Average Error: 0.8 → 0.6
Time: 23.9s
Precision: binary64
Cost: 768
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[e^{a - \sqrt[3]{{\log \left(e^{a} + e^{b}\right)}^{3}}}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
e^{a - \sqrt[3]{{\log \left(e^{a} + e^{b}\right)}^{3}}}
(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (exp (- a (cbrt (pow (log (+ (exp a) (exp b))) 3.0)))))
double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}
↓
double code(double a, double b) {
	return exp(a - cbrt(pow(log(exp(a) + exp(b)), 3.0)));
}

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
Herbie0.6
\[\frac{1}{1 + e^{b - a}}\]
Alternative 1
Accuracy0.8
Cost640
\[\frac{-e^{a}}{\left(-e^{a}\right) - e^{b}}\]
Alternative 2
Accuracy0.8
Cost640
\[\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}\]

Derivation

  1. Initial program 0.8

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

  3. Applied add-exp-log_binary64_28440.8

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

  4. Applied div-exp_binary64_28570.6

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

  6. Applied add-cbrt-cube_binary64_28420.6

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

  7. Simplified0.6

    \[\leadsto e^{a - \sqrt[3]{\color{blue}{{\log \left(e^{a} + e^{b}\right)}^{3}}}}\]
  8. Using strategy rm

  9. Applied *-un-lft-identity_binary64_28060.6

    \[\leadsto e^{a - \color{blue}{1 \cdot \sqrt[3]{{\log \left(e^{a} + e^{b}\right)}^{3}}}}\]

  10. Final simplification0.6

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

Reproduce

herbie shell --seed 2020322 
(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))))