Quotient of sum of exps

Average Error: 0.8 → 0.0

Time: 2.2s

Precision: binary64

Math

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

↓

\[\frac{1}{1 + e^{\left(-a\right) + b}}\]

Error

Bits error versus a

Bits error versus b

Target

Original	0.8
Target	0.0
Herbie	0.0

\[\frac{1}{1 + e^{b - a}}\]

Derivation

1. Initial program 0.8

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

3. Applied clear-num 0.8

   \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}}\]

4. Simplified 0.0

   \[\leadsto \frac{1}{\color{blue}{1 + e^{\left(-a\right) + b}}}\]

5. Final simplification 0.0

   \[\leadsto \frac{1}{1 + e^{\left(-a\right) + b}}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ 1.0 (exp (- b a))))

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