Average Error: 0.5 → 0.9
Time: 1.7s
Precision: binary64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\frac{\frac{e^{a}}{\sqrt{e^{a} + e^{b}}}}{\sqrt{e^{a} + e^{b}}}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\frac{\frac{e^{a}}{\sqrt{e^{a} + e^{b}}}}{\sqrt{e^{a} + e^{b}}}
(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (/ (/ (exp a) (sqrt (+ (exp a) (exp b)))) (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 (exp(a) / sqrt(exp(a) + exp(b))) / 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

Original0.5
Target0.0
Herbie0.9
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.5

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

  3. Applied add-sqr-sqrt_binary641.1

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

  4. Applied associate-/r*_binary640.9

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

  5. Final simplification0.9

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

Reproduce

herbie shell --seed 2020224 
(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))))