Average Error: 58.7 → 3.3
Time: 29.0s
Precision: 64
Internal Precision: 128
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.7
Target14.3
Herbie3.3
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.7

    \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  2. Taylor expanded around 0 3.3

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

  3. Final simplification3.3

    \[\leadsto \frac{1}{a} + \frac{1}{b}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1 eps) (< eps 1))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))

Details

Time bar (total: 28.2s)Debug log

sample648.0ms

Algorithm
intervals

simplify229.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

229.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

prune16.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.9b

localize69.0ms

Local error

Found 4 expressions with local error:

5.0b
(- (exp (* (+ a b) eps)) 1)
3.4b
(- (exp (* b eps)) 1)
3.1b
(- (exp (* a eps)) 1)
0.5b
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

rewrite78.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

40.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))
15.0ms
(- (exp (* a eps)) 1)
12.0ms
(- (exp (* (+ a b) eps)) 1)

series290.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

187.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))
44.0ms
(- (exp (* (+ a b) eps)) 1)
33.0ms
(- (exp (* b eps)) 1)
25.0ms
(- (exp (* a eps)) 1)

simplify4.3s

Counts
32 → 75
Calls

32 calls. Slowest were:

613.0ms
(* (+ (exp (* a eps)) 1) (+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1))))
390.0ms
(+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1)))
356.0ms
(/ (* (* (* eps (- (exp (* (+ a b) eps)) 1)) (* eps (- (exp (* (+ a b) eps)) 1))) (* eps (- (exp (* (+ a b) eps)) 1))) (* (* (* (- (exp (* a eps)) 1) (- (exp (* a eps)) 1)) (- (exp (* a eps)) 1)) (* (* (- (exp (* b eps)) 1) (- (exp (* b eps)) 1)) (- (exp (* b eps)) 1))))

prune1.1s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 1 expressions with local error:

0.0b
(+ (/ 1 a) (/ 1 b))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
1 → 16
Calls

1 calls. Slowest were:

6.0ms
(+ (/ 1 a) (/ 1 b))

series12.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

12.0ms
(+ (/ 1 a) (/ 1 b))

simplify38.0ms

Counts
10 → 19
Calls

10 calls. Slowest were:

13.0ms
(* (exp (/ 1 a)) (exp (/ 1 b)))
4.0ms
(+ (* 1 b) (* a 1))
4.0ms
(+ (/ 1 a) (/ 1 b))

prune222.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 1 expressions with local error:

13.8b
(/ (+ b a) (* a b))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
1 → 21
Calls

1 calls. Slowest were:

6.0ms
(/ (+ b a) (* a b))

series34.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

34.0ms
(/ (+ b a) (* a b))

simplify194.0ms

Counts
15 → 24
Calls

15 calls. Slowest were:

82.0ms
(/ (* (* (+ b a) (+ b a)) (+ b a)) (* (* (* a b) (* a b)) (* a b)))
35.0ms
(- (log (+ b a)) (log (* a b)))
21.0ms
(- (* b b) (* a a))

prune212.0ms

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0b

localize12.0ms

Local error

Found 4 expressions with local error:

3.2b
(* (/ (sqrt (+ b a)) a) (/ (sqrt (+ b a)) b))
0.1b
(/ (sqrt (+ b a)) a)
0.1b
(/ (sqrt (+ b a)) b)
0.0b
(sqrt (+ b a))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

6.0ms
(/ (sqrt (+ b a)) a)
6.0ms
(/ (sqrt (+ b a)) b)
5.0ms
(* (/ (sqrt (+ b a)) a) (/ (sqrt (+ b a)) b))

series129.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

39.0ms
(/ (sqrt (+ b a)) a)
37.0ms
(/ (sqrt (+ b a)) b)
35.0ms
(* (/ (sqrt (+ b a)) a) (/ (sqrt (+ b a)) b))
18.0ms
(sqrt (+ b a))

simplify590.0ms

Counts
45 → 84
Calls

45 calls. Slowest were:

301.0ms
(* (* (* (/ (sqrt (+ b a)) a) (/ (sqrt (+ b a)) a)) (/ (sqrt (+ b a)) a)) (* (* (/ (sqrt (+ b a)) b) (/ (sqrt (+ b a)) b)) (/ (sqrt (+ b a)) b)))
104.0ms
(+ (log (/ (sqrt (+ b a)) a)) (log (/ (sqrt (+ b a)) b)))
35.0ms
(sqrt (- (* b b) (* a a)))

prune638.0ms

Pruning

6 alts after pruning (3 fresh and 3 done)

Merged error: 0b

regimes217.0ms

Accuracy

0% (3.2b remaining)

Error of 3.3b against oracle of 0.0b and baseline of 3.3b

bsearch1.0ms

end0.0ms

sample19.1s

Algorithm
intervals