Average Error: 58.6 → 3.4
Time: 44.4s
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. Taylor expanded around 0 56.6

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

  3. Simplified55.2

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

  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 2019004 
(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: 43.7s)Debug log

sample649.0ms

Algorithm
intervals

simplify185.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

185.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 59.0b

localize57.0ms

Local error

Found 4 expressions with local error:

3.9b
(- (exp (* (+ a b) eps)) 1)
3.2b
(- (exp (* b eps)) 1)
1.7b
(- (exp (* a eps)) 1)
0.5b
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))

rewrite46.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

27.0ms
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))
9.0ms
(- (exp (* b eps)) 1)
6.0ms
(- (exp (* (+ a b) eps)) 1)

series180.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

61.0ms
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))
42.0ms
(- (exp (* a eps)) 1)
40.0ms
(- (exp (* b eps)) 1)
38.0ms
(- (exp (* (+ a b) eps)) 1)

simplify4.5s

Counts
41 → 79
Calls

41 calls. Slowest were:

579.0ms
(* (* (* (- (exp (* a eps)) 1) (- (exp (* a eps)) 1)) (- (exp (* a eps)) 1)) (* (* (- (exp (* b eps)) 1) (- (exp (* b eps)) 1)) (- (exp (* b eps)) 1)))
512.0ms
(* (+ (exp (* a eps)) 1) (+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1))))
398.0ms
(+ (* 1/2 (* (pow a 2) (pow eps 2))) (+ (* 1/6 (* (pow a 3) (pow eps 3))) (* a eps)))

prune1.2s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 45.3b

localize28.0ms

Local error

Found 4 expressions with local error:

3.9b
(- (exp (* (+ a b) eps)) 1)
3.5b
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))))
2.7b
(* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))
1.7b
(- (exp (* a eps)) 1)

rewrite77.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

30.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))))
24.0ms
(* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))
11.0ms
(- (exp (* a eps)) 1)

series1.9s

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)))))
175.0ms
(* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))
60.0ms
(- (exp (* (+ a b) eps)) 1)
38.0ms
(- (exp (* a eps)) 1)

simplify8.7s

Counts
54 → 91
Calls

54 calls. Slowest were:

765.0ms
(* (+ (* (exp (* a eps)) (exp (* a eps))) (+ (* 1 1) (* (exp (* a eps)) 1))) (+ (* (* eps b) (* eps b)) (- (* (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))) (* (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))))
495.0ms
(/ (* (* (* 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))))))
420.0ms
(* (+ (* (exp (* a eps)) (exp (* a eps))) (+ (* 1 1) (* (exp (* a eps)) 1))) (+ (* (* eps b) (* eps b)) (- (* (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))) (* (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))))

prune1.2s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 1 expressions with local error:

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

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
1 → 16
Calls

1 calls. Slowest were:

3.0ms
(+ (/ 1 a) (/ 1 b))

series7.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

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

simplify29.0ms

Counts
10 → 19
Calls

10 calls. Slowest were:

8.0ms
(* (exp (/ 1 a)) (exp (/ 1 b)))
2.0ms
(+ (/ 1 a) (/ 1 b))
2.0ms
(+ (/ 1 b) (/ 1 a))

prune146.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize14.0ms

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))))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 105
Calls

4 calls. Slowest were:

18.0ms
(* (cbrt (+ (/ 1 a) (/ 1 b))) (cbrt (+ (/ 1 a) (/ 1 b))))
3.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))
3.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))

series313.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

103.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))
76.0ms
(* (cbrt (+ (/ 1 a) (/ 1 b))) (cbrt (+ (/ 1 a) (/ 1 b))))
68.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))
66.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))

simplify4.8s

Counts
94 → 117
Calls

94 calls. Slowest were:

377.0ms
(- (+ (pow (/ 1 b) 1/3) (* 1/3 (* (/ 1 a) (pow (pow b 2) 1/3)))) (* 1/9 (* (/ 1 (pow a 2)) (pow (pow b 5) 1/3))))
321.0ms
(- (+ (pow (/ 1 b) 1/3) (* 1/3 (* (/ 1 a) (pow (pow b 2) 1/3)))) (* 1/9 (* (/ 1 (pow a 2)) (pow (pow b 5) 1/3))))
276.0ms
(- (* (cbrt -1) (pow (/ -1 b) 1/3)) (+ (* 1/9 (* (/ (cbrt -1) (pow a 2)) (pow (* (pow b 5) -1) 1/3))) (* 1/3 (* (/ (cbrt -1) a) (pow (pow b 2) 1/3)))))

prune882.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

regimes112.0ms

Accuracy

0% (3.2b remaining)

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

bsearch1.0ms

end0.0ms

sample18.6s

Algorithm
intervals