Average Error: 58.8 → 3.2
Time: 2.7m
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
Target14.2
Herbie3.2
\[\frac{a + b}{a \cdot b}\]

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. Taylor expanded around 0 56.5

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

  3. Simplified54.9

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

  4. Taylor expanded around 0 3.2

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

  5. Final simplification3.2

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

Reproduce

herbie shell --seed 1042149663 
(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: 2.7m)Debug log

sample825.0ms

Algorithm
intervals
Results
310×(pre true 80)
101×(body real 1280)
92×(body real 2560)
52×(body exit 10240)
34×(body real 640)
15×(body real 80)
9×(body real 320)
7×(body real 160)

simplify330.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
329.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

prune19.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.0b

localize101.0ms

Local error

Found 4 expressions with local error:

5.8b
(- (exp (* (+ a b) eps)) 1)
4.5b
(- (exp (* a eps)) 1)
3.1b
(- (exp (* b eps)) 1)
0.7b
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))

rewrite77.0ms

Algorithm
rewrite-expression-head
Rules
11×add-sqr-sqrt
9×flip--
9×flip3--
6×add-cube-cbrt
6×add-exp-log
6×add-cbrt-cube
6×*-un-lft-identity
6×pow1
5×difference-of-sqr-1
5×sub-neg
4×add-log-exp
4×associate-*l*
4×associate-*r*
4×frac-times
2×associate-*r/
2×associate-*l/
2×distribute-lft-in
2×distribute-rgt-in
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
Counts
4 → 67
Calls
4 calls:
Slowest
42.0ms
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))
14.0ms
(- (exp (* (+ a b) eps)) 1)
10.0ms
(- (exp (* a eps)) 1)
10.0ms
(- (exp (* b eps)) 1)

series249.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
76.0ms
(- (exp (* (+ a b) eps)) 1)
76.0ms
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))
49.0ms
(- (exp (* a eps)) 1)
48.0ms
(- (exp (* b eps)) 1)

simplify7.0s

Counts
41 → 79
Calls
41 calls:
Slowest
1.2s
(* (* (* (- (exp (* a eps)) 1) (- (exp (* a eps)) 1)) (- (exp (* a eps)) 1)) (* (* (- (exp (* b eps)) 1) (- (exp (* b eps)) 1)) (- (exp (* b eps)) 1)))
791.0ms
(* (+ (exp (* a eps)) 1) (+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1))))
494.0ms
(+ (* 1/2 (* (pow a 2) (pow eps 2))) (+ (* 1/6 (* (pow a 3) (pow eps 3))) (* a eps)))
492.0ms
(+ (* eps b) (+ (* 1/2 (* (pow eps 2) (pow b 2))) (* 1/6 (* (pow eps 3) (pow b 3)))))
433.0ms
(* (- (* (exp (* a eps)) (exp (* a eps))) (* 1 1)) (- (exp (* b eps)) 1))

prune1.8s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 46.0b

localize61.0ms

Local error

Found 4 expressions with local error:

5.8b
(- (exp (* (+ a b) eps)) 1)
4.0b
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1)))
3.1b
(- (exp (* b eps)) 1)
2.5b
(* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1))

rewrite161.0ms

Algorithm
rewrite-expression-head
Rules
11×add-exp-log
11×add-cbrt-cube
9×flip--
9×flip3--
9×add-sqr-sqrt
8×associate-/r/
8×frac-times
6×flip-+
6×add-cube-cbrt
6×associate-*r/
6×flip3-+
6×*-un-lft-identity
6×pow1
4×add-log-exp
4×associate-*l/
4×associate-*r*
4×sub-neg
3×associate-*l*
3×difference-of-sqr-1
2×associate-/l/
2×cbrt-unprod
2×prod-exp
2×div-exp
2×distribute-lft-in
2×cbrt-undiv
2×distribute-rgt-in
1×associate-/l*
1×div-inv
1×*-commutative
1×pow-prod-down
1×times-frac
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
4 → 81
Calls
4 calls:
Slowest
69.0ms
(* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1))
66.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1)))
14.0ms
(- (exp (* (+ a b) eps)) 1)
10.0ms
(- (exp (* b eps)) 1)

series1.4s

Counts
4 → 12
Calls
4 calls:
Slowest
1.1s
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1)))
155.0ms
(* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1))
73.0ms
(- (exp (* (+ a b) eps)) 1)
50.0ms
(- (exp (* b eps)) 1)

simplify14.3s

Counts
56 → 93
Calls
56 calls:
Slowest
669.0ms
(* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1))
610.0ms
(- (+ (/ (exp (* (+ a b) eps)) (* (pow a 3) (* (- (* 1/6 (exp (* eps b))) 1/6) (pow eps 2)))) (+ (* 1/2 (/ (exp (* eps b)) (* (pow a 4) (* (pow eps 3) (pow (- (* 1/6 (exp (* eps b))) 1/6) 2))))) (* 1/2 (/ (exp (* (+ a b) eps)) (* (pow a 4) (* (pow (- (* 1/6 (exp (* eps b))) 1/6) 2) (pow eps 3))))))) (+ (* 1/2 (/ (* (exp (* (+ a b) eps)) (exp (* eps b))) (* (pow a 4) (* (pow (- (* 1/6 (exp (* eps b))) 1/6) 2) (pow eps 3))))) (+ (* 1/2 (/ 1 (* (pow a 4) (* (pow eps 3) (pow (- (* 1/6 (exp (* eps b))) 1/6) 2))))) (/ 1 (* (pow a 3) (* (pow eps 2) (- (* 1/6 (exp (* eps b))) 1/6)))))))
608.0ms
(* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- 1))
608.0ms
(- (+ (/ (exp (* (+ a b) eps)) (* (pow a 3) (* (- (* 1/6 (exp (* eps b))) 1/6) (pow eps 2)))) (+ (* 1/2 (/ (exp (* eps b)) (* (pow a 4) (* (pow eps 3) (pow (- (* 1/6 (exp (* eps b))) 1/6) 2))))) (* 1/2 (/ (exp (* (+ a b) eps)) (* (pow a 4) (* (pow (- (* 1/6 (exp (* eps b))) 1/6) 2) (pow eps 3))))))) (+ (* 1/2 (/ (* (exp (* (+ a b) eps)) (exp (* eps b))) (* (pow a 4) (* (pow (- (* 1/6 (exp (* eps b))) 1/6) 2) (pow eps 3))))) (+ (* 1/2 (/ 1 (* (pow a 4) (* (pow eps 3) (pow (- (* 1/6 (exp (* eps b))) 1/6) 2))))) (/ 1 (* (pow a 3) (* (pow eps 2) (- (* 1/6 (exp (* eps b))) 1/6)))))))
605.0ms
(/ (* (* (* eps (- (exp (* (+ a b) eps)) 1)) (* eps (- (exp (* (+ a b) eps)) 1))) (* eps (- (exp (* (+ a b) eps)) 1))) (* (* (* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1)) (* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1))) (* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1))))

prune2.3s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 1 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
5×*-un-lft-identity
4×distribute-lft-out
4×div-inv
3×add-log-exp
1×flip-+
1×add-cube-cbrt
1×add-exp-log
1×frac-add
1×add-cbrt-cube
1×flip3-+
1×sum-log
1×+-commutative
1×pow1
1×add-sqr-sqrt
Counts
1 → 16
Calls
1 calls:
Slowest
8.0ms
(+ (/ 1 a) (/ 1 b))

series15.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
15.0ms
(+ (/ 1 a) (/ 1 b))

simplify58.0ms

Counts
10 → 19
Calls
10 calls:
Slowest
12.0ms
(* (exp (/ 1 a)) (exp (/ 1 b)))
5.0ms
(+ (/ 1 a) (/ 1 b))
5.0ms
(+ (/ 1 a) (/ 1 b))
5.0ms
(+ (/ 1 a) (/ 1 b))
5.0ms
(+ (/ 1 b) (/ 1 a))

prune263.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize70.0ms

Local error

Found 4 expressions with local error:

43.3b
(/ (+ (pow (/ 1 a) 3) (pow (/ 1 b) 3)) (+ (* (/ 1 a) (/ 1 a)) (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 b)))))
4.0b
(- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 b)))
2.8b
(+ (pow (/ 1 a) 3) (pow (/ 1 b) 3))
2.0b
(+ (* (/ 1 a) (/ 1 a)) (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 b))))

rewrite170.0ms

Algorithm
rewrite-expression-head
Rules
194×frac-add
165×un-div-inv
165×associate-*r/
144×frac-sub
138×associate-*l/
120×*-un-lft-identity
120×frac-times
98×associate-/r/
85×distribute-lft-out
81×div-inv
80×cube-prod
80×unpow-prod-down
63×times-frac
54×sub-div
30×add-cube-cbrt
30×add-sqr-sqrt
21×associate-/l*
13×add-log-exp
9×flip--
9×flip3--
6×add-exp-log
6×add-cbrt-cube
5×sum-cubes
4×flip-+
4×flip3-+
4×pow1
4×cube-div
3×associate-/l/
3×sum-log
3×associate-/r*
2×diff-log
2×sub-neg
2×associate-+r+
2×+-commutative
1×div-exp
1×distribute-rgt-out--
1×frac-2neg
1×associate-+r-
1×clear-num
1×cbrt-undiv
Counts
4 → 376
Calls
4 calls:
Slowest
89.0ms
(/ (+ (pow (/ 1 a) 3) (pow (/ 1 b) 3)) (+ (* (/ 1 a) (/ 1 a)) (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 b)))))
29.0ms
(+ (* (/ 1 a) (/ 1 a)) (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 b))))
15.0ms
(+ (pow (/ 1 a) 3) (pow (/ 1 b) 3))
13.0ms
(- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 b)))

series161.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
67.0ms
(/ (+ (pow (/ 1 a) 3) (pow (/ 1 b) 3)) (+ (* (/ 1 a) (/ 1 a)) (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 b)))))
42.0ms
(+ (pow (/ 1 a) 3) (pow (/ 1 b) 3))
29.0ms
(- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 b)))
24.0ms
(+ (* (/ 1 a) (/ 1 a)) (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 b))))

simplify1.6m

Counts
539 → 388
Calls
539 calls:
Slowest
1.1s
(+ (* (/ 1 a) (* b (* a b))) (* a (- (* (/ 1 b) (* a b)) (* b (* 1 1)))))
1.0s
(+ (* (* 1 1) (* b (* a b))) (* (* a a) (- (* (/ 1 b) (* a b)) (* b (* 1 1)))))
1.0s
(+ (* (/ 1 a) (* b b)) (* a (- (* (* 1 (/ 1 b)) b) (* b (/ 1 a)))))
997.0ms
(+ (* (* 1 (/ 1 a)) (* b (* a b))) (* a (- (* (/ 1 b) (* a b)) (* b (* 1 1)))))
988.0ms
(+ (* (/ 1 a) (* (* b b) a)) (* a (- (* (* 1 1) a) (* (* b b) (* 1 (/ 1 b))))))

prune7.5s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

regimes250.0ms

Accuracy

0% (3.1b remaining)

Error of 3.2b against oracle of 0.1b and baseline of 3.2b

bsearch2.0ms

end0.0ms

sample27.8s

Algorithm
intervals
Results
10048×(pre true 80)
3357×(body real 1280)
2640×(body real 2560)
2045×(body exit 10240)
1088×(body real 640)
428×(body real 80)
350×(body real 320)
140×(body real 160)