Average Error: 0.9 → 0.9
Time: 3.7s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\sqrt{e^{a}} \cdot \frac{\sqrt{e^{a}}}{e^{a} + e^{b}}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\sqrt{e^{a}} \cdot \frac{\sqrt{e^{a}}}{e^{a} + e^{b}}
double code(double a, double b) {
	return (exp(a) / (exp(a) + exp(b)));
}
↓
double code(double a, double b) {
	return (sqrt(exp(a)) * (sqrt(exp(a)) / (exp(a) + exp(b))));
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.9
Target0.0
Herbie0.9
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.9

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

  3. Applied *-un-lft-identity0.9

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

  4. Applied add-sqr-sqrt0.9

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

  5. Applied times-frac0.9

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

  6. Simplified0.9

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

  7. Final simplification0.9

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

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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