Average Error: 11.1 → 3.0
Time: 21.0s
Precision: 64
Internal Precision: 576
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1} \le -9.8055771676722 \cdot 10^{+298}:\\ \;\;\;\;\frac{a1}{b2 \cdot \frac{b1}{a2}}\\ \mathbf{if}\;\frac{a1 \cdot a2}{b1} \le -4.499636684067 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{if}\;\frac{a1 \cdot a2}{b1} \le 6.269427852969574 \cdot 10^{-266}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{if}\;\frac{a1 \cdot a2}{b1} \le 1.4859452468517468 \cdot 10^{+268}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2 \cdot \frac{b1}{a2}}\\ \end{array}\]

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

Original11.1
Target11.3
Herbie3.0
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* a1 a2) b1) < -9.8055771676722e+298 or 1.4859452468517468e+268 < (/ (* a1 a2) b1)

    1. Initial program 32.8

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

    3. Applied associate-/r*53.6

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

    5. Applied associate-/l*33.5

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

    7. Applied associate-/l/15.9

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

    if -9.8055771676722e+298 < (/ (* a1 a2) b1) < -4.499636684067e-311 or 6.269427852969574e-266 < (/ (* a1 a2) b1) < 1.4859452468517468e+268

    1. Initial program 7.4

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

    3. Applied associate-/r*0.6

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

    if -4.499636684067e-311 < (/ (* a1 a2) b1) < 6.269427852969574e-266

    1. Initial program 11.8

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

    3. Applied associate-/r*19.3

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

    5. Applied associate-/l*12.8

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

    7. Applied associate-/l/5.8

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

    9. Applied div-inv5.9

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

    10. Applied simplify3.9

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

Runtime

Time bar (total: 21.0s)Debug logProfile

herbie shell --seed 2018207 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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