Average Error: 11.3 → 3.0
Time: 3.6s
Precision: binary64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.0486969344217753 \cdot 10^{+269}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -7.268695233203412 \cdot 10^{-303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 9.191577884368106 \cdot 10^{+278}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.0486969344217753 \cdot 10^{+269}:\\
\;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -7.268695233203412 \cdot 10^{-303}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\
\;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\

\mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 9.191577884368106 \cdot 10^{+278}:\\
\;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\

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

\end{array}
(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))
↓
(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (/ (* a1 a2) (* b1 b2)) -1.0486969344217753e+269)
   (* (/ a1 b1) (/ a2 b2))
   (if (<= (/ (* a1 a2) (* b1 b2)) -7.268695233203412e-303)
     (/ (* a1 a2) (* b1 b2))
     (if (<= (/ (* a1 a2) (* b1 b2)) 0.0)
       (/ a1 (* b1 (/ b2 a2)))
       (if (<= (/ (* a1 a2) (* b1 b2)) 9.191577884368106e+278)
         (/ (* a1 a2) (* b1 b2))
         (* (/ a1 b1) (/ a2 b2)))))))
double code(double a1, double a2, double b1, double b2) {
	return (((double) (a1 * a2)) / ((double) (b1 * b2)));
}
↓
double code(double a1, double a2, double b1, double b2) {
	double tmp;
	if (((((double) (a1 * a2)) / ((double) (b1 * b2))) <= -1.0486969344217753e+269)) {
		tmp = ((double) ((a1 / b1) * (a2 / b2)));
	} else {
		double tmp_1;
		if (((((double) (a1 * a2)) / ((double) (b1 * b2))) <= -7.268695233203412e-303)) {
			tmp_1 = (((double) (a1 * a2)) / ((double) (b1 * b2)));
		} else {
			double tmp_2;
			if (((((double) (a1 * a2)) / ((double) (b1 * b2))) <= 0.0)) {
				tmp_2 = (a1 / ((double) (b1 * (b2 / a2))));
			} else {
				double tmp_3;
				if (((((double) (a1 * a2)) / ((double) (b1 * b2))) <= 9.191577884368106e+278)) {
					tmp_3 = (((double) (a1 * a2)) / ((double) (b1 * b2)));
				} else {
					tmp_3 = ((double) ((a1 / b1) * (a2 / b2)));
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	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

Original11.3
Target10.8
Herbie3.0
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* a1 a2) (* b1 b2)) < -1.04869693442177529e269 or 9.1915778843681061e278 < (/ (* a1 a2) (* b1 b2))

    1. Initial program Error: 54.3 bits

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

    3. Applied times-fracError: 10.6 bits

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

    if -1.04869693442177529e269 < (/ (* a1 a2) (* b1 b2)) < -7.2686952332034124e-303 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 9.1915778843681061e278

    1. Initial program Error: 0.8 bits

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

    if -7.2686952332034124e-303 < (/ (* a1 a2) (* b1 b2)) < 0.0

    1. Initial program Error: 13.2 bits

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

    3. Applied associate-/l*Error: 7.0 bits

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

    4. SimplifiedError: 3.8 bits

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

  4. Final simplificationError: 3.0 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.0486969344217753 \cdot 10^{+269}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -7.268695233203412 \cdot 10^{-303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 9.191577884368106 \cdot 10^{+278}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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