Average Error: 11.0 → 3.2
Time: 4.6s
Precision: binary64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\left(a1 \cdot \frac{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}\right) \cdot \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1}}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\left(a1 \cdot \frac{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}\right) \cdot \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1}}
double code(double a1, double a2, double b1, double b2) {
	return ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
}
↓
double code(double a1, double a2, double b1, double b2) {
	return ((double) (((double) (a1 * ((double) (((double) (((double) (((double) cbrt(a2)) * ((double) cbrt(a2)))) / ((double) (((double) cbrt(b2)) * ((double) cbrt(b2)))))) / ((double) (((double) cbrt(b1)) * ((double) cbrt(b1)))))))) * ((double) (((double) (((double) cbrt(a2)) / ((double) cbrt(b2)))) / ((double) cbrt(b1))))));
}

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

Original11.0
Target11.3
Herbie3.2
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Initial program 11.0

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

  3. Applied times-frac11.3

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

  5. Applied div-inv11.4

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

  6. Applied associate-*l*11.5

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

  7. Simplified11.4

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

  9. Applied add-cube-cbrt12.0

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

  10. Applied add-cube-cbrt12.2

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

  11. Applied add-cube-cbrt12.3

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

  12. Applied times-frac12.3

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

  13. Applied times-frac8.4

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

  14. Applied associate-*r*3.2

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

  15. Final simplification3.2

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

Reproduce

herbie shell --seed 2020174 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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