Average Error: 10.9 → 3.4
Time: 23.0s
Precision: 64
Internal Precision: 384
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot \frac{a2}{b1} \le -2.3396816010736717 \cdot 10^{+291}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{if}\;a1 \cdot \frac{a2}{b1} \le -8.030881417198774 \cdot 10^{-137}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \mathbf{if}\;a1 \cdot \frac{a2}{b1} \le 1.7115334070464407 \cdot 10^{-288}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{if}\;a1 \cdot \frac{a2}{b1} \le 2.291989826186601 \cdot 10^{+209}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original10.9
Target10.9
Herbie3.4
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a1 (/ a2 b1)) < -2.3396816010736717e+291 or -8.030881417198774e-137 < (* a1 (/ a2 b1)) < 1.7115334070464407e-288

    1. Initial program 7.8

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

    3. Applied times-frac7.4

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

    5. Applied associate-*l/5.0

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

    if -2.3396816010736717e+291 < (* a1 (/ a2 b1)) < -8.030881417198774e-137 or 1.7115334070464407e-288 < (* a1 (/ a2 b1)) < 2.291989826186601e+209

    1. Initial program 12.6

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

    3. Applied times-frac12.9

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

    5. Applied div-inv12.9

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

    6. Applied associate-*l*14.0

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

    7. Applied simplify7.7

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

    9. Applied div-inv7.8

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

    10. Applied associate-*r*0.5

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

    if 2.291989826186601e+209 < (* a1 (/ a2 b1))

    1. Initial program 15.8

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

    3. Applied times-frac15.6

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

Runtime

Time bar (total: 23.0s)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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