Average Error: 0.7 → 0.8
Time: 26.5s
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
Herbie0.8
\[\frac{1}{1 + e^{b - a}}\]

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-sqrt0.8

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

  7. Applied times-frac0.8

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

  8. Simplified0.8

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

  9. Final simplification0.8

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

Reproduce

herbie shell --seed 2019016 
(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: 25.3s)Debug log

sample132.0ms

Algorithm
intervals
Results
368×(pre true 80)
258×(body real 80)
110×(body exit 10240)

simplify2.0ms

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

prune17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize11.0ms

Local error

Found 2 expressions with local error:

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

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
21×*-un-lft-identity
12×times-frac
11×add-cube-cbrt
11×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
2×flip-+
2×associate-/r/
2×flip3-+
2×pow1
1×div-inv
1×div-exp
1×frac-2neg
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
2 → 45
Calls
2 calls:
Slowest
8.0ms
(/ (exp a) (+ (exp a) (exp b)))
6.0ms
(+ (exp a) (exp b))

series95.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
59.0ms
(/ (exp a) (+ (exp a) (exp b)))
35.0ms
(+ (exp a) (exp b))

simplify528.0ms

Counts
42 → 51
Calls
42 calls:
Slowest
201.0ms
(/ (* (* (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))
23.0ms
(/ (* (cbrt (exp a)) (cbrt (exp a))) (* (cbrt (+ (exp a) (exp b))) (cbrt (+ (exp a) (exp b)))))
14.0ms
(/ (* (cbrt (exp a)) (cbrt (exp a))) (sqrt (+ (exp a) (exp b))))

prune482.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.2b

localize8.0ms

Local error

Found 4 expressions with local error:

0.5b
(/ (sqrt (exp a)) (+ (exp a) (exp b)))
0.3b
(* (sqrt (exp a)) (/ (sqrt (exp a)) (+ (exp a) (exp b))))
0.0b
(+ (exp a) (exp b))
0.0b
(sqrt (exp a))

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
67×*-un-lft-identity
48×times-frac
40×add-sqr-sqrt
39×add-cube-cbrt
33×sqrt-prod
30×associate-*r*
14×distribute-lft-out
11×add-exp-log
11×add-cbrt-cube
6×add-log-exp
6×associate-/l*
6×associate-*l*
6×pow1
4×associate-/r/
4×associate-/r*
3×flip-+
3×flip3-+
2×div-inv
2×cbrt-unprod
2×prod-exp
2×div-exp
2×cbrt-undiv
1×*-commutative
1×associate-*r/
1×pow-prod-down
1×pow1/2
1×frac-2neg
1×sum-log
1×clear-num
1×rem-sqrt-square
1×+-commutative
Counts
4 → 122
Calls
4 calls:
Slowest
12.0ms
(* (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))
1.0ms
(sqrt (exp a))

series194.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
77.0ms
(/ (sqrt (exp a)) (+ (exp a) (exp b)))
59.0ms
(* (sqrt (exp a)) (/ (sqrt (exp a)) (+ (exp a) (exp b))))
40.0ms
(+ (exp a) (exp b))
18.0ms
(sqrt (exp a))

simplify1.8s

Counts
120 → 134
Calls
120 calls:
Slowest
527.0ms
(* (* (* (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)))))
320.0ms
(/ (* (* (sqrt (exp a)) (sqrt (exp a))) (sqrt (exp a))) (* (* (+ (exp a) (exp b)) (+ (exp a) (exp b))) (+ (exp a) (exp b))))
131.0ms
(* (* (* (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)))))
89.0ms
(- 1/2 (+ (* 1/16 (pow a 2)) (* 1/4 b)))
78.0ms
(+ (* (exp a) (exp a)) (- (* (exp b) (exp b)) (* (exp a) (exp b))))

prune1.4s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.2b

localize19.0ms

Local error

Found 4 expressions with local error:

0.5b
(/ (exp a) (+ (exp a) (exp b)))
0.5b
(/ (exp a) (+ (exp a) (exp b)))
0.5b
(/ (exp a) (+ (exp a) (exp b)))
0.3b
(* (cbrt (/ (exp a) (+ (exp a) (exp b)))) (cbrt (/ (exp a) (+ (exp a) (exp b)))))

rewrite48.0ms

Algorithm
rewrite-expression-head
Rules
85×*-un-lft-identity
60×times-frac
49×add-cube-cbrt
49×add-sqr-sqrt
36×cbrt-prod
21×associate-*l*
21×associate-*r*
18×distribute-lft-out
12×associate-/r*
10×associate-/r/
10×add-cbrt-cube
9×associate-/l*
9×add-exp-log
9×pow1
5×div-inv
5×flip-+
5×flip3-+
4×add-log-exp
4×pow1/3
4×cbrt-div
3×div-exp
3×frac-2neg
3×clear-num
3×cbrt-undiv
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×associate-*r/
1×prod-exp
1×associate-*l/
1×pow-plus
1×frac-times
1×pow2
Counts
4 → 160
Calls
4 calls:
Slowest
12.0ms
(/ (exp a) (+ (exp a) (exp b)))
12.0ms
(/ (exp a) (+ (exp a) (exp b)))
12.0ms
(* (cbrt (/ (exp a) (+ (exp a) (exp b)))) (cbrt (/ (exp a) (+ (exp a) (exp b)))))
9.0ms
(/ (exp a) (+ (exp a) (exp b)))

series258.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
114.0ms
(* (cbrt (/ (exp a) (+ (exp a) (exp b)))) (cbrt (/ (exp a) (+ (exp a) (exp b)))))
54.0ms
(/ (exp a) (+ (exp a) (exp b)))
47.0ms
(/ (exp a) (+ (exp a) (exp b)))
43.0ms
(/ (exp a) (+ (exp a) (exp b)))

simplify2.4s

Counts
168 → 172
Calls
168 calls:
Slowest
282.0ms
(/ (* (* (exp a) (exp a)) (exp a)) (* (* (+ (exp a) (exp b)) (+ (exp a) (exp b))) (+ (exp a) (exp b))))
254.0ms
(/ (* (* (exp a) (exp a)) (exp a)) (* (* (+ (exp a) (exp b)) (+ (exp a) (exp b))) (+ (exp a) (exp b))))
184.0ms
(/ (* (* (exp a) (exp a)) (exp a)) (* (* (+ (exp a) (exp b)) (+ (exp a) (exp b))) (+ (exp a) (exp b))))
157.0ms
(- (+ (pow 1/4 1/3) (* 1/3 (* a (pow 1/4 1/3)))) (* 1/3 (* (pow 1/4 1/3) b)))
85.0ms
(pow (/ (pow (exp a) 2) (pow (+ (exp b) (exp a)) 2)) 1/3)

prune2.1s

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.2b

localize21.0ms

Local error

Found 4 expressions with local error:

0.7b
(/ (* (exp a) (* (exp a) (exp a))) (pow (+ (exp b) (exp a)) 3))
0.3b
(cbrt (/ (* (exp a) (* (exp a) (exp a))) (pow (+ (exp b) (exp a)) 3)))
0.1b
(pow (+ (exp b) (exp a)) 3)
0.0b
(* (exp a) (exp a))

rewrite70.0ms

Algorithm
rewrite-expression-head
Rules
34×*-un-lft-identity
26×times-frac
19×cbrt-prod
18×add-cube-cbrt
18×add-sqr-sqrt
17×add-exp-log
16×cube-prod
16×add-cbrt-cube
16×unpow-prod-down
13×associate-/r*
10×prod-exp
10×pow1
9×div-exp
8×distribute-lft-out
6×cube-div
4×add-log-exp
4×cbrt-unprod
4×associate-/r/
4×pow-exp
4×unpow3
4×pow-to-exp
4×cube-mult
3×flip-+
3×associate-*l*
3×associate-*r*
3×flip3-+
3×cbrt-undiv
2×div-inv
1×rem-cube-cbrt
1×associate-/l*
1×pow-prod-up
1×*-commutative
1×pow-prod-down
1×pow1/3
1×pow-plus
1×frac-2neg
1×pow-pow
1×clear-num
1×cbrt-div
1×pow2
Counts
4 → 122
Calls
4 calls:
Slowest
36.0ms
(/ (* (exp a) (* (exp a) (exp a))) (pow (+ (exp b) (exp a)) 3))
13.0ms
(cbrt (/ (* (exp a) (* (exp a) (exp a))) (pow (+ (exp b) (exp a)) 3)))
11.0ms
(* (exp a) (exp a))
9.0ms
(pow (+ (exp b) (exp a)) 3)

series162.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
60.0ms
(/ (* (exp a) (* (exp a) (exp a))) (pow (+ (exp b) (exp a)) 3))
50.0ms
(cbrt (/ (* (exp a) (* (exp a) (exp a))) (pow (+ (exp b) (exp a)) 3)))
42.0ms
(pow (+ (exp b) (exp a)) 3)
9.0ms
(* (exp a) (exp a))

simplify9.6s

Counts
110 → 134
Calls
110 calls:
Slowest
550.0ms
(cbrt (/ (* (exp a) (* (exp a) (exp a))) (pow (- (* (exp b) (exp b)) (* (exp a) (exp a))) 3)))
542.0ms
(cbrt (pow (+ (exp b) (exp a)) 3))
457.0ms
(/ (* (exp a) (exp a)) (cbrt (pow (+ (exp b) (exp a)) 3)))
361.0ms
(cbrt (sqrt (/ (* (exp a) (* (exp a) (exp a))) (pow (+ (exp b) (exp a)) 3))))
361.0ms
(cbrt (* (cbrt (/ (* (exp a) (* (exp a) (exp a))) (pow (+ (exp b) (exp a)) 3))) (cbrt (/ (* (exp a) (* (exp a) (exp a))) (pow (+ (exp b) (exp a)) 3)))))

prune1.5s

Pruning

6 alts after pruning (3 fresh and 3 done)

Merged error: 0.2b

regimes238.0ms

Accuracy

0% (0.2b remaining)

Error of 0.8b against oracle of 0.6b and baseline of 0.8b

bsearch2.0ms

end1.0ms

sample4.3s

Algorithm
intervals
Results
11509×(pre true 80)
8003×(body real 80)
3506×(body exit 10240)