Average Error: 0.6 → 1.0
Time: 20.0s
Precision: 64
Internal Precision: 128
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\frac{\frac{e^{a}}{\sqrt{e^{a} + e^{b}}}}{\sqrt{e^{a} + e^{b}}}\]

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.6
Target0.0
Herbie1.0
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.6

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

  3. Applied add-sqr-sqrt1.2

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

  4. Applied associate-/r*1.0

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

  5. Final simplification1.0

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

Reproduce

herbie shell --seed 2018360 
(FPCore (a b)
  :name "Quotient of sum of exps"

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

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

Details

Time bar (total: 13.1s)Debug log

start112.0ms

Algorithm
intervals

setup10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize34.0ms

Local error

Found 2 expressions with local error:

0.5b
(/ (exp a) (+ (exp a) (exp b)))
0.0b
(+ (exp a) (exp b))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
2 → 45
Calls

2 calls. Slowest were:

15.0ms
(/ (exp a) (+ (exp a) (exp b)))
6.0ms
(+ (exp a) (exp b))

series78.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

55.0ms
(/ (exp a) (+ (exp a) (exp b)))
23.0ms
(+ (exp a) (exp b))

simplify445.0ms

Counts
42 → 51
Calls

42 calls. Slowest were:

190.0ms
(/ (* (* (exp a) (exp a)) (exp a)) (* (* (+ (exp a) (exp b)) (+ (exp a) (exp b))) (+ (exp a) (exp b))))
55.0ms
(+ (* (exp a) (exp a)) (- (* (exp b) (exp b)) (* (exp a) (exp b))))
28.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))

prune518.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.5b

localize17.0ms

Local error

Found 4 expressions with local error:

0.7b
(/ (exp a) (sqrt (+ (exp a) (exp b))))
0.5b
(/ (/ (exp a) (sqrt (+ (exp a) (exp b)))) (sqrt (+ (exp a) (exp b))))
0.0b
(+ (exp a) (exp b))
0.0b
(+ (exp a) (exp b))

rewrite58.0ms

Algorithm
rewrite-expression-head
Counts
4 → 309
Calls

4 calls. Slowest were:

28.0ms
(/ (/ (exp a) (sqrt (+ (exp a) (exp b)))) (sqrt (+ (exp a) (exp b))))
12.0ms
(/ (exp a) (sqrt (+ (exp a) (exp b))))
6.0ms
(+ (exp a) (exp b))

series195.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

102.0ms
(/ (exp a) (sqrt (+ (exp a) (exp b))))
36.0ms
(/ (/ (exp a) (sqrt (+ (exp a) (exp b)))) (sqrt (+ (exp a) (exp b))))
30.0ms
(+ (exp a) (exp b))
27.0ms
(+ (exp a) (exp b))

simplify7.7s

Counts
489 → 321
Calls

489 calls. Slowest were:

592.0ms
(/ (* (* (/ (exp a) (sqrt (+ (exp a) (exp b)))) (/ (exp a) (sqrt (+ (exp a) (exp b))))) (/ (exp a) (sqrt (+ (exp a) (exp b))))) (* (* (sqrt (+ (exp a) (exp b))) (sqrt (+ (exp a) (exp b)))) (sqrt (+ (exp a) (exp b)))))
304.0ms
(- (+ (* a (sqrt 1/2)) (sqrt 1/2)) (+ (* 1/2 (* a (sqrt 1/8))) (* 1/8 (/ b (sqrt 1/2)))))
265.0ms
(/ (/ (* (* (exp a) (exp a)) (exp a)) (* (* (sqrt (+ (exp a) (exp b))) (sqrt (+ (exp a) (exp b)))) (sqrt (+ (exp a) (exp b))))) (* (* (sqrt (+ (exp a) (exp b))) (sqrt (+ (exp a) (exp b)))) (sqrt (+ (exp a) (exp b)))))

prune3.7s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.5b

localize8.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Counts
0 → 0
Calls

0 calls. Slowest were:

series0.0ms

Counts
0 → 0
Calls

0 calls. Slowest were:

simplify0.0ms

Counts
0 → 0
Calls

0 calls. Slowest were:

prune0.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.5b

regimes134.0ms

Accuracy

0% (0.4b remaining)

Error of 1.0b against oracle of 0.6b and baseline of 1.0b

bsearch1.0ms