Quotient of products

Average Error: 11.3 → 2.6

Time: 12.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.331097236289201929838111566058261923975 \cdot 10^{-312} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 2.759064889840746628621735483350290847538 \cdot 10^{303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r189565 = a1;
        double r189566 = a2;
        double r189567 = r189565 * r189566;
        double r189568 = b1;
        double r189569 = b2;
        double r189570 = r189568 * r189569;
        double r189571 = r189567 / r189570;
        return r189571;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r189572 = a1;
        double r189573 = a2;
        double r189574 = r189572 * r189573;
        double r189575 = b1;
        double r189576 = b2;
        double r189577 = r189575 * r189576;
        double r189578 = r189574 / r189577;
        double r189579 = -inf.0;
        bool r189580 = r189578 <= r189579;
        double r189581 = r189574 / r189575;
        double r189582 = r189581 / r189576;
        double r189583 = -4.3310972362892e-312;
        bool r189584 = r189578 <= r189583;
        double r189585 = 0.0;
        bool r189586 = r189578 <= r189585;
        double r189587 = !r189586;
        double r189588 = 2.7590648898407466e+303;
        bool r189589 = r189578 <= r189588;
        bool r189590 = r189587 && r189589;
        bool r189591 = r189584 || r189590;
        double r189592 = r189573 / r189576;
        double r189593 = r189572 / r189575;
        double r189594 = r189592 * r189593;
        double r189595 = r189591 ? r189578 : r189594;
        double r189596 = r189580 ? r189582 : r189595;
        return r189596;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.3
Herbie	2.6

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0

   1. Initial program 64.0

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

   3. Applied associate-/r* 28.8

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -4.3310972362892e-312 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 2.7590648898407466e+303

   1. Initial program 3.4

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

   if -4.3310972362892e-312 < (/ (* a1 a2) (* b1 b2)) < 0.0 or 2.7590648898407466e+303 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 29.3

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

   2. Taylor expanded around 0 29.3

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

   3. Simplified 3.6

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

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.331097236289201929838111566058261923975 \cdot 10^{-312} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 2.759064889840746628621735483350290847538 \cdot 10^{303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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