Average Error: 58.6 → 3.4
Time: 37.1s
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

Original58.6
Target14.6
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.6

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

  2. Simplified27.8

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

  3. Taylor expanded around 0 3.4

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

  4. Final simplification3.4

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

Reproduce

herbie shell --seed 2019004 +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: 36.7s)Debug log

sample1.1s

Algorithm
intervals

simplify159.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

159.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 26.9b

localize29.0ms

Local error

Found 4 expressions with local error:

16.5b
(/ eps (expm1 (* eps a)))
8.7b
(/ (expm1 (* (+ a b) eps)) (expm1 (* eps b)))
6.2b
(* (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b))) (/ eps (expm1 (* eps a))))
0.0b
(expm1 (* (+ a b) eps))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 91
Calls

4 calls. Slowest were:

6.0ms
(* (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b))) (/ eps (expm1 (* eps a))))
3.0ms
(/ (expm1 (* (+ a b) eps)) (expm1 (* eps b)))
1.0ms
(/ eps (expm1 (* eps a)))

series686.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

502.0ms
(* (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b))) (/ eps (expm1 (* eps a))))
95.0ms
(/ (expm1 (* (+ a b) eps)) (expm1 (* eps b)))
47.0ms
(expm1 (* (+ a b) eps))
42.0ms
(/ eps (expm1 (* eps a)))

simplify2.8s

Counts
67 → 103
Calls

67 calls. Slowest were:

463.0ms
(* (* (* (/ (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)))))
297.0ms
(+ (- (log (expm1 (* (+ a b) eps))) (log (expm1 (* eps b)))) (log (/ eps (expm1 (* eps a)))))
226.0ms
(+ (* 1/2 (* (pow a 2) (pow eps 2))) (+ (* eps b) (* a eps)))

prune1.3s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 1 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
1 → 22
Calls

1 calls. Slowest were:

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

series12.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

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

simplify65.0ms

Counts
12 → 25
Calls

12 calls. Slowest were:

10.0ms
(* (exp (/ 1 a)) (exp (/ 1 b)))
8.0ms
(+ (* 1 b) (* a 1))
8.0ms
(+ (/ 1 a) (/ 1 b))

prune251.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize19.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (+ (/ 1 a) (/ 1 b)))
0.6b
(cbrt (+ (/ 1 a) (/ 1 b)))
0.6b
(cbrt (+ (/ 1 a) (/ 1 b)))
0.5b
(* (cbrt (+ (/ 1 a) (/ 1 b))) (cbrt (+ (/ 1 a) (/ 1 b))))

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
4 → 113
Calls

4 calls. Slowest were:

28.0ms
(* (cbrt (+ (/ 1 a) (/ 1 b))) (cbrt (+ (/ 1 a) (/ 1 b))))
4.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))
4.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))

series411.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

126.0ms
(* (cbrt (+ (/ 1 a) (/ 1 b))) (cbrt (+ (/ 1 a) (/ 1 b))))
104.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))
104.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))
76.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))

simplify6.7s

Counts
94 → 125
Calls

94 calls. Slowest were:

365.0ms
(* (cbrt (- (* (/ 1 a) (/ 1 a)) (* (/ 1 b) (/ 1 b)))) (cbrt (+ (* 1 b) (* a 1))))
364.0ms
(* (cbrt (+ (pow (/ 1 a) 3) (pow (/ 1 b) 3))) (cbrt (- (* (/ 1 a) (/ 1 a)) (* (/ 1 b) (/ 1 b)))))
360.0ms
(* (cbrt (* a b)) (cbrt (+ (* (/ 1 a) (/ 1 a)) (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 b))))))

prune1.1s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

16.5b
(/ eps (expm1 (* eps a)))
14.6b
(/ (* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a)))) (expm1 (* eps b)))
4.5b
(* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a))))
0.0b
(expm1 (* (+ a b) eps))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 71
Calls

4 calls. Slowest were:

15.0ms
(/ (* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a)))) (expm1 (* eps b)))
7.0ms
(* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a))))
1.0ms
(/ eps (expm1 (* eps a)))

series552.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

393.0ms
(/ (* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a)))) (expm1 (* eps b)))
79.0ms
(* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a))))
45.0ms
(/ eps (expm1 (* eps a)))
35.0ms
(expm1 (* (+ a b) eps))

simplify1.4s

Counts
41 → 83
Calls

41 calls. Slowest were:

199.0ms
(- (+ (log (expm1 (* (+ a b) eps))) (log (/ eps (expm1 (* eps a))))) (log (expm1 (* eps b))))
134.0ms
(+ (* 1/2 (* (pow a 2) (pow eps 2))) (+ (* eps b) (* a eps)))
120.0ms
(/ (* (- (exp (* (+ a b) eps)) 1) eps) (- (+ (* (exp (* a eps)) (exp (* eps b))) 1) (+ (exp (* eps b)) (exp (* a eps)))))

prune653.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0.0b

regimes85.0ms

Accuracy

0% (3.3b remaining)

Error of 3.4b against oracle of 0.0b and baseline of 3.4b

bsearch1.0ms

end0.0ms

sample19.3s

Algorithm
intervals