Average Error: 11.2 → 2.3
Time: 18.4s
Precision: 64
Internal Precision: 384
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot \frac{a2}{b1} = -\infty:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{if}\;a1 \cdot \frac{a2}{b1} \le -1.2338693843607816 \cdot 10^{-257}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \mathbf{if}\;a1 \cdot \frac{a2}{b1} \le -0.0:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{if}\;a1 \cdot \frac{a2}{b1} \le 1.2405521163106836 \cdot 10^{+238}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original11.2
Target10.5
Herbie2.3
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a1 (/ a2 b1)) or -1.2338693843607816e-257 < (* a1 (/ a2 b1)) < -0.0

    1. Initial program 7.1

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

    3. Applied times-frac6.9

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

    5. Applied associate-*l/3.5

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

    if (* a1 (/ a2 b1)) < -1.2338693843607816e-257 or -0.0 < (* a1 (/ a2 b1)) < 1.2405521163106836e+238

    1. Initial program 12.8

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

    3. Applied times-frac11.9

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

    5. Applied div-inv12.0

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

    6. Applied associate-*l*12.7

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

    7. Applied simplify7.1

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

    9. Applied div-inv7.1

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

    10. Applied associate-*r*0.6

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

    if 1.2405521163106836e+238 < (* a1 (/ a2 b1))

    1. Initial program 15.9

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

    3. Applied times-frac14.8

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

Runtime

Time bar (total: 18.4s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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