Average Error: 11.2 → 6.3
Time: 10.1s
Precision: 64
Internal Precision: 128
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -7.510710233104737 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{a2}{\frac{b2}{a1}}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.3382752892841924 \cdot 10^{-242}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1} \cdot \frac{1}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.5240948257229717 \cdot 10^{+90}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1} \cdot \frac{1}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

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

Original11.2
Target10.9
Herbie6.3
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a1 a2) < -7.510710233104737e+83

    1. Initial program 20.0

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

    3. Applied associate-/r*20.9

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

    5. Applied div-inv21.0

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

    7. Applied associate-*l/21.2

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

    8. Simplified15.9

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

    if -7.510710233104737e+83 < (* a1 a2) < -2.3382752892841924e-242 or 0.0 < (* a1 a2) < 2.5240948257229717e+90

    1. Initial program 4.6

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

    3. Applied associate-/r*4.0

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

    5. Applied div-inv4.1

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

    if -2.3382752892841924e-242 < (* a1 a2) < 0.0 or 2.5240948257229717e+90 < (* a1 a2)

    1. Initial program 19.2

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

    3. Applied times-frac6.1

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -7.510710233104737 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{a2}{\frac{b2}{a1}}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.3382752892841924 \cdot 10^{-242}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1} \cdot \frac{1}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.5240948257229717 \cdot 10^{+90}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1} \cdot \frac{1}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Runtime

Time bar (total: 10.1s)Debug logProfile

herbie shell --seed 2018346 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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