Average Error: 0.6 → 0.5
Time: 15.0s
Precision: 64
Internal Precision: 128
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[e^{a - \log \left(e^{a} + e^{b}\right)}\]

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
Herbie0.5
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.6

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

  2. Initial simplification0.6

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

  4. Applied add-exp-log0.6

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

  5. Applied div-exp0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2018362 +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: 7.7s)Debug log

start243.0ms

Algorithm
intervals

setup19.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.8b

localize20.0ms

Local error

Found 2 expressions with local error:

0.8b
(/ (exp a) (+ (exp a) (exp b)))
0.0b
(+ (exp a) (exp b))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
2 → 52
Calls

2 calls. Slowest were:

8.0ms
(/ (exp a) (+ (exp a) (exp b)))
3.0ms
(+ (exp a) (exp b))

series87.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

63.0ms
(/ (exp a) (+ (exp a) (exp b)))
25.0ms
(+ (exp a) (exp b))

simplify569.0ms

Counts
44 → 58
Calls

44 calls. Slowest were:

251.0ms
(/ (* (* (exp a) (exp a)) (exp a)) (* (* (+ (exp a) (exp b)) (+ (exp a) (exp b))) (+ (exp a) (exp b))))
124.0ms
(+ (* (exp a) (exp a)) (- (* (exp b) (exp b)) (* (exp a) (exp b))))
37.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))

prune563.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.5b

localize12.0ms

Local error

Found 4 expressions with local error:

2.7b
(log (+ (exp a) (exp b)))
0.2b
(- a (log (+ (exp a) (exp b))))
0.0b
(exp (- a (log (+ (exp a) (exp b)))))
0.0b
(+ (exp a) (exp b))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 71
Calls

4 calls. Slowest were:

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

series191.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

69.0ms
(exp (- a (log (+ (exp a) (exp b)))))
49.0ms
(- a (log (+ (exp a) (exp b))))
44.0ms
(log (+ (exp a) (exp b)))
29.0ms
(+ (exp a) (exp b))

simplify1.2s

Counts
38 → 83
Calls

38 calls. Slowest were:

410.0ms
(exp (- a (log (- (* (exp a) (exp a)) (* (exp b) (exp b))))))
272.0ms
(- (+ (exp (- (log 2))) (* 1/2 (* a (exp (- (log 2)))))) (* 1/2 (* (exp (- (log 2))) b)))
114.0ms
(- (* 1/2 a) (+ (* 1/2 b) (log 2)))

prune903.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.5b

localize8.0ms

Local error

Found 4 expressions with local error:

0.7b
(/ (sqrt (exp a)) (+ (exp a) (exp b)))
0.5b
(* (sqrt (exp a)) (/ (sqrt (exp a)) (+ (exp a) (exp b))))
0.2b
(sqrt (exp a))
0.2b
(sqrt (exp a))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 130
Calls

4 calls. Slowest were:

10.0ms
(* (sqrt (exp a)) (/ (sqrt (exp a)) (+ (exp a) (exp b))))
7.0ms
(/ (sqrt (exp a)) (+ (exp a) (exp b)))
1.0ms
(sqrt (exp a))

series165.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

79.0ms
(/ (sqrt (exp a)) (+ (exp a) (exp b)))
58.0ms
(* (sqrt (exp a)) (/ (sqrt (exp a)) (+ (exp a) (exp b))))
15.0ms
(sqrt (exp a))
13.0ms
(sqrt (exp a))

simplify2.0s

Counts
122 → 142
Calls

122 calls. Slowest were:

560.0ms
(* (* (* (sqrt (exp a)) (sqrt (exp a))) (sqrt (exp a))) (* (* (/ (sqrt (exp a)) (+ (exp a) (exp b))) (/ (sqrt (exp a)) (+ (exp a) (exp b)))) (/ (sqrt (exp a)) (+ (exp a) (exp b)))))
287.0ms
(/ (* (* (sqrt (exp a)) (sqrt (exp a))) (sqrt (exp a))) (* (* (+ (exp a) (exp b)) (+ (exp a) (exp b))) (+ (exp a) (exp b))))
160.0ms
(* (* (* (sqrt (exp a)) (sqrt (exp a))) (sqrt (exp a))) (/ (* (* (sqrt (exp a)) (sqrt (exp a))) (sqrt (exp a))) (* (* (+ (exp a) (exp b)) (+ (exp a) (exp b))) (+ (exp a) (exp b)))))

prune1.5s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.5b

localize14.0ms

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:

prune1.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0.5b

regimes192.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms