Average Error: 0.7 → 0.6
Time: 8.5s
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

Original0.7
Target0.0
Herbie0.6
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.7

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

  3. Applied add-exp-log0.7

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

  4. Applied div-exp0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019010 +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: 7.9s)Debug log

sample116.0ms

Algorithm
intervals

simplify4.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
4.0ms
(/ (exp a) (+ (exp a) (exp b)))

prune24.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize23.0ms

Local error

Found 2 expressions with local error:

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

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
22×*-un-lft-identity
12×add-cube-cbrt
12×times-frac
12×add-sqr-sqrt
5×distribute-lft-out
4×add-log-exp
4×add-cbrt-cube
4×associate-/r*
3×associate-/l*
3×add-exp-log
3×fma-def
2×log1p-expm1-u
2×flip-+
2×associate-/r/
2×flip3-+
2×pow1
2×expm1-log1p-u
1×div-inv
1×div-exp
1×frac-2neg
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
2 → 52
Calls
2 calls:
Slowest
10.0ms
(/ (exp a) (+ (exp a) (exp b)))
3.0ms
(+ (exp a) (exp b))

series70.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
46.0ms
(/ (exp a) (+ (exp a) (exp b)))
24.0ms
(+ (exp a) (exp b))

simplify607.0ms

Counts
44 → 58
Calls
44 calls:
Slowest
266.0ms
(/ (* (* (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))))
68.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))
20.0ms
(/ (* (cbrt (exp a)) (cbrt (exp a))) (* (cbrt (+ (exp a) (exp b))) (cbrt (+ (exp a) (exp b)))))
12.0ms
(+ a (+ b 2))

prune506.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize9.0ms

Local error

Found 4 expressions with local error:

2.8b
(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
Rules
14×*-un-lft-identity
8×add-cube-cbrt
8×log-prod
8×add-sqr-sqrt
7×add-log-exp
6×log-div
5×add-exp-log
5×pow1
4×associate--r-
4×associate--r+
4×log1p-expm1-u
4×flip-+
4×exp-sum
4×add-cbrt-cube
4×flip3-+
4×expm1-log1p-u
3×distribute-lft-out
3×fma-def
3×exp-prod
2×sub-neg
1×rem-exp-log
1×flip--
1×log-pow
1×rem-log-exp
1×exp-diff
1×flip3--
1×sum-log
1×+-commutative
Counts
4 → 71
Calls
4 calls:
Slowest
4.0ms
(+ (exp a) (exp b))
3.0ms
(log (+ (exp a) (exp b)))
2.0ms
(- a (log (+ (exp a) (exp b))))
2.0ms
(exp (- a (log (+ (exp a) (exp b)))))

series169.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
57.0ms
(log (+ (exp a) (exp b)))
44.0ms
(- a (log (+ (exp a) (exp b))))
38.0ms
(exp (- a (log (+ (exp a) (exp b)))))
30.0ms
(+ (exp a) (exp b))

simplify1.1s

Counts
38 → 83
Calls
38 calls:
Slowest
353.0ms
(exp (- a (log (- (* (exp a) (exp a)) (* (exp b) (exp b))))))
310.0ms
(- (+ (exp (- (log 2))) (* 1/2 (* a (exp (- (log 2)))))) (* 1/2 (* (exp (- (log 2))) b)))
97.0ms
(- (* 1/2 a) (+ (* 1/2 b) (log 2)))
85.0ms
(log (- (* (exp a) (exp a)) (* (exp b) (exp b))))
59.0ms
(log (+ (* (exp a) (exp a)) (- (* (exp b) (exp b)) (* (exp a) (exp b)))))

prune963.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.2b

localize16.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Rules
Counts
0 → 0
Calls
0 calls:
Slowest

series0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

simplify0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

prune1.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.2b

regimes91.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample4.2s

Algorithm
intervals