Average Error: 58.5 → 3.4
Time: 44.7s
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

Original58.5
Target14.4
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 56.2

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\varepsilon \cdot b + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right)\right)\right)}}\]

  3. Simplified54.7

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\varepsilon \cdot b + \left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) \cdot \left(b \cdot \left(\frac{1}{6} \cdot \varepsilon\right) + \frac{1}{2}\right)\right)}}\]

  4. Taylor expanded around 0 3.4

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

  5. Final simplification3.4

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

Reproduce

herbie shell --seed 2018362 
(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: 22.0s)Debug log

start596.0ms

Algorithm
intervals

setup260.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 58.6b

localize39.0ms

Local error

Found 4 expressions with local error:

5.6b
(- (exp (* (+ a b) eps)) 1)
3.7b
(- (exp (* a eps)) 1)
2.8b
(- (exp (* b eps)) 1)
0.7b
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))

rewrite52.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

34.0ms
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))
7.0ms
(- (exp (* b eps)) 1)
6.0ms
(- (exp (* (+ a b) eps)) 1)

series174.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

59.0ms
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))
57.0ms
(- (exp (* (+ a b) eps)) 1)
36.0ms
(- (exp (* a eps)) 1)
23.0ms
(- (exp (* b eps)) 1)

simplify4.2s

Counts
41 → 79
Calls

41 calls. Slowest were:

642.0ms
(* (* (* (- (exp (* a eps)) 1) (- (exp (* a eps)) 1)) (- (exp (* a eps)) 1)) (* (* (- (exp (* b eps)) 1) (- (exp (* b eps)) 1)) (- (exp (* b eps)) 1)))
434.0ms
(* (+ (exp (* a eps)) 1) (+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1))))
338.0ms
(+ (* 1/2 (* (pow a 2) (pow eps 2))) (+ (* 1/6 (* (pow a 3) (pow eps 3))) (* a eps)))

prune1.4s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 45.7b

localize38.0ms

Local error

Found 4 expressions with local error:

5.6b
(- (exp (* (+ a b) eps)) 1)
3.7b
(- (exp (* a eps)) 1)
3.6b
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))))
2.5b
(* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))

rewrite87.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

38.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))))
27.0ms
(* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))
12.0ms
(- (exp (* (+ a b) eps)) 1)

series1.8s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.6s
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))))
166.0ms
(* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))
63.0ms
(- (exp (* (+ a b) eps)) 1)
41.0ms
(- (exp (* a eps)) 1)

simplify10.7s

Counts
54 → 91
Calls

54 calls. Slowest were:

609.0ms
(/ (* (* (* eps (- (exp (* (+ a b) eps)) 1)) (* eps (- (exp (* (+ a b) eps)) 1))) (* eps (- (exp (* (+ a b) eps)) 1))) (* (* (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))))
532.0ms
(* (* (* (- (exp (* a eps)) 1) (- (exp (* a eps)) 1)) (- (exp (* a eps)) 1)) (* (* (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))))
520.0ms
(- (+ (/ (exp (* (+ a b) eps)) (* (- (* 1/6 (exp (* a eps))) 1/6) (* (pow b 3) (pow eps 2)))) (+ (* 1/2 (/ (exp (* a eps)) (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (* (pow b 4) (pow eps 3))))) (* 1/2 (/ (exp (* (+ a b) eps)) (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (* (pow b 4) (pow eps 3))))))) (+ (* 1/2 (/ (* (exp (* (+ a b) eps)) (exp (* a eps))) (* (pow eps 3) (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (pow b 4))))) (+ (/ 1 (* (- (* 1/6 (exp (* a eps))) 1/6) (* (pow eps 2) (pow b 3)))) (* 1/2 (/ 1 (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (* (pow eps 3) (pow b 4))))))))

prune1.9s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize19.0ms

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))

simplify52.0ms

Counts
10 → 19
Calls

10 calls. Slowest were:

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

prune149.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize18.0ms

Local error

Found 1 expressions with local error:

13.4b
(/ (+ b a) (* a b))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
1 → 21
Calls

1 calls. Slowest were:

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

series36.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

36.0ms
(/ (+ b a) (* a b))

simplify151.0ms

Counts
15 → 24
Calls

15 calls. Slowest were:

77.0ms
(/ (* (* (+ b a) (+ b a)) (+ b a)) (* (* (* a b) (* a b)) (* a b)))
31.0ms
(- (log (+ b a)) (log (* a b)))
10.0ms
(- (* b b) (* a a))

prune133.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0b

regimes168.0ms

Accuracy

0% (3.4b remaining)

Error of 3.4b against oracle of 0.1b and baseline of 3.4b

bsearch1.0ms