Average Error: 10.6 → 4.8
Time: 18.1s
Precision: 64
Internal Precision: 128
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -5.0946409181618005 \cdot 10^{+209}:\\ \;\;\;\;\left(\frac{1}{b2} \cdot a1\right) \cdot \frac{a2}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -1.7345924279536303 \cdot 10^{-230}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.916600784605298 \cdot 10^{-276}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.7101214766568924 \cdot 10^{+251}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(\frac{1}{b2} \cdot \frac{a2}{b1}\right)\\ \end{array}\]

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

Original10.6
Target10.9
Herbie4.8
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 4 regimes

  2. if (* a1 a2) < -5.0946409181618005e+209

    1. Initial program 34.8

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

    2. Initial simplification10.0

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

    4. Applied div-inv10.1

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

    5. Applied associate-*l*9.9

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

    7. Applied associate-*r*10.1

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

    if -5.0946409181618005e+209 < (* a1 a2) < -1.7345924279536303e-230 or 1.916600784605298e-276 < (* a1 a2) < 2.7101214766568924e+251

    1. Initial program 4.7

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

    2. Initial simplification13.8

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

    4. Applied div-inv13.9

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

    5. Applied associate-*l*14.1

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

    7. Applied associate-*r*13.9

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

    8. Taylor expanded around inf 4.7

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

    if -1.7345924279536303e-230 < (* a1 a2) < 1.916600784605298e-276

    1. Initial program 15.9

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

    2. Initial simplification3.5

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

    4. Applied div-inv3.5

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

    5. Applied associate-*l*4.0

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

    7. Applied associate-*r*3.5

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

    9. Applied pow13.5

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

    10. Applied pow13.5

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

    11. Applied pow-prod-down3.5

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

    12. Simplified3.0

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

    if 2.7101214766568924e+251 < (* a1 a2)

    1. Initial program 43.2

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

    2. Initial simplification9.2

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

    4. Applied div-inv9.3

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

    5. Applied associate-*l*7.9

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

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -5.0946409181618005 \cdot 10^{+209}:\\ \;\;\;\;\left(\frac{1}{b2} \cdot a1\right) \cdot \frac{a2}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -1.7345924279536303 \cdot 10^{-230}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.916600784605298 \cdot 10^{-276}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.7101214766568924 \cdot 10^{+251}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(\frac{1}{b2} \cdot \frac{a2}{b1}\right)\\ \end{array}\]

Runtime

Time bar (total: 18.1s)Debug logProfile

herbie shell --seed 2018285 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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