Average Error: 0.7 → 0.8
Time: 3.7s
Precision: binary64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[e^{a - \left(\log \left(1 + e^{b}\right) + \frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\frac{1 + e^{b}}{\sqrt[3]{a}}}\right)}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
e^{a - \left(\log \left(1 + e^{b}\right) + \frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\frac{1 + e^{b}}{\sqrt[3]{a}}}\right)}
(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (exp
  (-
   a
   (+
    (log (+ 1.0 (exp b)))
    (/ (* (cbrt a) (cbrt a)) (/ (+ 1.0 (exp b)) (cbrt a)))))))
double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}
↓
double code(double a, double b) {
	return exp(a - (log(1.0 + exp(b)) + ((cbrt(a) * cbrt(a)) / ((1.0 + exp(b)) / cbrt(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

Original0.7
Target0.0
Herbie0.8
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.7

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

  3. Applied add-exp-log_binary640.7

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

  4. Applied div-exp_binary640.6

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

  5. Simplified0.6

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

  6. Taylor expanded around 0 0.7

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

  7. Simplified0.7

    \[\leadsto e^{a - \color{blue}{\left(\log \left(1 + e^{b}\right) + \frac{a}{1 + e^{b}}\right)}}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt_binary640.8

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

  10. Applied associate-/l*_binary640.8

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

  11. Final simplification0.8

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

Reproduce

herbie shell --seed 2021173 
(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))))