Average Error: 11.2 → 7.1
Time: 8.7s
Precision: 64
Internal Precision: 128
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -1.6805196993943107 \cdot 10^{+107}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{1}{b2} \cdot a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.241101594633656 \cdot 10^{-95}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.1040403110207 \cdot 10^{-317}:\\ \;\;\;\;\frac{\frac{a2}{\frac{b1}{a1}}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.185658120574032 \cdot 10^{-174}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{1}{b2} \cdot a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.8640250125590844 \cdot 10^{+26}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b1} \cdot \left(a2 \cdot \frac{a1}{b2}\right)\\ \end{array}\]

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
Target11.2
Herbie7.1
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 4 regimes

  2. if (* a1 a2) < -1.6805196993943107e+107 or -2.1040403110207e-317 < (* a1 a2) < 5.185658120574032e-174

    1. Initial program 17.7

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

    2. Initial simplification7.1

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

    4. Applied associate-*r/11.0

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

    6. Applied div-inv11.1

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

    7. Applied associate-*l*10.9

      \[\leadsto \frac{\color{blue}{a1 \cdot \left(\frac{1}{b2} \cdot a2\right)}}{b1}\]
    8. Using strategy rm

    9. Applied associate-/l*6.4

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

    if -1.6805196993943107e+107 < (* a1 a2) < -2.241101594633656e-95 or 5.185658120574032e-174 < (* a1 a2) < 5.8640250125590844e+26

    1. Initial program 3.1

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

    2. Initial simplification14.7

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

    3. Taylor expanded around inf 3.1

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

    if -2.241101594633656e-95 < (* a1 a2) < -2.1040403110207e-317

    1. Initial program 6.3

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

    2. Initial simplification9.7

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

    4. Applied associate-*r/9.7

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

    6. Applied div-inv9.7

      \[\leadsto \color{blue}{\left(\frac{a1}{b2} \cdot a2\right) \cdot \frac{1}{b1}}\]
    7. Using strategy rm

    8. Applied associate-*l/6.9

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

    9. Applied associate-*l/6.6

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

    10. Simplified8.6

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

    if 5.8640250125590844e+26 < (* a1 a2)

    1. Initial program 17.0

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

    2. Initial simplification12.7

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

    4. Applied associate-*r/14.1

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

    6. Applied div-inv14.2

      \[\leadsto \color{blue}{\left(\frac{a1}{b2} \cdot a2\right) \cdot \frac{1}{b1}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -1.6805196993943107 \cdot 10^{+107}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{1}{b2} \cdot a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.241101594633656 \cdot 10^{-95}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.1040403110207 \cdot 10^{-317}:\\ \;\;\;\;\frac{\frac{a2}{\frac{b1}{a1}}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.185658120574032 \cdot 10^{-174}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{1}{b2} \cdot a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.8640250125590844 \cdot 10^{+26}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b1} \cdot \left(a2 \cdot \frac{a1}{b2}\right)\\ \end{array}\]

Runtime

Time bar (total: 8.7s)Debug logProfile

herbie shell --seed 2018277 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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