Average Error: 0.7 → 0.7
Time: 15.1s
Precision: 64
Internal Precision: 128
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[(e^{\log_* (1 + \frac{e^{a}}{e^{a} + e^{b}})} - 1)^*\]

Error

Bits error versus a

Bits error versus b

Target

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

Derivation

  1. Initial program 0.7

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

  3. Applied expm1-log1p-u0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019053 +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))))