Average Error: 0.7 → 0.6
Time: 9.7s
Precision: 64
Internal Precision: 128
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[e^{a - \log \left(e^{a} + e^{b}\right)}\]

Error

Bits error versus a

Bits error versus b

Target

Original0.7
Target0.0
Herbie0.6
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.7

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

  3. Applied add-exp-log0.7

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

  4. Applied div-exp0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019050 
(FPCore (a b)
  :name "Quotient of sum of exps"

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

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