Average Error: 58.8 → 3.1
Time: 30.5s
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

Original58.8
Target13.8
Herbie3.1
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.8

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

  2. Simplified28.0

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

  3. Taylor expanded around 0 3.1

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

  4. Final simplification3.1

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

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(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: 29.9s)Debug log

sample515.0ms

Algorithm
intervals

simplify172.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

172.0ms
(/ (* 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: 26.7b

localize31.0ms

Local error

Found 4 expressions with local error:

14.8b
(/ eps (expm1 (* eps a)))
11.0b
(/ (expm1 (* (+ a b) eps)) (expm1 (* eps b)))
5.1b
(* (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b))) (/ eps (expm1 (* eps a))))
0.0b
(expm1 (* (+ a b) eps))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 91
Calls

4 calls. Slowest were:

19.0ms
(* (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b))) (/ eps (expm1 (* eps a))))
6.0ms
(/ (expm1 (* (+ a b) eps)) (expm1 (* eps b)))
1.0ms
(/ eps (expm1 (* eps a)))

series665.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

489.0ms
(* (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b))) (/ eps (expm1 (* eps a))))
62.0ms
(/ (expm1 (* (+ a b) eps)) (expm1 (* eps b)))
62.0ms
(expm1 (* (+ a b) eps))
52.0ms
(/ eps (expm1 (* eps a)))

simplify2.6s

Counts
67 → 103
Calls

67 calls. Slowest were:

348.0ms
(* (* (* (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b))) (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b)))) (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b)))) (* (* (/ eps (expm1 (* eps a))) (/ eps (expm1 (* eps a)))) (/ eps (expm1 (* eps a)))))
314.0ms
(+ (- (log (expm1 (* (+ a b) eps))) (log (expm1 (* eps b)))) (log (/ eps (expm1 (* eps a)))))
240.0ms
(+ (* 1/2 (* (pow a 2) (pow eps 2))) (+ (* eps b) (* a eps)))

prune1.2s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize13.0ms

Local error

Found 1 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
1 → 22
Calls

1 calls. Slowest were:

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

series7.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

7.0ms
(+ (/ 1 a) (/ 1 b))

simplify30.0ms

Counts
12 → 25
Calls

12 calls. Slowest were:

5.0ms
(* (exp (/ 1 a)) (exp (/ 1 b)))
4.0ms
(+ (* 1 b) (* a 1))
2.0ms
(+ (/ 1 a) (/ 1 b))

prune154.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize6.0ms

Local error

Found 4 expressions with local error:

15.6b
(/ (* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a)))) (expm1 (* eps b)))
14.8b
(/ eps (expm1 (* eps a)))
4.3b
(* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a))))
0.0b
(expm1 (* (+ a b) eps))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 71
Calls

4 calls. Slowest were:

7.0ms
(/ (* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a)))) (expm1 (* eps b)))
4.0ms
(* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a))))
1.0ms
(expm1 (* (+ a b) eps))

series739.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

496.0ms
(/ (* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a)))) (expm1 (* eps b)))
144.0ms
(* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a))))
53.0ms
(/ eps (expm1 (* eps a)))
46.0ms
(expm1 (* (+ a b) eps))

simplify2.1s

Counts
41 → 83
Calls

41 calls. Slowest were:

358.0ms
(- (+ (log (expm1 (* (+ a b) eps))) (log (/ eps (expm1 (* eps a))))) (log (expm1 (* eps b))))
208.0ms
(+ (* 1/2 (* (pow a 2) (pow eps 2))) (+ (* eps b) (* a eps)))
191.0ms
(* (* (* (expm1 (* (+ a b) eps)) (expm1 (* (+ a b) eps))) (expm1 (* (+ a b) eps))) (* (* (/ eps (expm1 (* eps a))) (/ eps (expm1 (* eps a)))) (/ eps (expm1 (* eps a)))))

prune1.1s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize18.0ms

Local error

Found 3 expressions with local error:

15.3b
(/ (fma (/ b a) eps eps) (expm1 (* eps b)))
5.0b
(fma (/ b a) eps eps)
0.0b
(expm1 (* eps b))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
3 → 50
Calls

3 calls. Slowest were:

7.0ms
(/ (fma (/ b a) eps eps) (expm1 (* eps b)))
0.0ms
(expm1 (* eps b))
0.0ms
(fma (/ b a) eps eps)

series144.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

93.0ms
(/ (fma (/ b a) eps eps) (expm1 (* eps b)))
26.0ms
(fma (/ b a) eps eps)
25.0ms
(expm1 (* eps b))

simplify1.3s

Counts
35 → 59
Calls

35 calls. Slowest were:

193.0ms
(/ (* (* (fma (/ b a) eps eps) (fma (/ b a) eps eps)) (fma (/ b a) eps eps)) (* (* (expm1 (* eps b)) (expm1 (* eps b))) (expm1 (* eps b))))
141.0ms
(+ (/ eps (- (exp (* eps b)) 1)) (/ (* eps b) (* a (- (exp (* eps b)) 1))))
127.0ms
(+ (* eps b) (+ (* 1/2 (* (pow eps 2) (pow b 2))) (* 1/6 (* (pow eps 3) (pow b 3)))))

prune486.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes106.0ms

Accuracy

0% (3.1b remaining)

Error of 3.1b against oracle of 0.0b and baseline of 3.1b

bsearch2.0ms

end0.0ms

sample18.6s

Algorithm
intervals