Average Error: 0.6 → 0.6
Time: 7.0s
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

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

Derivation

  1. Initial program 0.6

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

  2. Final simplification0.6

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

Reproduce

herbie shell --seed 2019022 +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: 6.6s)Debug log

sample110.0ms

Algorithm
intervals
Results
77.0ms105×body10240exit
23.0ms256×body80valid

simplify2.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
2.0ms
(/ (exp a) (+ (exp a) (exp b)))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.0b

localize14.0ms

Local error

Found 2 expressions with local error:

1.0b
(/ (exp a) (+ (exp a) (exp b)))
0.0b
(+ (exp a) (exp b))

rewrite21.0ms

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
13.0ms
(/ (exp a) (+ (exp a) (exp b)))
7.0ms
(+ (exp a) (exp b))

series96.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
72.0ms
(/ (exp a) (+ (exp a) (exp b)))
24.0ms
(+ (exp a) (exp b))

simplify554.0ms

Counts
44 → 58
Calls
44 calls:
Slowest
208.0ms
(/ (* (* (exp a) (exp a)) (exp a)) (* (* (+ (exp a) (exp b)) (+ (exp a) (exp b))) (+ (exp a) (exp b))))
82.0ms
(+ (* (exp a) (exp a)) (- (* (exp b) (exp b)) (* (exp a) (exp b))))
49.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))
36.0ms
(/ (* (cbrt (exp a)) (cbrt (exp a))) (* (cbrt (+ (exp a) (exp b))) (cbrt (+ (exp a) (exp b)))))
17.0ms
(/ (* (cbrt (exp a)) (cbrt (exp a))) 1)

prune602.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.7b

localize15.0ms

Local error

Found 1 expressions with local error:

0.0b
(/ (exp a) (+ a (+ b 2)))

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
9×add-cube-cbrt
9×times-frac
9×*-un-lft-identity
9×add-sqr-sqrt
3×associate-/l*
3×add-cbrt-cube
3×associate-/r*
2×associate-/r/
2×add-exp-log
1×add-log-exp
1×div-inv
1×log1p-expm1-u
1×flip-+
1×div-exp
1×flip3-+
1×frac-2neg
1×clear-num
1×cbrt-undiv
1×pow1
1×expm1-log1p-u
Counts
1 → 31
Calls
1 calls:
Slowest
4.0ms
(/ (exp a) (+ a (+ b 2)))

series83.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
83.0ms
(/ (exp a) (+ a (+ b 2)))

simplify604.0ms

Counts
30 → 34
Calls
30 calls:
Slowest
144.0ms
(+ (* a a) (- (* (+ b 2) (+ b 2)) (* a (+ b 2))))
119.0ms
(/ (* (* (exp a) (exp a)) (exp a)) (* (* (+ a (+ b 2)) (+ a (+ b 2))) (+ a (+ b 2))))
72.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))
20.0ms
(/ (* (cbrt (exp a)) (cbrt (exp a))) (* (cbrt (+ a (+ b 2))) (cbrt (+ a (+ b 2)))))
20.0ms
(- a (+ b 2))

prune280.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.7b

localize8.0ms

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

prune0.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0.7b

regimes174.0ms

Accuracy

0% (0.2b remaining)

Error of 0.6b against oracle of 0.4b and baseline of 0.6b

bsearch1.0ms

end0.0ms

sample4.0s

Algorithm
intervals
Results
3.1s3481×body10240exit
723.0ms8000×body80valid