Average Error: 11.2 → 4.9
Time: 5.2s
Precision: binary64
\[[a1, a2] = \mathsf{sort}([a1, a2]) \[b1, b2] = \mathsf{sort}([b1, b2]) \\]
\[\frac{a1 \cdot a2}{b1 \cdot b2} \]
↓
\[\begin{array}{l} t_0 := \frac{1}{b1} \cdot \left(\frac{a2}{b2} \cdot a1\right)\\ \mathbf{if}\;b1 \cdot b2 \leq -9.823918728322016 \cdot 10^{+138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq -7.712653371308522 \cdot 10^{-200}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{a1}\right)}^{2}}{b1 \cdot b2} \cdot \frac{\sqrt[3]{a1}}{\frac{1}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 4.8465554479546714 \cdot 10^{-225}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5.461558340952208 \cdot 10^{+254}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
t_0 := \frac{1}{b1} \cdot \left(\frac{a2}{b2} \cdot a1\right)\\
\mathbf{if}\;b1 \cdot b2 \leq -9.823918728322016 \cdot 10^{+138}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;b1 \cdot b2 \leq -7.712653371308522 \cdot 10^{-200}:\\
\;\;\;\;\frac{{\left(\sqrt[3]{a1}\right)}^{2}}{b1 \cdot b2} \cdot \frac{\sqrt[3]{a1}}{\frac{1}{a2}}\\

\mathbf{elif}\;b1 \cdot b2 \leq 4.8465554479546714 \cdot 10^{-225}:\\
\;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\

\mathbf{elif}\;b1 \cdot b2 \leq 5.461558340952208 \cdot 10^{+254}:\\
\;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))
↓
(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 b1) (* (/ a2 b2) a1))))
   (if (<= (* b1 b2) -9.823918728322016e+138)
     t_0
     (if (<= (* b1 b2) -7.712653371308522e-200)
       (* (/ (pow (cbrt a1) 2.0) (* b1 b2)) (/ (cbrt a1) (/ 1.0 a2)))
       (if (<= (* b1 b2) 4.8465554479546714e-225)
         (/ (* a2 (/ a1 b1)) b2)
         (if (<= (* b1 b2) 5.461558340952208e+254)
           (/ a1 (/ (* b1 b2) a2))
           t_0))))))
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 = (1.0 / b1) * ((a2 / b2) * a1);
	double tmp;
	if ((b1 * b2) <= -9.823918728322016e+138) {
		tmp = t_0;
	} else if ((b1 * b2) <= -7.712653371308522e-200) {
		tmp = (pow(cbrt(a1), 2.0) / (b1 * b2)) * (cbrt(a1) / (1.0 / a2));
	} else if ((b1 * b2) <= 4.8465554479546714e-225) {
		tmp = (a2 * (a1 / b1)) / b2;
	} else if ((b1 * b2) <= 5.461558340952208e+254) {
		tmp = a1 / ((b1 * b2) / a2);
	} else {
		tmp = t_0;
	}
	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

Original11.2
Target11.6
Herbie4.9
\[\frac{a1}{b1} \cdot \frac{a2}{b2} \]

Derivation

  1. Split input into 4 regimes

  2. if (*.f64 b1 b2) < -9.8239187283220161e138 or 5.461558340952208e254 < (*.f64 b1 b2)

    1. Initial program 15.7

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

    2. Applied associate-/l*_binary6415.0

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

    3. Applied *-un-lft-identity_binary6415.0

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

    4. Applied times-frac_binary647.7

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

    5. Applied *-un-lft-identity_binary647.7

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

    6. Applied times-frac_binary645.4

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

    7. Simplified5.4

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

    8. Simplified5.3

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

    if -9.8239187283220161e138 < (*.f64 b1 b2) < -7.7126533713085219e-200

    1. Initial program 3.7

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

    2. Applied associate-/l*_binary644.4

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

    3. Applied div-inv_binary644.4

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

    4. Applied add-cube-cbrt_binary645.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \sqrt[3]{a1}}}{\left(b1 \cdot b2\right) \cdot \frac{1}{a2}} \]

    5. Applied times-frac_binary642.0

      \[\leadsto \color{blue}{\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{b1 \cdot b2} \cdot \frac{\sqrt[3]{a1}}{\frac{1}{a2}}} \]

    6. Applied pow2_binary642.0

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a1}\right)}^{2}}}{b1 \cdot b2} \cdot \frac{\sqrt[3]{a1}}{\frac{1}{a2}} \]

    if -7.7126533713085219e-200 < (*.f64 b1 b2) < 4.84655544795467135e-225

    1. Initial program 33.3

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

    2. Applied associate-/r*_binary6416.6

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

    3. Simplified9.7

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

    if 4.84655544795467135e-225 < (*.f64 b1 b2) < 5.461558340952208e254

    1. Initial program 5.4

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

    2. Applied associate-/l*_binary645.0

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

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -9.823918728322016 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{b1} \cdot \left(\frac{a2}{b2} \cdot a1\right)\\ \mathbf{elif}\;b1 \cdot b2 \leq -7.712653371308522 \cdot 10^{-200}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{a1}\right)}^{2}}{b1 \cdot b2} \cdot \frac{\sqrt[3]{a1}}{\frac{1}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 4.8465554479546714 \cdot 10^{-225}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5.461558340952208 \cdot 10^{+254}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b1} \cdot \left(\frac{a2}{b2} \cdot a1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022055 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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