Quotient of products

Average Error: 11.5 → 3.8

Time: 3.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.2602034138658774 \cdot 10^{-231} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 7.10047474805390751 \cdot 10^{-168} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 3.70834244534979421 \cdot 10^{294}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r151039 = a1;
        double r151040 = a2;
        double r151041 = r151039 * r151040;
        double r151042 = b1;
        double r151043 = b2;
        double r151044 = r151042 * r151043;
        double r151045 = r151041 / r151044;
        return r151045;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r151046 = a1;
        double r151047 = a2;
        double r151048 = r151046 * r151047;
        double r151049 = b1;
        double r151050 = b2;
        double r151051 = r151049 * r151050;
        double r151052 = r151048 / r151051;
        double r151053 = -inf.0;
        bool r151054 = r151052 <= r151053;
        double r151055 = -4.2602034138658774e-231;
        bool r151056 = r151052 <= r151055;
        double r151057 = 7.1004747480539075e-168;
        bool r151058 = r151052 <= r151057;
        double r151059 = 3.708342445349794e+294;
        bool r151060 = r151052 <= r151059;
        double r151061 = !r151060;
        bool r151062 = r151058 || r151061;
        double r151063 = !r151062;
        bool r151064 = r151056 || r151063;
        double r151065 = !r151064;
        bool r151066 = r151054 || r151065;
        double r151067 = r151046 / r151049;
        double r151068 = r151047 / r151050;
        double r151069 = r151067 * r151068;
        double r151070 = r151066 ? r151069 : r151052;
        return r151070;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.5
Target	11.5
Herbie	3.8

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

Derivation

1. Split input into 2 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or -4.2602034138658774e-231 < (/ (* a1 a2) (* b1 b2)) < 7.1004747480539075e-168 or 3.708342445349794e+294 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 21.8

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

   3. Applied times-frac 6.7

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -4.2602034138658774e-231 or 7.1004747480539075e-168 < (/ (* a1 a2) (* b1 b2)) < 3.708342445349794e+294

   1. Initial program 0.8

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

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.2602034138658774 \cdot 10^{-231} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 7.10047474805390751 \cdot 10^{-168} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 3.70834244534979421 \cdot 10^{294}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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