Average Error: 11.0 → 3.4
Time: 15.4s
Precision: 64
Internal Precision: 576
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{b2 \cdot b1}{a2} \le -3.103582166838207 \cdot 10^{+257}:\\ \;\;\;\;\frac{\frac{a2 \cdot a1}{b1}}{b2}\\ \mathbf{elif}\;\frac{b2 \cdot b1}{a2} \le -6.074142513375316 \cdot 10^{-259}:\\ \;\;\;\;\frac{a1}{\frac{b2 \cdot b1}{a2}}\\ \mathbf{elif}\;\frac{b2 \cdot b1}{a2} \le 2.337037482566 \cdot 10^{-312}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;\frac{b2 \cdot b1}{a2} \le 1.2879245740414301 \cdot 10^{+250}:\\ \;\;\;\;\frac{a1}{\frac{b2 \cdot b1}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array}\]

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

Original11.0
Target11.5
Herbie3.4
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* b1 b2) a2) < -3.103582166838207e+257

    1. Initial program 11.2

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

    3. Applied associate-/r*5.0

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

    if -3.103582166838207e+257 < (/ (* b1 b2) a2) < -6.074142513375316e-259 or 2.337037482566e-312 < (/ (* b1 b2) a2) < 1.2879245740414301e+250

    1. Initial program 8.2

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

    3. Applied associate-/l*0.5

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

    if -6.074142513375316e-259 < (/ (* b1 b2) a2) < 2.337037482566e-312 or 1.2879245740414301e+250 < (/ (* b1 b2) a2)

    1. Initial program 18.0

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

    3. Applied associate-/l*30.6

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

    5. Applied div-inv30.6

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

    6. Applied associate-/r*18.0

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

    8. Applied *-un-lft-identity18.0

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

    9. Applied *-un-lft-identity18.0

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

    10. Applied times-frac18.0

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

    11. Simplified18.0

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

    12. Simplified9.5

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

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b2 \cdot b1}{a2} \le -3.103582166838207 \cdot 10^{+257}:\\ \;\;\;\;\frac{\frac{a2 \cdot a1}{b1}}{b2}\\ \mathbf{elif}\;\frac{b2 \cdot b1}{a2} \le -6.074142513375316 \cdot 10^{-259}:\\ \;\;\;\;\frac{a1}{\frac{b2 \cdot b1}{a2}}\\ \mathbf{elif}\;\frac{b2 \cdot b1}{a2} \le 2.337037482566 \cdot 10^{-312}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;\frac{b2 \cdot b1}{a2} \le 1.2879245740414301 \cdot 10^{+250}:\\ \;\;\;\;\frac{a1}{\frac{b2 \cdot b1}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array}\]

Runtime

Time bar (total: 15.4s)Debug logProfile

herbie shell --seed 2018215 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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