Average Error: 10.9 → 3.4
Time: 31.5s
Precision: 64
Internal Precision: 576
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{a1 \cdot a2}{b1}}{b2} = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;\frac{\frac{a1 \cdot a2}{b1}}{b2} \le -2.5376693767344 \cdot 10^{-319}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{if}\;\frac{\frac{a1 \cdot a2}{b1}}{b2} \le 0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;\frac{\frac{a1 \cdot a2}{b1}}{b2} \le 2.22292973192715 \cdot 10^{+275}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original10.9
Target11.1
Herbie3.4
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (/ (* a1 a2) b1) b2) < -inf.0

    1. Initial program 36.3

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

    3. Applied times-frac14.2

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

    if -inf.0 < (/ (/ (* a1 a2) b1) b2) < -2.5376693767344e-319 or 0.0 < (/ (/ (* a1 a2) b1) b2) < 2.22292973192715e+275

    1. Initial program 7.8

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

    3. Applied associate-/r*0.8

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

    if -2.5376693767344e-319 < (/ (/ (* a1 a2) b1) b2) < 0.0 or 2.22292973192715e+275 < (/ (/ (* a1 a2) b1) b2)

    1. Initial program 12.5

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

    3. Applied times-frac5.9

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

Runtime

Time bar (total: 31.5s)Debug logProfile

herbie shell --seed '#(1071119240 1686926585 3481876196 78132896 2080707795 3185793749)' +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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