Average Error: 0.8 → 1.3
Time: 3.3s
Precision: binary64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\frac{\sqrt{e^{a}}}{\sqrt{e^{a} + e^{b}}} \cdot \frac{\sqrt{e^{a}}}{\sqrt{e^{a} + e^{b}}}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\frac{\sqrt{e^{a}}}{\sqrt{e^{a} + e^{b}}} \cdot \frac{\sqrt{e^{a}}}{\sqrt{e^{a} + e^{b}}}
(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (*
  (/ (sqrt (exp a)) (sqrt (+ (exp a) (exp b))))
  (/ (sqrt (exp a)) (sqrt (+ (exp a) (exp 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(b))) * (sqrt(exp(a)) / sqrt(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

Original0.8
Target0.0
Herbie1.3
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.8

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

  3. Applied add-sqr-sqrt_binary641.3

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

  4. Applied add-sqr-sqrt_binary641.3

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

  5. Applied times-frac_binary641.3

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

  6. Final simplification1.3

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

Reproduce

herbie shell --seed 2020233 
(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))))