Average Error: 58.7 → 3.3
Time: 32.8s
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
Target13.7
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. Initial simplification27.5

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

  3. Taylor expanded around 0 3.3

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

  4. Final simplification3.3

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

Reproduce

herbie shell --seed 2018360 +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: 8.2s)Debug log

start788.0ms

Algorithm
intervals

setup256.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 28.9b

localize58.0ms

Local error

Found 4 expressions with local error:

18.6b
(/ eps (expm1 (* eps a)))
13.0b
(/ (expm1 (* (+ a b) eps)) (expm1 (* eps b)))
3.4b
(* (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b))) (/ eps (expm1 (* eps a))))
0.0b
(* (+ a b) eps)

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 95
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)))
2.0ms
(* (+ a b) eps)

series688.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

547.0ms
(* (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b))) (/ eps (expm1 (* eps a))))
83.0ms
(/ (expm1 (* (+ a b) eps)) (expm1 (* eps b)))
44.0ms
(/ eps (expm1 (* eps a)))
15.0ms
(* (+ a b) eps)

simplify2.4s

Counts
70 → 107
Calls

70 calls. Slowest were:

443.0ms
(* (* (* (/ (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)))))
196.0ms
(/ (* (- (exp (* (+ a b) eps)) 1) eps) (- (+ (* (exp (* a eps)) (exp (* eps b))) 1) (+ (exp (* eps b)) (exp (* a eps)))))
182.0ms
(* (- (* a a) (* b b)) eps)

prune1.5s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 1 expressions with local error:

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

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
1 → 22
Calls

1 calls. Slowest were:

10.0ms
(+ (/ 1 a) (/ 1 b))

series12.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

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

simplify60.0ms

Counts
12 → 25
Calls

12 calls. Slowest were:

10.0ms
(* (exp (/ 1 a)) (exp (/ 1 b)))
5.0ms
(+ (/ 1 a) (/ 1 b))
5.0ms
(+ (/ 1 a) (/ 1 b))

prune193.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize7.0ms

Local error

Found 1 expressions with local error:

13.7b
(/ (+ b a) (* a b))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
1 → 23
Calls

1 calls. Slowest were:

3.0ms
(/ (+ b a) (* a b))

series26.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

26.0ms
(/ (+ b a) (* a b))

simplify198.0ms

Counts
15 → 26
Calls

15 calls. Slowest were:

102.0ms
(/ (* (* (+ b a) (+ b a)) (+ b a)) (* (* (* a b) (* a b)) (* a b)))
36.0ms
(- (log (+ b a)) (log (* a b)))
13.0ms
(- (* b b) (* a a))

prune261.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize5.0ms

Local error

Found 2 expressions with local error:

5.0b
(* (/ 1 a) (/ (+ b a) b))
0.0b
(/ (+ b a) b)

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
2 → 41
Calls

2 calls. Slowest were:

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

series34.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

21.0ms
(* (/ 1 a) (/ (+ b a) b))
13.0ms
(/ (+ b a) b)

simplify1.1s

Counts
22 → 47
Calls

22 calls. Slowest were:

798.0ms
(* (* (* (/ 1 a) (/ 1 a)) (/ 1 a)) (* (* (/ (+ b a) b) (/ (+ b a) b)) (/ (+ b a) b)))
145.0ms
(* (/ 1 a) (/ (+ b a) b))
50.0ms
(+ (log (/ 1 a)) (log (/ (+ b a) b)))

prune379.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

regimes145.0ms

Accuracy

0% (3.3b remaining)

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

bsearch1.0ms