Average Error: 0.7 → 0.7
Time: 20.3s
Precision: 64
Internal Precision: 128
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\frac{\sqrt{e^{a}}}{e^{a} + e^{b}} \cdot \sqrt{e^{a}}\]

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

Derivation

  1. Initial program 0.7

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

  3. Applied *-un-lft-identity0.7

    \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{1 \cdot e^{b}}}\]

  4. Applied *-un-lft-identity0.7

    \[\leadsto \frac{e^{a}}{\color{blue}{1 \cdot e^{a}} + 1 \cdot e^{b}}\]

  5. Applied distribute-lft-out0.7

    \[\leadsto \frac{e^{a}}{\color{blue}{1 \cdot \left(e^{a} + e^{b}\right)}}\]

  6. Applied add-sqr-sqrt0.7

    \[\leadsto \frac{\color{blue}{\sqrt{e^{a}} \cdot \sqrt{e^{a}}}}{1 \cdot \left(e^{a} + e^{b}\right)}\]

  7. Applied times-frac0.7

    \[\leadsto \color{blue}{\frac{\sqrt{e^{a}}}{1} \cdot \frac{\sqrt{e^{a}}}{e^{a} + e^{b}}}\]

  8. Simplified0.7

    \[\leadsto \color{blue}{\sqrt{e^{a}}} \cdot \frac{\sqrt{e^{a}}}{e^{a} + e^{b}}\]

  9. Final simplification0.7

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

Reproduce

herbie shell --seed 2019005 
(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: 19.8s)Debug log

sample128.0ms

Algorithm
intervals

simplify4.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

4.0ms
(/ (exp a) (+ (exp a) (exp b)))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.7b

localize20.0ms

Local error

Found 2 expressions with local error:

0.7b
(/ (exp a) (+ (exp a) (exp b)))
0.0b
(+ (exp a) (exp b))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
2 → 45
Calls

2 calls. Slowest were:

12.0ms
(/ (exp a) (+ (exp a) (exp b)))
6.0ms
(+ (exp a) (exp b))

series99.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

62.0ms
(/ (exp a) (+ (exp a) (exp b)))
36.0ms
(+ (exp a) (exp b))

simplify478.0ms

Counts
42 → 51
Calls

42 calls. Slowest were:

175.0ms
(/ (* (* (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))))
53.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))

prune419.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.7b

localize14.0ms

Local error

Found 4 expressions with local error:

0.7b
(/ (sqrt (exp a)) (+ (exp a) (exp b)))
0.7b
(* (sqrt (exp a)) (/ (sqrt (exp a)) (+ (exp a) (exp b))))
0.0b
(+ (exp a) (exp b))
0.0b
(sqrt (exp a))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 122
Calls

4 calls. Slowest were:

10.0ms
(/ (sqrt (exp a)) (+ (exp a) (exp b)))
10.0ms
(* (sqrt (exp a)) (/ (sqrt (exp a)) (+ (exp a) (exp b))))
3.0ms
(+ (exp a) (exp b))

series177.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

66.0ms
(* (sqrt (exp a)) (/ (sqrt (exp a)) (+ (exp a) (exp b))))
55.0ms
(/ (sqrt (exp a)) (+ (exp a) (exp b)))
38.0ms
(+ (exp a) (exp b))
19.0ms
(sqrt (exp a))

simplify1.9s

Counts
120 → 134
Calls

120 calls. Slowest were:

501.0ms
(* (* (* (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)))))
379.0ms
(/ (* (* (sqrt (exp a)) (sqrt (exp a))) (sqrt (exp a))) (* (* (+ (exp a) (exp b)) (+ (exp a) (exp b))) (+ (exp a) (exp b))))
140.0ms
(* (* (* (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)))))

prune1.3s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.7b

localize27.0ms

Local error

Found 4 expressions with local error:

4.4b
(- (exp b) (exp a))
1.5b
(* (/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3))) (- (* (exp b) (- (exp b) (exp a))) (* (exp a) (- (exp a)))))
0.8b
(/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3)))
0.4b
(- (* (exp b) (- (exp b) (exp a))) (* (exp a) (- (exp a))))

rewrite133.0ms

Algorithm
rewrite-expression-head
Counts
4 → 149
Calls

4 calls. Slowest were:

85.0ms
(* (/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3))) (- (* (exp b) (- (exp b) (exp a))) (* (exp a) (- (exp a)))))
19.0ms
(/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3)))
19.0ms
(- (* (exp b) (- (exp b) (exp a))) (* (exp a) (- (exp a))))

series347.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

188.0ms
(* (/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3))) (- (* (exp b) (- (exp b) (exp a))) (* (exp a) (- (exp a)))))
77.0ms
(/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3)))
55.0ms
(- (* (exp b) (- (exp b) (exp a))) (* (exp a) (- (exp a))))
27.0ms
(- (exp b) (exp a))

simplify8.4s

Counts
149 → 161
Calls

149 calls. Slowest were:

513.0ms
(+ (log (/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3)))) (log (- (* (exp b) (- (exp b) (exp a))) (* (exp a) (- (exp a))))))
382.0ms
(* (exp a) (- (pow (* (exp b) (- (exp b) (exp a))) 3) (pow (* (exp a) (- (exp a))) 3)))
343.0ms
(* (- (* (exp a) (- (exp a)))) (/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3))))

prune2.0s

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
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:

prune0.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0.7b

regimes114.0ms

Accuracy

0% (0.0b remaining)

Error of 0.7b against oracle of 0.7b and baseline of 0.7b

bsearch2.0ms

end0.0ms

sample4.1s

Algorithm
intervals