Average Error: 0.6 → 0.5
Time: 3.3s
Precision: binary64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[e^{a - \sqrt[3]{\log \left(e^{b} + e^{a}\right) \cdot \left(\log \left(e^{b} + e^{a}\right) \cdot \log \left(e^{b} + e^{a}\right)\right)}}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
e^{a - \sqrt[3]{\log \left(e^{b} + e^{a}\right) \cdot \left(\log \left(e^{b} + e^{a}\right) \cdot \log \left(e^{b} + e^{a}\right)\right)}}
(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (exp
  (-
   a
   (cbrt
    (*
     (log (+ (exp b) (exp a)))
     (* (log (+ (exp b) (exp a))) (log (+ (exp b) (exp a)))))))))
double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}
↓
double code(double a, double b) {
	return exp(a - cbrt(log(exp(b) + exp(a)) * (log(exp(b) + exp(a)) * log(exp(b) + exp(a)))));
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.6
Target0.0
Herbie0.5
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.6

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

  3. Applied add-exp-log_binary640.6

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

  4. Applied div-exp_binary640.5

    \[\leadsto \color{blue}{e^{a - \log \left(e^{a} + e^{b}\right)}}\]

  5. Simplified0.5

    \[\leadsto e^{\color{blue}{a - \log \left(e^{b} + e^{a}\right)}}\]
  6. Using strategy rm

  7. Applied add-cbrt-cube_binary640.5

    \[\leadsto e^{a - \color{blue}{\sqrt[3]{\left(\log \left(e^{b} + e^{a}\right) \cdot \log \left(e^{b} + e^{a}\right)\right) \cdot \log \left(e^{b} + e^{a}\right)}}}\]

  8. Final simplification0.5

    \[\leadsto e^{a - \sqrt[3]{\log \left(e^{b} + e^{a}\right) \cdot \left(\log \left(e^{b} + e^{a}\right) \cdot \log \left(e^{b} + e^{a}\right)\right)}}\]

Reproduce

herbie shell --seed 2021174 
(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))))