Average Error: 10.8 → 3.0
Time: 15.0s
Precision: 64
Internal Precision: 384
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\frac{b2}{\frac{a2}{b1}}} = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;\frac{1}{\frac{b2}{\frac{a2}{b1}}} \le -4.7363803091445446 \cdot 10^{-268}:\\ \;\;\;\;a1 \cdot \frac{1}{\frac{b2}{\frac{a2}{b1}}}\\ \mathbf{if}\;\frac{1}{\frac{b2}{\frac{a2}{b1}}} \le 6.182262960128672 \cdot 10^{-280}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;\frac{1}{\frac{b2}{\frac{a2}{b1}}} \le 3.3051914724971476 \cdot 10^{+280}:\\ \;\;\;\;a1 \cdot \frac{1}{\frac{b2}{\frac{a2}{b1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a1 \cdot a2\right) \cdot \frac{1}{b2}}{b1}\\ \end{array}\]

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original10.8
Target10.6
Herbie3.0
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ 1 (/ b2 (/ a2 b1))) < -inf.0 or -4.7363803091445446e-268 < (/ 1 (/ b2 (/ a2 b1))) < 6.182262960128672e-280

    1. Initial program 6.9

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

    3. Applied times-frac6.9

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

    if -inf.0 < (/ 1 (/ b2 (/ a2 b1))) < -4.7363803091445446e-268 or 6.182262960128672e-280 < (/ 1 (/ b2 (/ a2 b1))) < 3.3051914724971476e+280

    1. Initial program 12.6

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

    3. Applied times-frac12.3

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

    5. Applied div-inv12.4

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

    6. Applied associate-*l*6.1

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

    7. Applied simplify0.5

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

    9. Applied clear-num0.5

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

    if 3.3051914724971476e+280 < (/ 1 (/ b2 (/ a2 b1)))

    1. Initial program 15.2

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

    3. Applied times-frac14.4

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

    5. Applied associate-*l/13.4

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

    7. Applied div-inv13.4

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

    8. Applied associate-*r*7.4

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

Runtime

Time bar (total: 15.0s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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