Average Error: 11.0 → 3.5
Time: 30.7s
Precision: 64
Internal Precision: 576
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}} = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}} \le -9.849108785460754 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{b2 \cdot \frac{b1}{a1 \cdot a2}}\\ \mathbf{if}\;\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}} \le 0.0:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{if}\;\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}} \le 2.663226549992245 \cdot 10^{+285}:\\ \;\;\;\;\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original11.0
Target10.8
Herbie3.5
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ 1 (/ b2 (/ (* a1 a2) b1))) < -inf.0 or 2.663226549992245e+285 < (/ 1 (/ b2 (/ (* a1 a2) b1)))

    1. Initial program 35.2

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

    3. Applied times-frac15.8

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

    if -inf.0 < (/ 1 (/ b2 (/ (* a1 a2) b1))) < -9.849108785460754e-309

    1. Initial program 8.8

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

    3. Applied associate-/r*0.9

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

    5. Applied clear-num1.0

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

    7. Applied div-inv1.3

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

    8. Applied simplify1.2

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

    if -9.849108785460754e-309 < (/ 1 (/ b2 (/ (* a1 a2) b1))) < 0.0

    1. Initial program 9.2

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

    3. Applied associate-/r*14.6

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

    5. Applied associate-/l*9.1

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

    7. Applied associate-/r/9.1

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

    8. Applied associate-/l*4.2

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

    if 0.0 < (/ 1 (/ b2 (/ (* a1 a2) b1))) < 2.663226549992245e+285

    1. Initial program 7.3

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

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

Runtime

Time bar (total: 30.7s)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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