Average Error: 9.2 → 5.3
Time: 5.8s
Precision: binary64
\[[a1, a2] = \mathsf{sort}([a1, a2]) \[b1, b2] = \mathsf{sort}([b1, b2]) \\]
\[\frac{a1 \cdot a2}{b1 \cdot b2} \]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -3.4267994419717365 \cdot 10^{+205} \lor \neg \left(a1 \cdot a2 \leq -1.6104718543684284 \cdot 10^{-288}\right) \land a1 \cdot a2 \leq 5.851337924507556 \cdot 10^{-237}:\\ \;\;\;\;\frac{a1}{\frac{b1}{a2} \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \end{array} \]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;a1 \cdot a2 \leq -3.4267994419717365 \cdot 10^{+205} \lor \neg \left(a1 \cdot a2 \leq -1.6104718543684284 \cdot 10^{-288}\right) \land a1 \cdot a2 \leq 5.851337924507556 \cdot 10^{-237}:\\
\;\;\;\;\frac{a1}{\frac{b1}{a2} \cdot b2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\


\end{array}

(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))
↓
(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (or (<= (* a1 a2) -3.4267994419717365e+205)
         (and (not (<= (* a1 a2) -1.6104718543684284e-288))
              (<= (* a1 a2) 5.851337924507556e-237)))
   (/ a1 (* (/ b1 a2) b2))
   (/ (/ (* a1 a2) b1) b2)))
double code(double a1, double a2, double b1, double b2) {
	return (a1 * a2) / (b1 * b2);
}
↓
double code(double a1, double a2, double b1, double b2) {
	double tmp;
	if (((a1 * a2) <= -3.4267994419717365e+205) || (!((a1 * a2) <= -1.6104718543684284e-288) && ((a1 * a2) <= 5.851337924507556e-237))) {
		tmp = a1 / ((b1 / a2) * b2);
	} else {
		tmp = ((a1 * a2) / b1) / b2;
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.2
Target8.6
Herbie5.3
\[\frac{a1}{b1} \cdot \frac{a2}{b2} \]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 a1 a2) < -3.42679944197173654e205 or -1.61047185436842835e-288 < (*.f64 a1 a2) < 5.85133792450755605e-237

    1. Initial program 17.4

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

    2. Applied associate-/l*_binary648.2

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

    3. Simplified4.0

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

    4. Applied associate-/r/_binary644.0

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

    if -3.42679944197173654e205 < (*.f64 a1 a2) < -1.61047185436842835e-288 or 5.85133792450755605e-237 < (*.f64 a1 a2)

    1. Initial program 6.0

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

    2. Applied associate-/r*_binary645.8

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

    3. Simplified9.0

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

    4. Applied *-un-lft-identity_binary649.0

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

    5. Applied associate-*l*_binary649.0

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

    6. Simplified5.8

      \[\leadsto \frac{1 \cdot \color{blue}{\frac{a1 \cdot a2}{b1}}}{b2} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -3.4267994419717365 \cdot 10^{+205} \lor \neg \left(a1 \cdot a2 \leq -1.6104718543684284 \cdot 10^{-288}\right) \land a1 \cdot a2 \leq 5.851337924507556 \cdot 10^{-237}:\\ \;\;\;\;\frac{a1}{\frac{b1}{a2} \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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