Average Error: 11.4 → 7.2
Time: 8.3s
Precision: 64
Internal Precision: 128
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -4.246112204752491 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.169416492823567 \cdot 10^{-274}:\\ \;\;\;\;a1 \cdot \frac{a2 \cdot \frac{1}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.5166109207776906 \cdot 10^{+109}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \end{array}\]

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

Original11.4
Target10.9
Herbie7.2
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 4 regimes

  2. if (* a1 a2) < -4.246112204752491e-125

    1. Initial program 10.9

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

    2. Initial simplification14.5

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

    4. Applied associate-*l/12.9

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

    5. Taylor expanded around inf 10.7

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

    if -4.246112204752491e-125 < (* a1 a2) < 5.169416492823567e-274

    1. Initial program 14.1

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

    2. Initial simplification5.4

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

    4. Applied div-inv5.4

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

    5. Applied associate-*l*5.3

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

    7. Applied associate-*r/5.0

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

    if 5.169416492823567e-274 < (* a1 a2) < 3.5166109207776906e+109

    1. Initial program 4.0

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

    2. Initial simplification13.2

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

    4. Applied associate-*l/11.1

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

    6. Applied associate-*r/4.4

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

    7. Applied associate-/l/4.0

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

    if 3.5166109207776906e+109 < (* a1 a2)

    1. Initial program 24.1

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

    2. Initial simplification12.0

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

    4. Applied associate-*l/14.7

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

    6. Applied associate-/l*11.4

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -4.246112204752491 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.169416492823567 \cdot 10^{-274}:\\ \;\;\;\;a1 \cdot \frac{a2 \cdot \frac{1}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.5166109207776906 \cdot 10^{+109}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \end{array}\]

Runtime

Time bar (total: 8.3s)Debug logProfile

herbie shell --seed 2018256 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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