Average Error: 0.7 → 0.7
Time: 12.0s
Precision: 64
Internal Precision: 128
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\frac{e^{a}}{e^{a} + e^{b}}\]

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 0.7

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

  2. Taylor expanded around -inf 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(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: 11.4s)Debug log

sample121.0ms

Algorithm
intervals

simplify2.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

2.0ms
(/ (exp a) (+ (exp a) (exp b)))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.7b

localize11.0ms

Local error

Found 2 expressions with local error:

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

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
2 → 52
Calls

2 calls. Slowest were:

7.0ms
(/ (exp a) (+ (exp a) (exp b)))
3.0ms
(+ (exp a) (exp b))

series76.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

48.0ms
(/ (exp a) (+ (exp a) (exp b)))
28.0ms
(+ (exp a) (exp b))

simplify527.0ms

Counts
44 → 58
Calls

44 calls. Slowest were:

220.0ms
(/ (* (* (exp a) (exp a)) (exp a)) (* (* (+ (exp a) (exp b)) (+ (exp a) (exp b))) (+ (exp a) (exp b))))
78.0ms
(+ (* (exp a) (exp a)) (- (* (exp b) (exp b)) (* (exp a) (exp b))))
36.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))

prune601.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.4b

localize11.0ms

Local error

Found 2 expressions with local error:

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

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
2 → 52
Calls

2 calls. Slowest were:

12.0ms
(/ (exp a) (+ (exp b) (exp a)))
7.0ms
(+ (exp b) (exp a))

series101.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

64.0ms
(/ (exp a) (+ (exp b) (exp a)))
37.0ms
(+ (exp b) (exp a))

simplify663.0ms

Counts
44 → 58
Calls

44 calls. Slowest were:

230.0ms
(/ (* (* (exp a) (exp a)) (exp a)) (* (* (+ (exp b) (exp a)) (+ (exp b) (exp a))) (+ (exp b) (exp a))))
114.0ms
(+ (* (exp b) (exp b)) (- (* (exp a) (exp a)) (* (exp b) (exp a))))
73.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))

prune654.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.4b

localize8.0ms

Local error

Found 4 expressions with local error:

2.4b
(log (+ (exp b) (exp a)))
1.0b
(- a (log (+ (exp b) (exp a))))
0.0b
(exp (- a (log (+ (exp b) (exp a)))))
0.0b
(+ (exp b) (exp a))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 71
Calls

4 calls. Slowest were:

3.0ms
(+ (exp b) (exp a))
3.0ms
(log (+ (exp b) (exp a)))
2.0ms
(- a (log (+ (exp b) (exp a))))

series139.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

42.0ms
(- a (log (+ (exp b) (exp a))))
38.0ms
(log (+ (exp b) (exp a)))
37.0ms
(exp (- a (log (+ (exp b) (exp a)))))
22.0ms
(+ (exp b) (exp a))

simplify997.0ms

Counts
38 → 83
Calls

38 calls. Slowest were:

249.0ms
(exp (- a (log (- (* (exp b) (exp b)) (* (exp a) (exp a))))))
235.0ms
(- (+ (exp (- (log 2))) (* 1/2 (* a (exp (- (log 2)))))) (* 1/2 (* (exp (- (log 2))) b)))
127.0ms
(log (- (* (exp b) (exp b)) (* (exp a) (exp a))))

prune885.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.4b

localize13.0ms

Local error

Found 2 expressions with local error:

0.0b
(exp (- (* 1/2 (- a b)) (log 2)))
0.0b
(- (* 1/2 (- a b)) (log 2))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
2 → 46
Calls

2 calls. Slowest were:

8.0ms
(exp (- (* 1/2 (- a b)) (log 2)))
6.0ms
(- (* 1/2 (- a b)) (log 2))

series61.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

35.0ms
(- (* 1/2 (- a b)) (log 2))
26.0ms
(exp (- (* 1/2 (- a b)) (log 2)))

simplify1.9s

Counts
32 → 52
Calls

32 calls. Slowest were:

264.0ms
(- (+ (exp (- (log 2))) (* 1/2 (* a (exp (- (log 2)))))) (* 1/2 (* (exp (- (log 2))) b)))
163.0ms
(exp (- (* 1/2 a) (+ (log 2) (* 1/2 b))))
156.0ms
(exp (- (* 1/2 a) (+ (* 1/2 b) (log 2))))

prune465.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.4b

regimes147.0ms

Accuracy

0% (0.2b remaining)

Error of 0.7b against oracle of 0.5b and baseline of 0.7b

bsearch1.0ms

end0.0ms

sample4.0s

Algorithm
intervals