Average Error: 10.6 → 4.1
Time: 6.9s
Precision: binary64
\[[a1, a2]=\mathsf{sort}([a1, a2])\]
\[[b1, b2]=\mathsf{sort}([b1, b2])\]
\[\frac{a1 \cdot a2}{b1 \cdot b2} \]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -1.736041581919262 \cdot 10^{+209}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt[3]{b2} \cdot \sqrt[3]{b2}\\ \mathbf{if}\;a1 \cdot a2 \leq -2.5746168329260578 \cdot 10^{-222}:\\ \;\;\;\;\frac{1}{\frac{t_0}{a1 \cdot a2} \cdot \left(b1 \cdot \sqrt[3]{b2}\right)}\\ \mathbf{elif}\;a1 \cdot a2 \leq 9.600530308119492 \cdot 10^{-256}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{t_0} \cdot \frac{\frac{a2}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;a1 \cdot a2 \leq 5.8338690603176334 \cdot 10^{+253}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \end{array}\\ \end{array} \]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;a1 \cdot a2 \leq -1.736041581919262 \cdot 10^{+209}:\\
\;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := \sqrt[3]{b2} \cdot \sqrt[3]{b2}\\
\mathbf{if}\;a1 \cdot a2 \leq -2.5746168329260578 \cdot 10^{-222}:\\
\;\;\;\;\frac{1}{\frac{t_0}{a1 \cdot a2} \cdot \left(b1 \cdot \sqrt[3]{b2}\right)}\\

\mathbf{elif}\;a1 \cdot a2 \leq 9.600530308119492 \cdot 10^{-256}:\\
\;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{t_0} \cdot \frac{\frac{a2}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\

\mathbf{elif}\;a1 \cdot a2 \leq 5.8338690603176334 \cdot 10^{+253}:\\
\;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\

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


\end{array}\\


\end{array}

(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))
↓
(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (* a1 a2) -1.736041581919262e+209)
   (* a1 (/ (/ a2 b1) b2))
   (let* ((t_0 (* (cbrt b2) (cbrt b2))))
     (if (<= (* a1 a2) -2.5746168329260578e-222)
       (/ 1.0 (* (/ t_0 (* a1 a2)) (* b1 (cbrt b2))))
       (if (<= (* a1 a2) 9.600530308119492e-256)
         (*
          (/ (/ a1 (* (cbrt b1) (cbrt b1))) t_0)
          (/ (/ a2 (cbrt b1)) (cbrt b2)))
         (if (<= (* a1 a2) 5.8338690603176334e+253)
           (/ (/ (* a1 a2) b2) b1)
           (/ a1 (/ b2 (/ a2 b1)))))))))
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) <= -1.736041581919262e+209) {
		tmp = a1 * ((a2 / b1) / b2);
	} else {
		double t_0 = cbrt(b2) * cbrt(b2);
		double tmp_1;
		if ((a1 * a2) <= -2.5746168329260578e-222) {
			tmp_1 = 1.0 / ((t_0 / (a1 * a2)) * (b1 * cbrt(b2)));
		} else if ((a1 * a2) <= 9.600530308119492e-256) {
			tmp_1 = ((a1 / (cbrt(b1) * cbrt(b1))) / t_0) * ((a2 / cbrt(b1)) / cbrt(b2));
		} else if ((a1 * a2) <= 5.8338690603176334e+253) {
			tmp_1 = ((a1 * a2) / b2) / b1;
		} else {
			tmp_1 = a1 / (b2 / (a2 / b1));
		}
		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

Original10.6
Target11.3
Herbie4.1
\[\frac{a1}{b1} \cdot \frac{a2}{b2} \]

Derivation

  1. Split input into 5 regimes

  2. if (*.f64 a1 a2) < -1.7360415819192621e209

    1. Initial program 35.5

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

    2. Applied associate-/r*_binary6434.3

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

    3. Applied clear-num_binary6434.3

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

    4. Applied *-un-lft-identity_binary6434.3

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

    5. Applied times-frac_binary6416.3

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

    6. Applied *-un-lft-identity_binary6416.3

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

    7. Applied times-frac_binary648.5

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

    8. Applied add-cube-cbrt_binary648.5

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

    9. Applied times-frac_binary648.6

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

    10. Simplified8.6

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

    11. Simplified8.1

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

    if -1.7360415819192621e209 < (*.f64 a1 a2) < -2.5746168329260578e-222

    1. Initial program 4.7

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

    2. Applied associate-/r*_binary645.0

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

    3. Applied clear-num_binary645.4

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

    4. Applied div-inv_binary645.4

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

    5. Applied add-cube-cbrt_binary646.1

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

    6. Applied times-frac_binary643.3

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

    7. Simplified3.2

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

    if -2.5746168329260578e-222 < (*.f64 a1 a2) < 9.60053030811949162e-256

    1. Initial program 15.5

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

    2. Applied associate-/r*_binary6415.2

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

    3. Applied add-cube-cbrt_binary6415.4

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

    4. Applied add-cube-cbrt_binary6415.4

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

    5. Applied times-frac_binary647.6

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

    6. Applied times-frac_binary642.3

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

    if 9.60053030811949162e-256 < (*.f64 a1 a2) < 5.83386906031763342e253

    1. Initial program 4.8

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

    2. Applied associate-/r*_binary645.5

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

    3. Applied clear-num_binary646.0

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

    4. Applied associate-/r/_binary645.5

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

    5. Applied associate-/r*_binary645.3

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

    6. Simplified10.8

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

    7. Taylor expanded in a2 around 0 4.9

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

    if 5.83386906031763342e253 < (*.f64 a1 a2)

    1. Initial program 43.6

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

    2. Applied associate-/r*_binary6445.2

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

    3. Applied clear-num_binary6445.3

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

    4. Applied *-un-lft-identity_binary6445.3

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

    5. Applied times-frac_binary6420.6

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

    6. Applied *-un-lft-identity_binary6420.6

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

    7. Applied times-frac_binary648.5

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

    8. Applied associate-/r*_binary648.3

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

    9. Simplified8.3

      \[\leadsto \frac{\color{blue}{a1}}{\frac{b2}{\frac{a2}{b1}}} \]
  3. Recombined 5 regimes into one program.

  4. Final simplification4.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -1.736041581919262 \cdot 10^{+209}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq -2.5746168329260578 \cdot 10^{-222}:\\ \;\;\;\;\frac{1}{\frac{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}{a1 \cdot a2} \cdot \left(b1 \cdot \sqrt[3]{b2}\right)}\\ \mathbf{elif}\;a1 \cdot a2 \leq 9.600530308119492 \cdot 10^{-256}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;a1 \cdot a2 \leq 5.8338690603176334 \cdot 10^{+253}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022005 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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