Average Error: 11.1 → 3.4
Time: 7.0s
Precision: binary64
\[[a1, a2]=\mathsf{sort}([a1, a2])\]
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}} \cdot \frac{\sqrt[3]{a1}}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}} \cdot \frac{\sqrt[3]{a1}}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}
(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))
↓
(FPCore (a1 a2 b1 b2)
 :precision binary64
 (*
  (/
   (* (cbrt a1) (cbrt a1))
   (/
    (* (cbrt b1) (cbrt b1))
    (/ (* (cbrt a2) (cbrt a2)) (* (cbrt b2) (cbrt b2)))))
  (/ (cbrt a1) (/ (cbrt b1) (/ (cbrt a2) (cbrt 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) {
	return ((cbrt(a1) * cbrt(a1)) / ((cbrt(b1) * cbrt(b1)) / ((cbrt(a2) * cbrt(a2)) / (cbrt(b2) * cbrt(b2))))) * (cbrt(a1) / (cbrt(b1) / (cbrt(a2) / cbrt(b2))));
}

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

Original11.1
Target11.5
Herbie3.4
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Initial program 11.1

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

  3. Applied associate-/l*_binary64_241010.9

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

  4. Simplified11.2

    \[\leadsto \frac{a1}{\color{blue}{\frac{b1}{\frac{a2}{b2}}}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt_binary64_250011.8

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

  7. Applied add-cube-cbrt_binary64_250012.0

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

  8. Applied times-frac_binary64_247112.0

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

  9. Applied add-cube-cbrt_binary64_250012.1

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

  10. Applied times-frac_binary64_24718.2

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

  11. Applied add-cube-cbrt_binary64_25008.3

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

  12. Applied times-frac_binary64_24713.4

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

  13. Final simplification3.4

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

Reproduce

herbie shell --seed 2021084 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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