Average Error: 0.6 → 0.5
Time: 22.3s
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 2018365 
(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: 15.4s)Debug log

start229.0ms

Algorithm
intervals

setup20.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize11.0ms

Local error

Found 2 expressions with local error:

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
2 → 45
Calls

2 calls. Slowest were:

6.0ms
(/ (exp a) (+ (exp a) (exp b)))
3.0ms
(+ (exp a) (exp b))

series406.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

380.0ms
(/ (exp a) (+ (exp a) (exp b)))
25.0ms
(+ (exp a) (exp b))

simplify576.0ms

Counts
42 → 51
Calls

42 calls. Slowest were:

244.0ms
(/ (* (* (exp a) (exp a)) (exp a)) (* (* (+ (exp a) (exp b)) (+ (exp a) (exp b))) (+ (exp a) (exp b))))
67.0ms
(+ (* (exp a) (exp a)) (- (* (exp b) (exp b)) (* (exp a) (exp b))))
48.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))

prune481.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.2b

localize17.0ms

Local error

Found 4 expressions with local error:

4.3b
(log (+ (exp a) (exp b)))
1.7b
(- a (log (+ (exp a) (exp b))))
0.0b
(+ (exp a) (exp b))
0.0b
(exp (- a (log (+ (exp a) (exp b)))))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

7.0ms
(+ (exp a) (exp b))
5.0ms
(log (+ (exp a) (exp b)))
4.0ms
(- a (log (+ (exp a) (exp b))))

series160.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

62.0ms
(exp (- a (log (+ (exp a) (exp b)))))
39.0ms
(log (+ (exp a) (exp b)))
33.0ms
(- a (log (+ (exp a) (exp b))))
26.0ms
(+ (exp a) (exp b))

simplify954.0ms

Counts
36 → 72
Calls

36 calls. Slowest were:

338.0ms
(- (+ (exp (- (log 2))) (* 1/2 (* a (exp (- (log 2)))))) (* 1/2 (* (exp (- (log 2))) b)))
190.0ms
(exp (- a (log (- (* (exp a) (exp a)) (* (exp b) (exp b))))))
83.0ms
(log (- (* (exp a) (exp a)) (* (exp b) (exp b))))

prune799.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.2b

localize29.0ms

Local error

Found 4 expressions with local error:

4.6b
(- (exp b) (exp a))
0.5b
(/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3)))
0.5b
(* (/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3))) (- (* (exp b) (- (exp b) (exp a))) (* (exp a) (- (exp a)))))
0.2b
(- (* (exp b) (- (exp b) (exp a))) (* (exp a) (- (exp a))))

rewrite116.0ms

Algorithm
rewrite-expression-head
Counts
4 → 149
Calls

4 calls. Slowest were:

75.0ms
(* (/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3))) (- (* (exp b) (- (exp b) (exp a))) (* (exp a) (- (exp a)))))
22.0ms
(- (* (exp b) (- (exp b) (exp a))) (* (exp a) (- (exp a))))
10.0ms
(/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3)))

series352.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

164.0ms
(* (/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3))) (- (* (exp b) (- (exp b) (exp a))) (* (exp a) (- (exp a)))))
88.0ms
(/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3)))
55.0ms
(- (* (exp b) (- (exp b) (exp a))) (* (exp a) (- (exp a))))
45.0ms
(- (exp b) (exp a))

simplify8.5s

Counts
149 → 161
Calls

149 calls. Slowest were:

455.0ms
(* (exp a) (- (pow (* (exp b) (- (exp b) (exp a))) 3) (pow (* (exp a) (- (exp a))) 3)))
363.0ms
(* (- (* (- (exp a)) (exp b)) (* (exp a) (- (exp a)))) (/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3))))
352.0ms
(* (- (* (exp a) (- (exp a)))) (/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3))))

prune2.2s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.2b

localize7.0ms

Local error

Found 1 expressions with local error:

0.0b
(- 1/2 (* 1/4 (- b a)))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 15
Calls

1 calls. Slowest were:

1.0ms
(- 1/2 (* 1/4 (- b a)))

series15.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

15.0ms
(- 1/2 (* 1/4 (- b a)))

simplify158.0ms

Counts
7 → 18
Calls

7 calls. Slowest were:

49.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))
47.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))
39.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))

prune188.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.2b

regimes165.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms