Average Error: 0.7 → 0.9
Time: 5.1s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\log \left({\left(e^{\frac{e^{a}}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}}\right)}^{\left({\left(e^{b}\right)}^{2} + \left({\left(e^{a}\right)}^{2} - e^{b + a}\right)\right)}\right)\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\log \left({\left(e^{\frac{e^{a}}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}}\right)}^{\left({\left(e^{b}\right)}^{2} + \left({\left(e^{a}\right)}^{2} - e^{b + a}\right)\right)}\right)
double f(double a, double b) {
        double r137566 = a;
        double r137567 = exp(r137566);
        double r137568 = b;
        double r137569 = exp(r137568);
        double r137570 = r137567 + r137569;
        double r137571 = r137567 / r137570;
        return r137571;
}
↓
double f(double a, double b) {
        double r137572 = a;
        double r137573 = exp(r137572);
        double r137574 = 3.0;
        double r137575 = pow(r137573, r137574);
        double r137576 = b;
        double r137577 = exp(r137576);
        double r137578 = pow(r137577, r137574);
        double r137579 = r137575 + r137578;
        double r137580 = r137573 / r137579;
        double r137581 = exp(r137580);
        double r137582 = 2.0;
        double r137583 = pow(r137577, r137582);
        double r137584 = pow(r137573, r137582);
        double r137585 = r137576 + r137572;
        double r137586 = exp(r137585);
        double r137587 = r137584 - r137586;
        double r137588 = r137583 + r137587;
        double r137589 = pow(r137581, r137588);
        double r137590 = log(r137589);
        return r137590;
}

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

Derivation

  1. Initial program 0.7

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

  3. Applied flip3-+17.6

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

  4. Applied associate-/r/17.6

    \[\leadsto \color{blue}{\frac{e^{a}}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}} \cdot \left(e^{a} \cdot e^{a} + \left(e^{b} \cdot e^{b} - e^{a} \cdot e^{b}\right)\right)}\]
  5. Using strategy rm

  6. Applied add-log-exp17.8

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

  7. Simplified0.9

    \[\leadsto \log \color{blue}{\left({\left(e^{\frac{e^{a}}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}}\right)}^{\left({\left(e^{b}\right)}^{2} + \left({\left(e^{a}\right)}^{2} - e^{b + a}\right)\right)}\right)}\]

  8. Final simplification0.9

    \[\leadsto \log \left({\left(e^{\frac{e^{a}}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}}\right)}^{\left({\left(e^{b}\right)}^{2} + \left({\left(e^{a}\right)}^{2} - e^{b + a}\right)\right)}\right)\]

Reproduce

herbie shell --seed 2020046 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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