Average Error: 0.6 → 0.6
Time: 19.6s
Precision: 64
Internal Precision: 128
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\frac{e^{a}}{(\left(\sqrt{e^{a}}\right) \cdot \left(\sqrt{e^{a}}\right) + \left(e^{b}\right))_*}\]

Error

Bits error versus a

Bits error versus b

Target

Original0.6
Target0.0
Herbie0.6
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.6

    \[\frac{e^{a}}{e^{a} + e^{b}}\]

  2. Initial simplification0.6

    \[\leadsto \frac{e^{a}}{e^{a} + e^{b}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.6

    \[\leadsto \frac{e^{a}}{\color{blue}{\sqrt{e^{a}} \cdot \sqrt{e^{a}}} + e^{b}}\]

  5. Applied fma-def0.6

    \[\leadsto \frac{e^{a}}{\color{blue}{(\left(\sqrt{e^{a}}\right) \cdot \left(\sqrt{e^{a}}\right) + \left(e^{b}\right))_*}}\]

  6. Final simplification0.6

    \[\leadsto \frac{e^{a}}{(\left(\sqrt{e^{a}}\right) \cdot \left(\sqrt{e^{a}}\right) + \left(e^{b}\right))_*}\]

Runtime

Time bar (total: 19.6s)Debug logProfile

herbie shell --seed 2018336 +o rules:numerics
(FPCore (a b)
  :name "Quotient of sum of exps"

  :herbie-target
  (/ 1 (+ 1 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))