Average Error: 11.2 → 3.5
Time: 37.7s
Precision: 64
Internal Precision: 576
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1}{b1} \cdot a2 \le -5.505268534644017 \cdot 10^{+219}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{if}\;\frac{a1}{b1} \cdot a2 \le -2.631708285375282 \cdot 10^{-224}:\\ \;\;\;\;\left(\frac{a1}{b1} \cdot a2\right) \cdot \frac{1}{b2}\\ \mathbf{if}\;\frac{a1}{b1} \cdot a2 \le 1.4821475551002457 \cdot 10^{-297}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{if}\;\frac{a1}{b1} \cdot a2 \le 5.912666292838447 \cdot 10^{+288}:\\ \;\;\;\;\left(\frac{a1}{b1} \cdot a2\right) \cdot \frac{1}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array}\]

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.2
Target10.7
Herbie3.5
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 4 regimes

  2. if (* (/ a1 b1) a2) < -5.505268534644017e+219

    1. Initial program 16.5

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

    3. Applied times-frac25.2

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

    5. Applied div-inv25.3

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

    6. Applied associate-*l*12.7

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

    7. Applied simplify12.6

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

    if -5.505268534644017e+219 < (* (/ a1 b1) a2) < -2.631708285375282e-224 or 1.4821475551002457e-297 < (* (/ a1 b1) a2) < 5.912666292838447e+288

    1. Initial program 12.8

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

    3. Applied times-frac7.2

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

    5. Applied div-inv7.3

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

    6. Applied associate-*r*0.6

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

    if -2.631708285375282e-224 < (* (/ a1 b1) a2) < 1.4821475551002457e-297

    1. Initial program 5.9

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

    3. Applied times-frac10.2

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

    5. Applied div-inv10.2

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

    6. Applied associate-*l*5.3

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

    7. Applied simplify5.2

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

    if 5.912666292838447e+288 < (* (/ a1 b1) a2)

    1. Initial program 16.7

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

    3. Applied times-frac35.0

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

    5. Applied div-inv35.1

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

    6. Applied associate-*r*55.9

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

    7. Taylor expanded around 0 35.1

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

    8. Applied simplify15.9

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

Runtime

Time bar (total: 37.7s)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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