Average Error: 0.7 → 0.7
Time: 3.8s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\log \left({\left(e^{e^{a}}\right)}^{\left(\frac{1}{e^{a} + e^{b}}\right)}\right)\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\log \left({\left(e^{e^{a}}\right)}^{\left(\frac{1}{e^{a} + e^{b}}\right)}\right)
double f(double a, double b) {
        double r140991 = a;
        double r140992 = exp(r140991);
        double r140993 = b;
        double r140994 = exp(r140993);
        double r140995 = r140992 + r140994;
        double r140996 = r140992 / r140995;
        return r140996;
}
↓
double f(double a, double b) {
        double r140997 = a;
        double r140998 = exp(r140997);
        double r140999 = exp(r140998);
        double r141000 = 1.0;
        double r141001 = b;
        double r141002 = exp(r141001);
        double r141003 = r140998 + r141002;
        double r141004 = r141000 / r141003;
        double r141005 = pow(r140999, r141004);
        double r141006 = log(r141005);
        return r141006;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.7
Target0.0
Herbie0.7
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.7

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

  3. Applied add-log-exp0.9

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

  5. Applied div-inv0.9

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

  6. Applied exp-prod0.7

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

  7. Final simplification0.7

    \[\leadsto \log \left({\left(e^{e^{a}}\right)}^{\left(\frac{1}{e^{a} + e^{b}}\right)}\right)\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1 (+ 1 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))