Average Error: 11.6 → 2.5
Time: 5.6s
Precision: binary64
\[[a1, a2]=\mathsf{sort}([a1, a2])\]
\[[b1, b2]=\mathsf{sort}([b1, b2])\]
\[\frac{a1 \cdot a2}{b1 \cdot b2} \]
↓
\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{if}\;t_0 \leq -4.038047391351627 \cdot 10^{+281}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -3.3157554142466014 \cdot 10^{-276}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 7.694643325017655 \cdot 10^{+299}:\\ \;\;\;\;\frac{a1 \cdot a2}{\frac{b1}{\frac{1}{b2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\
t_1 := \frac{a2}{b1} \cdot \frac{a1}{b2}\\
\mathbf{if}\;t_0 \leq -4.038047391351627 \cdot 10^{+281}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t_0 \leq -3.3157554142466014 \cdot 10^{-276}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;t_0 \leq 0:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t_0 \leq 7.694643325017655 \cdot 10^{+299}:\\
\;\;\;\;\frac{a1 \cdot a2}{\frac{b1}{\frac{1}{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
 (let* ((t_0 (/ (* a1 a2) (* b1 b2))) (t_1 (* (/ a2 b1) (/ a1 b2))))
   (if (<= t_0 -4.038047391351627e+281)
     t_1
     (if (<= t_0 -3.3157554142466014e-276)
       t_0
       (if (<= t_0 0.0)
         t_1
         (if (<= t_0 7.694643325017655e+299)
           (/ (* a1 a2) (/ b1 (/ 1.0 b2)))
           (* (/ a1 b1) (/ a2 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 t_0 = (a1 * a2) / (b1 * b2);
	double t_1 = (a2 / b1) * (a1 / b2);
	double tmp;
	if (t_0 <= -4.038047391351627e+281) {
		tmp = t_1;
	} else if (t_0 <= -3.3157554142466014e-276) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 7.694643325017655e+299) {
		tmp = (a1 * a2) / (b1 / (1.0 / b2));
	} else {
		tmp = (a1 / b1) * (a2 / 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

Original11.6
Target11.3
Herbie2.5
\[\frac{a1}{b1} \cdot \frac{a2}{b2} \]

Derivation

  1. Split input into 4 regimes

  2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -4.03804739135162685e281 or -3.3157554142466014e-276 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

    1. Initial program 18.1

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

    2. Applied associate-/l*_binary6410.1

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

    3. Simplified5.8

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

    4. Applied associate-/r/_binary645.7

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

    5. Applied *-un-lft-identity_binary645.7

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

    6. Applied times-frac_binary644.4

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

    7. Simplified4.2

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

    if -4.03804739135162685e281 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -3.3157554142466014e-276

    1. Initial program 0.7

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

    if 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 7.6946433250176545e299

    1. Initial program 1.0

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

    2. Applied associate-/l*_binary648.4

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

    3. Simplified14.7

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

    4. Applied div-inv_binary6414.8

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

    5. Applied *-un-lft-identity_binary6414.8

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

    6. Applied times-frac_binary648.5

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

    7. Applied associate-/r*_binary641.1

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

    8. Simplified1.0

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

    if 7.6946433250176545e299 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

    1. Initial program 61.9

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

    2. Applied times-frac_binary646.8

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -4.038047391351627 \cdot 10^{+281}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -3.3157554142466014 \cdot 10^{-276}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 7.694643325017655 \cdot 10^{+299}:\\ \;\;\;\;\frac{a1 \cdot a2}{\frac{b1}{\frac{1}{b2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2021310 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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