Average Error: 11.0 → 3.5
Time: 24.7s
Precision: 64
Internal Precision: 384
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a1}{\frac{b1}{a2}} \le -5.358362692026326 \cdot 10^{+96}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{if}\;\frac{a1}{\frac{b1}{a2}} \le -2.0532625218660933 \cdot 10^{-258}:\\ \;\;\;\;\frac{\frac{a1}{\frac{b1}{a2}}}{b2}\\ \mathbf{if}\;\frac{a1}{\frac{b1}{a2}} \le 1.023010417144655 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{a1}{b1 \cdot b2}}{\frac{1}{a2}}\\ \mathbf{if}\;\frac{a1}{\frac{b1}{a2}} \le 2.259869264089033 \cdot 10^{+299}:\\ \;\;\;\;\frac{\frac{a1}{\frac{b1}{a2}}}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \end{array}\]

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original11.0
Target11.3
Herbie3.5
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ a1 (/ b1 a2)) < -5.358362692026326e+96

    1. Initial program 15.8

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

    3. Applied associate-/l*11.4

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

    5. Applied associate-/l*12.1

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

    7. Applied div-inv12.5

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

    8. Applied simplify12.2

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

    if -5.358362692026326e+96 < (/ a1 (/ b1 a2)) < -2.0532625218660933e-258 or 1.023010417144655e-309 < (/ a1 (/ b1 a2)) < 2.259869264089033e+299

    1. Initial program 12.5

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

    3. Applied associate-/l*13.9

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

    5. Applied associate-/l*13.7

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

    7. Applied associate-/r/8.0

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

    8. Applied associate-/r*0.5

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

    if -2.0532625218660933e-258 < (/ a1 (/ b1 a2)) < 1.023010417144655e-309

    1. Initial program 5.4

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

    3. Applied associate-/l*5.6

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

    5. Applied div-inv5.6

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

    6. Applied associate-/r*3.3

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

    if 2.259869264089033e+299 < (/ a1 (/ b1 a2))

    1. Initial program 14.8

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

    3. Applied associate-/l*15.5

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

    5. Applied div-inv16.9

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

    6. Applied simplify16.8

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

Runtime

Time bar (total: 24.7s)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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