Average Error: 10.8 → 2.8
Time: 52.3s
Precision: 64
Internal Precision: 576
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{b2}{\frac{a1 \cdot a2}{b1}} \le -1.0577402428026127 \cdot 10^{+289}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{if}\;\frac{b2}{\frac{a1 \cdot a2}{b1}} \le -2.9055806012477387 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}}\\ \mathbf{if}\;\frac{b2}{\frac{a1 \cdot a2}{b1}} \le 6.919332692316653 \cdot 10^{-308}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{if}\;\frac{b2}{\frac{a1 \cdot a2}{b1}} \le 7.213076902919156 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.8
Target10.9
Herbie2.8
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ b2 (/ (* a1 a2) b1)) < -1.0577402428026127e+289

    1. Initial program 9.8

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

    3. Applied associate-/r*13.4

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

    5. Applied associate-/l*8.4

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

    7. Applied associate-/l/5.1

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

    9. Applied associate-/r*4.9

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

    if -1.0577402428026127e+289 < (/ b2 (/ (* a1 a2) b1)) < -2.9055806012477387e-308 or 6.919332692316653e-308 < (/ b2 (/ (* a1 a2) b1)) < 7.213076902919156e+304

    1. Initial program 7.9

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

    3. Applied associate-/r*0.8

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

    5. Applied clear-num0.9

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

    if -2.9055806012477387e-308 < (/ b2 (/ (* a1 a2) b1)) < 6.919332692316653e-308

    1. Initial program 35.4

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

    3. Applied associate-/r*59.8

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

    5. Applied associate-/l*37.3

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

    7. Applied associate-/l/15.7

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

    9. Applied div-inv15.8

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

    10. Applied simplify8.8

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

    if 7.213076902919156e+304 < (/ b2 (/ (* a1 a2) b1))

    1. Initial program 8.4

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

    3. Applied times-frac4.3

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

Runtime

Time bar (total: 52.3s)Debug logProfile

herbie shell --seed '#(1072743783 989954326 4239155542 3782239461 3602631542 1719177920)' 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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