Average Error: 0.7 → 0.7
Time: 2.3s
Precision: binary64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\sqrt{e^{a}} \cdot \frac{\sqrt{e^{a}}}{e^{a} + e^{b}}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\sqrt{e^{a}} \cdot \frac{\sqrt{e^{a}}}{e^{a} + e^{b}}
double code(double a, double b) {
	return ((double) (((double) exp(a)) / ((double) (((double) exp(a)) + ((double) exp(b))))));
}
↓
double code(double a, double b) {
	return ((double) (((double) sqrt(((double) exp(a)))) * ((double) (((double) sqrt(((double) exp(a)))) / ((double) (((double) exp(a)) + ((double) exp(b))))))));
}

Error

Bits error versus a

Bits error versus b

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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 *-un-lft-identity0.7

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

  4. Applied add-sqr-sqrt0.7

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

  5. Applied times-frac0.7

    \[\leadsto \color{blue}{\frac{\sqrt{e^{a}}}{1} \cdot \frac{\sqrt{e^{a}}}{e^{a} + e^{b}}}\]

  6. Simplified0.7

    \[\leadsto \color{blue}{\sqrt{e^{a}}} \cdot \frac{\sqrt{e^{a}}}{e^{a} + e^{b}}\]

  7. Final simplification0.7

    \[\leadsto \sqrt{e^{a}} \cdot \frac{\sqrt{e^{a}}}{e^{a} + e^{b}}\]

Reproduce

herbie shell --seed 2020173 
(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))))