Average Error: 10.7 → 3.8
Time: 20.6s
Precision: 64
Internal Precision: 576
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{a1}{b2} \cdot a2}{b1} \le -2.615710032324173 \cdot 10^{+291}:\\ \;\;\;\;\left(\frac{1}{b2} \cdot \frac{a2}{b1}\right) \cdot a1\\ \mathbf{elif}\;\frac{\frac{a1}{b2} \cdot a2}{b1} \le -6.41222932898153 \cdot 10^{-261}:\\ \;\;\;\;\frac{\frac{a1}{b2} \cdot a2}{b1}\\ \mathbf{elif}\;\frac{\frac{a1}{b2} \cdot a2}{b1} \le 1.3452419404965 \cdot 10^{-319}:\\ \;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{\frac{a1}{b2} \cdot a2}{b1} \le 2.27836726326273 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{a1}{b2} \cdot a2}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{1}{b2} \cdot a2}}\\ \end{array}\]

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

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Target

Original10.7
Target10.5
Herbie3.8
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ (* (/ a1 b2) a2) b1) < -2.615710032324173e+291

    1. Initial program 14.9

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

    2. Initial simplification35.0

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

    4. Applied div-inv35.0

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

    5. Applied associate-*l*15.9

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

    if -2.615710032324173e+291 < (/ (* (/ a1 b2) a2) b1) < -6.41222932898153e-261 or 1.3452419404965e-319 < (/ (* (/ a1 b2) a2) b1) < 2.27836726326273e+261

    1. Initial program 13.6

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

    2. Initial simplification7.4

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

    4. Applied associate-*r/0.8

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

    if -6.41222932898153e-261 < (/ (* (/ a1 b2) a2) b1) < 1.3452419404965e-319

    1. Initial program 4.6

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

    2. Initial simplification10.0

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

    3. Taylor expanded around -inf 4.6

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

    if 2.27836726326273e+261 < (/ (* (/ a1 b2) a2) b1)

    1. Initial program 17.1

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

    2. Initial simplification33.1

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

    4. Applied associate-*r/45.5

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

    6. Applied div-inv45.5

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

    7. Applied associate-*l*28.6

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

    9. Applied associate-/l*16.0

      \[\leadsto \color{blue}{\frac{a1}{\frac{b1}{\frac{1}{b2} \cdot a2}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification3.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{a1}{b2} \cdot a2}{b1} \le -2.615710032324173 \cdot 10^{+291}:\\ \;\;\;\;\left(\frac{1}{b2} \cdot \frac{a2}{b1}\right) \cdot a1\\ \mathbf{elif}\;\frac{\frac{a1}{b2} \cdot a2}{b1} \le -6.41222932898153 \cdot 10^{-261}:\\ \;\;\;\;\frac{\frac{a1}{b2} \cdot a2}{b1}\\ \mathbf{elif}\;\frac{\frac{a1}{b2} \cdot a2}{b1} \le 1.3452419404965 \cdot 10^{-319}:\\ \;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{\frac{a1}{b2} \cdot a2}{b1} \le 2.27836726326273 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{a1}{b2} \cdot a2}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{1}{b2} \cdot a2}}\\ \end{array}\]

Runtime

Time bar (total: 20.6s)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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