herbie	3.2

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

Derivation

1. Split input into 4 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -1.7259511238474872e+296

   1. Initial program 57.8

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

   3. Applied times-frac 10.9

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

   5. Applied div-inv 11.0

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

   6. Applied associate-*l* 17.4

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

   7. Simplified 17.4

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

   if -1.7259511238474872e+296 < (/ (* a1 a2) (* b1 b2)) < -3.5493029940074664e-307 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 8.09671095346136e+300

   1. Initial program 0.9

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

   if -3.5493029940074664e-307 < (/ (* a1 a2) (* b1 b2)) < 0.0

   1. Initial program 13.7

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

   3. Applied times-frac 2.2

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

   5. Applied add-cube-cbrt 2.4

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

   6. Applied *-un-lft-identity 2.4

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

   7. Applied times-frac 2.4

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

   8. Applied associate-*r* 3.1

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

   9. Simplified 3.1

      \[\leadsto \color{blue}{\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}} \cdot \frac{a2}{\sqrt[3]{b2}}\]

   if 8.09671095346136e+300 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 61.3

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

   3. Applied times-frac 6.3

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

   5. Applied associate-*l/ 13.8

      \[\leadsto \color{blue}{\frac{a1 \cdot \frac{a2}{b2}}{b1}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.72595112384748723 \cdot 10^{296}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.5493029940074664 \cdot 10^{-307}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 8.09671095346136017 \cdot 10^{300}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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