Average Error: 0.6 → 1.1
Time: 3.4s
Precision: binary64
\[\frac{e^{a}}{e^{a} + e^{b}} \]
↓
\[\frac{e^{a}}{\mathsf{fma}\left(0.5, a \cdot a, 1 + \left(a + e^{b}\right)\right)} \]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\frac{e^{a}}{\mathsf{fma}\left(0.5, a \cdot a, 1 + \left(a + e^{b}\right)\right)}

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (/ (exp a) (fma 0.5 (* a a) (+ 1.0 (+ a (exp b))))))
double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}
↓
double code(double a, double b) {
	return exp(a) / fma(0.5, (a * a), (1.0 + (a + exp(b))));
}

Error

Bits error versus a

Bits error versus b

Target

Original0.6
Target0.0
Herbie1.1
\[\frac{1}{1 + e^{b - a}} \]

Derivation

  1. Initial program 0.6

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

  2. Taylor expanded in a around 0 1.1

    \[\leadsto \frac{e^{a}}{\color{blue}{0.5 \cdot {a}^{2} + \left(1 + \left(e^{b} + a\right)\right)}} \]

  3. Simplified1.1

    \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(0.5, a \cdot a, 1 + \left(e^{b} + a\right)\right)}} \]

  4. Final simplification1.1

    \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(0.5, a \cdot a, 1 + \left(a + e^{b}\right)\right)} \]

Reproduce

herbie shell --seed 2021310 
(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))))