Quotient of products

Average Error: 11.1 → 2.9

Time: 4.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -6.28254 \cdot 10^{-319} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.16422584143949291 \cdot 10^{-290}\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 6.1160527718578736 \cdot 10^{209}\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r200730 = a1;
        double r200731 = a2;
        double r200732 = r200730 * r200731;
        double r200733 = b1;
        double r200734 = b2;
        double r200735 = r200733 * r200734;
        double r200736 = r200732 / r200735;
        return r200736;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r200737 = a1;
        double r200738 = a2;
        double r200739 = r200737 * r200738;
        double r200740 = b1;
        double r200741 = b2;
        double r200742 = r200740 * r200741;
        double r200743 = r200739 / r200742;
        double r200744 = -inf.0;
        bool r200745 = r200743 <= r200744;
        double r200746 = -6.2825387525173e-319;
        bool r200747 = r200743 <= r200746;
        double r200748 = 1.1642258414394929e-290;
        bool r200749 = r200743 <= r200748;
        double r200750 = !r200749;
        double r200751 = 6.1160527718578736e+209;
        bool r200752 = r200743 <= r200751;
        bool r200753 = r200750 && r200752;
        bool r200754 = r200747 || r200753;
        double r200755 = !r200754;
        bool r200756 = r200745 || r200755;
        double r200757 = r200737 / r200740;
        double r200758 = r200738 / r200741;
        double r200759 = r200757 * r200758;
        double r200760 = r200756 ? r200759 : r200743;
        return r200760;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.1
Target	11.6
Herbie	2.9

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

Derivation

1. Split input into 2 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or -6.2825387525173e-319 < (/ (* a1 a2) (* b1 b2)) < 1.1642258414394929e-290 or 6.1160527718578736e+209 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 23.1

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

   3. Applied times-frac 5.4

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -6.2825387525173e-319 or 1.1642258414394929e-290 < (/ (* a1 a2) (* b1 b2)) < 6.1160527718578736e+209

   1. Initial program 0.8

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -6.28254 \cdot 10^{-319} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.16422584143949291 \cdot 10^{-290}\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 6.1160527718578736 \cdot 10^{209}\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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