Average Error: 10.7 → 2.9
Time: 8.3s
Precision: 64
Internal Precision: 576
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -5.181953352930447 \cdot 10^{-302}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 5.449514474752433 \cdot 10^{-302}:\\ \;\;\;\;\left(a2 \cdot \frac{a1}{b2}\right) \cdot \frac{1}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 8.81624581596053 \cdot 10^{+228}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \end{array}\]

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

Original10.7
Target11.2
Herbie2.9
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or 8.81624581596053e+228 < (/ (* a1 a2) (* b1 b2))

    1. Initial program 48.4

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

    2. Initial simplification10.7

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

    4. Applied associate-*r/15.8

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

    6. Applied associate-/l*10.3

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

    if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -5.181953352930447e-302 or 5.449514474752433e-302 < (/ (* a1 a2) (* b1 b2)) < 8.81624581596053e+228

    1. Initial program 0.7

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

    if -5.181953352930447e-302 < (/ (* a1 a2) (* b1 b2)) < 5.449514474752433e-302

    1. Initial program 12.6

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

    2. Initial simplification2.6

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

    4. Applied div-inv2.6

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

    5. Applied associate-*r*3.8

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

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -5.181953352930447 \cdot 10^{-302}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 5.449514474752433 \cdot 10^{-302}:\\ \;\;\;\;\left(a2 \cdot \frac{a1}{b2}\right) \cdot \frac{1}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 8.81624581596053 \cdot 10^{+228}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \end{array}\]

Runtime

Time bar (total: 8.3s)Debug logProfile

herbie shell --seed 2018220 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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