Average Error: 0.7 → 0.7
Time: 27.5s
Precision: 64
Internal Precision: 128
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\log_* (1 + (e^{e^{\sqrt[3]{\left(\left(a - \log \left(e^{a} + e^{b}\right)\right) \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)}}} - 1)^*)\]

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 add-exp-log0.7

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

  4. Applied div-exp0.7

    \[\leadsto \color{blue}{e^{a - \log \left(e^{a} + e^{b}\right)}}\]
  5. Using strategy rm

  6. Applied log1p-expm1-u0.7

    \[\leadsto \color{blue}{\log_* (1 + (e^{e^{a - \log \left(e^{a} + e^{b}\right)}} - 1)^*)}\]
  7. Using strategy rm

  8. Applied add-cbrt-cube0.7

    \[\leadsto \log_* (1 + (e^{e^{\color{blue}{\sqrt[3]{\left(\left(a - \log \left(e^{a} + e^{b}\right)\right) \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)}}}} - 1)^*)\]

  9. Final simplification0.7

    \[\leadsto \log_* (1 + (e^{e^{\sqrt[3]{\left(\left(a - \log \left(e^{a} + e^{b}\right)\right) \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)}}} - 1)^*)\]

Reproduce

herbie shell --seed 2019016 +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: 26.7s)Debug log

sample107.0ms

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)))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize12.0ms

Local error

Found 2 expressions with local error:

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

rewrite12.0ms

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
8.0ms
(/ (exp a) (+ (exp a) (exp b)))
4.0ms
(+ (exp a) (exp b))

series74.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
51.0ms
(/ (exp a) (+ (exp a) (exp b)))
23.0ms
(+ (exp a) (exp b))

simplify959.0ms

Counts
44 → 58
Calls
44 calls:
Slowest
594.0ms
(/ (* (* (exp a) (exp a)) (exp a)) (* (* (+ (exp a) (exp b)) (+ (exp a) (exp b))) (+ (exp a) (exp b))))
85.0ms
(+ (* (exp a) (exp a)) (- (* (exp b) (exp b)) (* (exp a) (exp b))))
50.0ms
(- (+ (* 1/4 a) 1/2) (* 1/4 b))
32.0ms
(/ (* (cbrt (exp a)) (cbrt (exp a))) (* (cbrt (+ (exp a) (exp b))) (cbrt (+ (exp a) (exp b)))))
18.0ms
(/ (* (cbrt (exp a)) (cbrt (exp a))) 1)

prune546.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.2b

localize9.0ms

Local error

Found 4 expressions with local error:

3.3b
(log (+ (exp a) (exp b)))
0.5b
(- a (log (+ (exp a) (exp b))))
0.0b
(+ (exp a) (exp b))
0.0b
(exp (- a (log (+ (exp a) (exp b)))))

rewrite15.0ms

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
5.0ms
(log (+ (exp a) (exp b)))
4.0ms
(+ (exp a) (exp b))
3.0ms
(- a (log (+ (exp a) (exp b))))
2.0ms
(exp (- a (log (+ (exp a) (exp b)))))

series166.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
48.0ms
(log (+ (exp a) (exp b)))
44.0ms
(exp (- a (log (+ (exp a) (exp b)))))
38.0ms
(+ (exp a) (exp b))
36.0ms
(- a (log (+ (exp a) (exp b))))

simplify1.0s

Counts
38 → 83
Calls
38 calls:
Slowest
305.0ms
(exp (- a (log (- (* (exp a) (exp a)) (* (exp b) (exp b))))))
212.0ms
(- (+ (exp (- (log 2))) (* 1/2 (* a (exp (- (log 2)))))) (* 1/2 (* (exp (- (log 2))) b)))
143.0ms
(log (- (* (exp a) (exp a)) (* (exp b) (exp b))))
96.0ms
(log (+ (* (exp a) (exp a)) (- (* (exp b) (exp b)) (* (exp a) (exp b)))))
54.0ms
(- (* 1/2 a) (+ (* 1/2 b) (log 2)))

prune817.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.2b

localize7.0ms

Local error

Found 4 expressions with local error:

3.3b
(log (+ (exp a) (exp b)))
0.5b
(- a (log (+ (exp a) (exp b))))
0.2b
(expm1 (exp (- a (log (+ (exp a) (exp b))))))
0.0b
(log1p (expm1 (exp (- a (log (+ (exp a) (exp b)))))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
10×*-un-lft-identity
8×log-prod
6×add-cube-cbrt
6×add-sqr-sqrt
5×log1p-expm1-u
5×add-exp-log
5×pow1
4×add-log-exp
4×associate--r+
4×add-cbrt-cube
4×log-div
4×expm1-log1p-u
2×associate--r-
2×distribute-lft-out
2×flip-+
2×flip3-+
1×expm1-udef
1×flip--
1×log-pow
1×log1p-expm1
1×rem-log-exp
1×flip3--
1×sub-neg
1×expm1-log1p
1×log1p-udef
Counts
4 → 58
Calls
4 calls:
Slowest
3.0ms
(log (+ (exp a) (exp b)))
3.0ms
(- a (log (+ (exp a) (exp b))))
0.0ms
(expm1 (exp (- a (log (+ (exp a) (exp b))))))
0.0ms
(log1p (expm1 (exp (- a (log (+ (exp a) (exp b)))))))

series279.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
103.0ms
(expm1 (exp (- a (log (+ (exp a) (exp b))))))
89.0ms
(log1p (expm1 (exp (- a (log (+ (exp a) (exp b)))))))
49.0ms
(log (+ (exp a) (exp b)))
37.0ms
(- a (log (+ (exp a) (exp b))))

simplify1.3s

Counts
27 → 70
Calls
27 calls:
Slowest
560.0ms
(- (+ (* 1/2 (* a (* (exp (- (log 2))) (exp (exp (- (log 2))))))) (exp (exp (- (log 2))))) (+ 1 (* 1/2 (* (exp (exp (- (log 2)))) (* b (exp (- (log 2))))))))
268.0ms
(- (+ (exp (- (log 2))) (* 1/2 (* a (exp (- (log 2)))))) (* 1/2 (* (exp (- (log 2))) b)))
115.0ms
(log (- (* (exp a) (exp a)) (* (exp b) (exp b))))
80.0ms
(log (+ (* (exp a) (exp a)) (- (* (exp b) (exp b)) (* (exp a) (exp b)))))
78.0ms
(- (* 1/2 a) (+ (* 1/2 b) (log 2)))

prune967.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.2b

localize29.0ms

Local error

Found 4 expressions with local error:

17.5b
(cbrt (* (* (- a (log (+ (exp a) (exp b)))) (- a (log (+ (exp a) (exp b))))) (- a (log (+ (exp a) (exp b))))))
3.3b
(log (+ (exp a) (exp b)))
3.3b
(log (+ (exp a) (exp b)))
3.3b
(log (+ (exp a) (exp b)))

rewrite20.0ms

Algorithm
rewrite-expression-head
Rules
28×frac-times
27×flip--
27×flip3--
26×cbrt-div
14×associate-*l/
13×*-un-lft-identity
12×log-prod
8×associate-*r/
7×add-cube-cbrt
7×add-exp-log
7×pow1
7×add-sqr-sqrt
6×log-div
4×add-log-exp
4×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
3×distribute-lft-out
3×flip-+
3×log-pow
3×rem-log-exp
3×flip3-+
1×rem-cbrt-cube
1×pow3
1×pow1/3
1×cbrt-prod
Counts
4 → 89
Calls
4 calls:
Slowest
9.0ms
(cbrt (* (* (- a (log (+ (exp a) (exp b)))) (- a (log (+ (exp a) (exp b))))) (- a (log (+ (exp a) (exp b))))))
3.0ms
(log (+ (exp a) (exp b)))
3.0ms
(log (+ (exp a) (exp b)))
3.0ms
(log (+ (exp a) (exp b)))

series206.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
64.0ms
(log (+ (exp a) (exp b)))
50.0ms
(log (+ (exp a) (exp b)))
47.0ms
(cbrt (* (* (- a (log (+ (exp a) (exp b)))) (- a (log (+ (exp a) (exp b))))) (- a (log (+ (exp a) (exp b))))))
44.0ms
(log (+ (exp a) (exp b)))

simplify13.9s

Counts
60 → 101
Calls
60 calls:
Slowest
988.0ms
(cbrt (* (* (- (* a a) (* (log (+ (exp a) (exp b))) (log (+ (exp a) (exp b))))) (- (* a a) (* (log (+ (exp a) (exp b))) (log (+ (exp a) (exp b)))))) (- a (log (+ (exp a) (exp b))))))
865.0ms
(cbrt (* (* (- a (log (+ (exp a) (exp b)))) (- (* a a) (* (log (+ (exp a) (exp b))) (log (+ (exp a) (exp b)))))) (- (pow a 3) (pow (log (+ (exp a) (exp b))) 3))))
862.0ms
(cbrt (* (* (- a (log (+ (exp a) (exp b)))) (- (* a a) (* (log (+ (exp a) (exp b))) (log (+ (exp a) (exp b)))))) (- (* a a) (* (log (+ (exp a) (exp b))) (log (+ (exp a) (exp b)))))))
803.0ms
(cbrt (* (* (- (pow a 3) (pow (log (+ (exp a) (exp b))) 3)) (- (* a a) (* (log (+ (exp a) (exp b))) (log (+ (exp a) (exp b)))))) (- a (log (+ (exp a) (exp b))))))
753.0ms
(cbrt (* (* (- (pow a 3) (pow (log (+ (exp a) (exp b))) 3)) (- (* a a) (* (log (+ (exp a) (exp b))) (log (+ (exp a) (exp b)))))) (- (* a a) (* (log (+ (exp a) (exp b))) (log (+ (exp a) (exp b)))))))

prune1.7s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.2b

regimes135.0ms

Accuracy

0% (0.2b remaining)

Error of 0.7b against oracle of 0.5b and baseline of 0.7b

bsearch2.0ms

end0.0ms

sample4.4s

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