Quotient of products

Average Error: 11.4 → 2.5

Time: 7.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -6.6844398141843 \cdot 10^{-312}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 8.55317884451173137 \cdot 10^{276}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r201551 = a1;
        double r201552 = a2;
        double r201553 = r201551 * r201552;
        double r201554 = b1;
        double r201555 = b2;
        double r201556 = r201554 * r201555;
        double r201557 = r201553 / r201556;
        return r201557;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r201558 = a1;
        double r201559 = a2;
        double r201560 = r201558 * r201559;
        double r201561 = b1;
        double r201562 = b2;
        double r201563 = r201561 * r201562;
        double r201564 = r201560 / r201563;
        double r201565 = -inf.0;
        bool r201566 = r201564 <= r201565;
        double r201567 = r201558 / r201561;
        double r201568 = r201559 / r201562;
        double r201569 = r201567 * r201568;
        double r201570 = -6.6844398141843e-312;
        bool r201571 = r201564 <= r201570;
        double r201572 = 1.0;
        double r201573 = r201572 / r201561;
        double r201574 = r201573 / r201562;
        double r201575 = r201560 * r201574;
        double r201576 = 0.0;
        bool r201577 = r201564 <= r201576;
        double r201578 = 8.553178844511731e+276;
        bool r201579 = r201564 <= r201578;
        double r201580 = r201579 ? r201564 : r201569;
        double r201581 = r201577 ? r201569 : r201580;
        double r201582 = r201571 ? r201575 : r201581;
        double r201583 = r201566 ? r201569 : r201582;
        return r201583;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.5
Herbie	2.5

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or -6.6844398141843e-312 < (/ (* a1 a2) (* b1 b2)) < 0.0 or 8.553178844511731e+276 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 25.7

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

   3. Applied times-frac 4.3

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -6.6844398141843e-312

   1. Initial program 0.9

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

   3. Applied associate-/r* 8.9

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

   5. Applied *-un-lft-identity 8.9

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

   6. Applied div-inv 9.0

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

   7. Applied times-frac 1.4

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

   8. Simplified 1.4

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

   if 0.0 < (/ (* a1 a2) (* b1 b2)) < 8.553178844511731e+276

   1. Initial program 0.8

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -6.6844398141843 \cdot 10^{-312}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 8.55317884451173137 \cdot 10^{276}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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