Average Error: 0.6 → 0.5
Time: 16.3s
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 +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: 9.7s)Debug log

start159.0ms

Algorithm
intervals

setup11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize10.0ms

Local error

Found 2 expressions with local error:

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

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
2 → 52
Calls

2 calls. Slowest were:

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

series86.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

56.0ms
(/ (exp a) (+ (exp a) (exp b)))
30.0ms
(+ (exp a) (exp b))

simplify605.0ms

Counts
44 → 58
Calls

44 calls. Slowest were:

244.0ms
(/ (* (* (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))))
54.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))

prune630.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.2b

localize16.0ms

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)))))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 71
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))))

series187.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

62.0ms
(log (+ (exp a) (exp b)))
52.0ms
(- a (log (+ (exp a) (exp b))))
38.0ms
(exp (- a (log (+ (exp a) (exp b)))))
35.0ms
(+ (exp a) (exp b))

simplify1.6s

Counts
38 → 83
Calls

38 calls. Slowest were:

732.0ms
(exp (- a (log (- (* (exp a) (exp a)) (* (exp b) (exp b))))))
325.0ms
(- (+ (exp (- (log 2))) (* 1/2 (* a (exp (- (log 2)))))) (* 1/2 (* (exp (- (log 2))) b)))
119.0ms
(log (- (* (exp a) (exp a)) (* (exp b) (exp b))))

prune919.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.2b

localize34.0ms

Local error

Found 4 expressions with local error:

8.6b
(log (fma (- (exp b) (exp a)) (exp b) (* (exp a) (exp a))))
4.6b
(- (exp b) (exp a))
4.2b
(log (+ (pow (exp a) 3) (pow (exp b) 3)))
1.9b
(+ (- a (log (+ (pow (exp a) 3) (pow (exp b) 3)))) (log (fma (- (exp b) (exp a)) (exp b) (* (exp a) (exp a)))))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

5.0ms
(log (+ (pow (exp a) 3) (pow (exp b) 3)))
5.0ms
(+ (- a (log (+ (pow (exp a) 3) (pow (exp b) 3)))) (log (fma (- (exp b) (exp a)) (exp b) (* (exp a) (exp a)))))
4.0ms
(- (exp b) (exp a))

series396.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

154.0ms
(+ (- a (log (+ (pow (exp a) 3) (pow (exp b) 3)))) (log (fma (- (exp b) (exp a)) (exp b) (* (exp a) (exp a)))))
138.0ms
(log (fma (- (exp b) (exp a)) (exp b) (* (exp a) (exp a))))
77.0ms
(log (+ (pow (exp a) 3) (pow (exp b) 3)))
27.0ms
(- (exp b) (exp a))

simplify2.8s

Counts
62 → 102
Calls

62 calls. Slowest were:

307.0ms
(- a (log (+ (pow (pow (exp a) 3) 3) (pow (pow (exp b) 3) 3))))
247.0ms
(* (exp (- a (log (+ (pow (exp a) 3) (pow (exp b) 3))))) (fma (- (exp b) (exp a)) (exp b) (* (exp a) (exp a))))
237.0ms
(- a (log (- (* (pow (exp a) 3) (pow (exp a) 3)) (* (pow (exp b) 3) (pow (exp b) 3)))))

prune1.8s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.2b

localize14.0ms

Local error

Found 1 expressions with local error:

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

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 10
Calls

1 calls. Slowest were:

0.0ms
(fma 1/4 (- a b) 1/2)

series18.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

18.0ms
(fma 1/4 (- a b) 1/2)

simplify189.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

71.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))
62.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))
56.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))

prune117.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.2b

regimes87.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms