Quotient of products

Average Error: 10.5 → 6.8

Time: 23.5s

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 r24578107 = a1;
        double r24578108 = a2;
        double r24578109 = r24578107 * r24578108;
        double r24578110 = b1;
        double r24578111 = b2;
        double r24578112 = r24578110 * r24578111;
        double r24578113 = r24578109 / r24578112;
        return r24578113;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r24578114 = a1;
        double r24578115 = a2;
        double r24578116 = r24578114 * r24578115;
        double r24578117 = -1.4413717353503713e+166;
        bool r24578118 = r24578116 <= r24578117;
        double r24578119 = b1;
        double r24578120 = r24578114 / r24578119;
        double r24578121 = b2;
        double r24578122 = r24578115 / r24578121;
        double r24578123 = r24578120 * r24578122;
        double r24578124 = -7.442078385181948e-96;
        bool r24578125 = r24578116 <= r24578124;
        double r24578126 = r24578116 / r24578119;
        double r24578127 = 1.0;
        double r24578128 = r24578127 / r24578121;
        double r24578129 = r24578126 * r24578128;
        double r24578130 = 0.0;
        bool r24578131 = r24578116 <= r24578130;
        double r24578132 = r24578121 * r24578119;
        double r24578133 = r24578132 / r24578115;
        double r24578134 = r24578114 / r24578133;
        double r24578135 = 4.15369418345865e+157;
        bool r24578136 = r24578116 <= r24578135;
        double r24578137 = r24578136 ? r24578129 : r24578134;
        double r24578138 = r24578131 ? r24578134 : r24578137;
        double r24578139 = r24578125 ? r24578129 : r24578138;
        double r24578140 = r24578118 ? r24578123 : r24578139;
        return r24578140;
}

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 0 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 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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