Average Error: 0.7 → 0.7
Time: 22.0s
Precision: 64
Internal Precision: 128
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\frac{\sqrt{e^{a}}}{(\left(\sqrt{e^{a}}\right) \cdot \left(\sqrt{e^{a}}\right) + \left(e^{b}\right))_*} \cdot \sqrt{e^{a}}\]

Error

Bits error versus a

Bits error versus b

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. Using strategy rm

  10. Applied add-sqr-sqrt0.7

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

  11. Applied fma-def0.7

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

  12. Final simplification0.7

    \[\leadsto \frac{\sqrt{e^{a}}}{(\left(\sqrt{e^{a}}\right) \cdot \left(\sqrt{e^{a}}\right) + \left(e^{b}\right))_*} \cdot \sqrt{e^{a}}\]

Reproduce

herbie shell --seed 2019005 +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: 21.5s)Debug log

sample80.0ms

Algorithm
intervals

simplify3.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.7b

localize41.0ms

Local error

Found 2 expressions with local error:

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

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
2 → 52
Calls

2 calls. Slowest were:

14.0ms
(/ (exp a) (+ (exp a) (exp b)))
6.0ms
(+ (exp a) (exp b))

series80.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

53.0ms
(/ (exp a) (+ (exp a) (exp b)))
27.0ms
(+ (exp a) (exp b))

simplify586.0ms

Counts
44 → 58
Calls

44 calls. Slowest were:

294.0ms
(/ (* (* (exp a) (exp a)) (exp a)) (* (* (+ (exp a) (exp b)) (+ (exp a) (exp b))) (+ (exp a) (exp b))))
65.0ms
(+ (* (exp a) (exp a)) (- (* (exp b) (exp b)) (* (exp a) (exp b))))
40.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))

prune466.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.7b

localize8.0ms

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))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 133
Calls

4 calls. Slowest were:

10.0ms
(* (sqrt (exp a)) (/ (sqrt (exp a)) (+ (exp a) (exp b))))
7.0ms
(/ (sqrt (exp a)) (+ (exp a) (exp b)))
3.0ms
(+ (exp a) (exp b))

series154.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

56.0ms
(/ (sqrt (exp a)) (+ (exp a) (exp b)))
47.0ms
(* (sqrt (exp a)) (/ (sqrt (exp a)) (+ (exp a) (exp b))))
36.0ms
(+ (exp a) (exp b))
16.0ms
(sqrt (exp a))

simplify2.1s

Counts
122 → 145
Calls

122 calls. Slowest were:

459.0ms
(* (* (* (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)))))
356.0ms
(/ (* (* (sqrt (exp a)) (sqrt (exp a))) (sqrt (exp a))) (* (* (+ (exp a) (exp b)) (+ (exp a) (exp b))) (+ (exp a) (exp b))))
151.0ms
(* (* (* (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)))))

prune1.7s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.7b

localize18.0ms

Local error

Found 4 expressions with local error:

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

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 109
Calls

4 calls. Slowest were:

12.0ms
(* (sqrt (exp a)) (/ (sqrt (exp a)) (fma (sqrt (exp a)) (sqrt (exp a)) (exp b))))
8.0ms
(/ (sqrt (exp a)) (fma (sqrt (exp a)) (sqrt (exp a)) (exp b)))
2.0ms
(sqrt (exp a))

series237.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

114.0ms
(/ (sqrt (exp a)) (fma (sqrt (exp a)) (sqrt (exp a)) (exp b)))
64.0ms
(* (sqrt (exp a)) (/ (sqrt (exp a)) (fma (sqrt (exp a)) (sqrt (exp a)) (exp b))))
45.0ms
(fma (sqrt (exp a)) (sqrt (exp a)) (exp b))
14.0ms
(sqrt (exp a))

simplify2.5s

Counts
95 → 121
Calls

95 calls. Slowest were:

490.0ms
(/ (* (* (sqrt (exp a)) (sqrt (exp a))) (sqrt (exp a))) (* (* (fma (sqrt (exp a)) (sqrt (exp a)) (exp b)) (fma (sqrt (exp a)) (sqrt (exp a)) (exp b))) (fma (sqrt (exp a)) (sqrt (exp a)) (exp b))))
465.0ms
(* (* (* (sqrt (exp a)) (sqrt (exp a))) (sqrt (exp a))) (* (* (/ (sqrt (exp a)) (fma (sqrt (exp a)) (sqrt (exp a)) (exp b))) (/ (sqrt (exp a)) (fma (sqrt (exp a)) (sqrt (exp a)) (exp b)))) (/ (sqrt (exp a)) (fma (sqrt (exp a)) (sqrt (exp a)) (exp b)))))
385.0ms
(* (* (* (sqrt (exp a)) (sqrt (exp a))) (sqrt (exp a))) (/ (* (* (sqrt (exp a)) (sqrt (exp a))) (sqrt (exp a))) (* (* (fma (sqrt (exp a)) (sqrt (exp a)) (exp b)) (fma (sqrt (exp a)) (sqrt (exp a)) (exp b))) (fma (sqrt (exp a)) (sqrt (exp a)) (exp b)))))

prune1.4s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.7b

localize51.0ms

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))) (fma (exp b) (- (exp b) (exp a)) (exp (+ a a))))
0.8b
(/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3)))
0.6b
(fma (exp b) (- (exp b) (exp a)) (exp (+ a a)))

rewrite60.0ms

Algorithm
rewrite-expression-head
Counts
4 → 147
Calls

4 calls. Slowest were:

27.0ms
(* (/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3))) (fma (exp b) (- (exp b) (exp a)) (exp (+ a a))))
20.0ms
(/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3)))
8.0ms
(- (exp b) (exp a))

series372.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

174.0ms
(* (/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3))) (fma (exp b) (- (exp b) (exp a)) (exp (+ a a))))
89.0ms
(fma (exp b) (- (exp b) (exp a)) (exp (+ a a)))
70.0ms
(/ (exp a) (+ (pow (exp a) 3) (pow (exp b) 3)))
40.0ms
(- (exp b) (exp a))

simplify5.2s

Counts
148 → 159
Calls

148 calls. Slowest were:

323.0ms
(/ (* (- (+ (pow (exp b) 2) (exp (* 2 a))) (* (exp b) (exp a))) (exp a)) (+ (pow (exp b) 3) (pow (exp a) 3)))
250.0ms
(+ (* (pow (exp a) 3) (pow (exp a) 3)) (- (* (pow (exp b) 3) (pow (exp b) 3)) (* (pow (exp a) 3) (pow (exp b) 3))))
220.0ms
(/ (exp a) (+ (pow (pow (exp a) 3) 3) (pow (pow (exp b) 3) 3)))

prune2.1s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.7b

regimes123.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample4.2s

Algorithm
intervals