Quotient of products

Average Error: 10.5 → 6.8

Time: 24.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -1.4413717353503713 \cdot 10^{+166}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -7.442078385181948 \cdot 10^{-96}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1} \cdot \frac{1}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 0.0:\\ \;\;\;\;\frac{a1}{\frac{b2 \cdot b1}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 4.15369418345865 \cdot 10^{+157}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1} \cdot \frac{1}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2 \cdot b1}{a2}}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r4511086 = a1;
        double r4511087 = a2;
        double r4511088 = r4511086 * r4511087;
        double r4511089 = b1;
        double r4511090 = b2;
        double r4511091 = r4511089 * r4511090;
        double r4511092 = r4511088 / r4511091;
        return r4511092;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r4511093 = a1;
        double r4511094 = a2;
        double r4511095 = r4511093 * r4511094;
        double r4511096 = -1.4413717353503713e+166;
        bool r4511097 = r4511095 <= r4511096;
        double r4511098 = b1;
        double r4511099 = r4511093 / r4511098;
        double r4511100 = b2;
        double r4511101 = r4511094 / r4511100;
        double r4511102 = r4511099 * r4511101;
        double r4511103 = -7.442078385181948e-96;
        bool r4511104 = r4511095 <= r4511103;
        double r4511105 = r4511095 / r4511098;
        double r4511106 = 1.0;
        double r4511107 = r4511106 / r4511100;
        double r4511108 = r4511105 * r4511107;
        double r4511109 = 0.0;
        bool r4511110 = r4511095 <= r4511109;
        double r4511111 = r4511100 * r4511098;
        double r4511112 = r4511111 / r4511094;
        double r4511113 = r4511093 / r4511112;
        double r4511114 = 4.15369418345865e+157;
        bool r4511115 = r4511095 <= r4511114;
        double r4511116 = r4511115 ? r4511108 : r4511113;
        double r4511117 = r4511110 ? r4511113 : r4511116;
        double r4511118 = r4511104 ? r4511108 : r4511117;
        double r4511119 = r4511097 ? r4511102 : r4511118;
        return r4511119;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	10.5
Target	11.4
Herbie	6.8

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

Derivation

1. Split input into 3 regimes

2. if (* a1 a2) < -1.4413717353503713e+166

   1. Initial program 28.4

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

   3. Applied times-frac 11.5

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

   if -1.4413717353503713e+166 < (* a1 a2) < -7.442078385181948e-96 or 0.0 < (* a1 a2) < 4.15369418345865e+157

   1. Initial program 4.3

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

   3. Applied associate-/r* 4.3

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

   4. Taylor expanded around inf 4.3

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

   6. Applied div-inv 4.4

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

   if -7.442078385181948e-96 < (* a1 a2) < 0.0 or 4.15369418345865e+157 < (* a1 a2)

   1. Initial program 15.7

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

   3. Applied associate-/l* 9.2

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

4. Final simplification 6.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -1.4413717353503713 \cdot 10^{+166}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -7.442078385181948 \cdot 10^{-96}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1} \cdot \frac{1}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 0.0:\\ \;\;\;\;\frac{a1}{\frac{b2 \cdot b1}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 4.15369418345865 \cdot 10^{+157}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1} \cdot \frac{1}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2 \cdot b1}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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