Average Error: 58.7 → 3.3
Time: 26.1s
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

Original58.7
Target14.1
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. Simplified27.6

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

  3. Taylor expanded around 0 3.3

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

  4. Final simplification3.3

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

Reproduce

herbie shell --seed 2019022 +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: 25.7s)Debug log

sample618.0ms

Algorithm
intervals
Results
220.0ms85×body2560valid
189.0ms81×body10240exit
147.0ms123×body1280valid
19.0ms25×body640valid
13.0ms338×pre80true
4.0ms10×body320valid
2.0ms13×body80valid
0.0ms1×body160valid

simplify418.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
417.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

prune21.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 29.4b

localize59.0ms

Local error

Found 4 expressions with local error:

16.5b
(/ eps (expm1 (* eps a)))
12.2b
(/ (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))

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
21×add-cube-cbrt
21×*-un-lft-identity
21×add-sqr-sqrt
18×times-frac
13×associate-*l*
11×add-exp-log
11×add-cbrt-cube
6×associate-/r*
6×pow1
5×log1p-expm1-u
4×add-log-exp
4×div-inv
4×associate-*r*
4×expm1-log1p-u
3×associate-/l*
2×expm1-udef
2×cbrt-unprod
2×prod-exp
2×div-exp
2×frac-2neg
2×clear-num
2×cbrt-undiv
1×*-commutative
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×div-sub
1×expm1-log1p
1×frac-times
Counts
4 → 91
Calls
4 calls:
Slowest
12.0ms
(* (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b))) (/ eps (expm1 (* eps a))))
6.0ms
(/ (expm1 (* (+ a b) eps)) (expm1 (* eps b)))
1.0ms
(/ eps (expm1 (* eps a)))
1.0ms
(expm1 (* (+ a b) eps))

series558.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
365.0ms
(* (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b))) (/ eps (expm1 (* eps a))))
87.0ms
(/ (expm1 (* (+ a b) eps)) (expm1 (* eps b)))
56.0ms
(expm1 (* (+ a b) eps))
50.0ms
(/ eps (expm1 (* eps a)))

simplify2.5s

Counts
67 → 103
Calls
67 calls:
Slowest
407.0ms
(* (* (* (/ (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)))))
285.0ms
(+ (- (log (expm1 (* (+ a b) eps))) (log (expm1 (* eps b)))) (log (/ eps (expm1 (* eps a)))))
227.0ms
(+ (* 1/2 (* (pow a 2) (pow eps 2))) (+ (* eps b) (* a eps)))
188.0ms
(/ (* (- (exp (* (+ a b) eps)) 1) eps) (- (+ (* (exp (* a eps)) (exp (* eps b))) 1) (+ (exp (* eps b)) (exp (* a eps)))))
140.0ms
(/ (* (- (exp (* (+ a b) eps)) 1) eps) (- (+ (* (exp (* a eps)) (exp (* eps b))) 1) (+ (exp (* eps b)) (exp (* a eps)))))

prune1.3s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 1 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
6×*-un-lft-identity
5×div-inv
4×distribute-lft-out
4×fma-def
3×add-log-exp
2×add-cube-cbrt
2×add-sqr-sqrt
1×log1p-expm1-u
1×flip-+
1×add-exp-log
1×frac-add
1×add-cbrt-cube
1×flip3-+
1×sum-log
1×+-commutative
1×pow1
1×expm1-log1p-u
Counts
1 → 22
Calls
1 calls:
Slowest
7.0ms
(+ (/ 1 a) (/ 1 b))

series12.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
12.0ms
(+ (/ 1 a) (/ 1 b))

simplify65.0ms

Counts
12 → 25
Calls
12 calls:
Slowest
10.0ms
(* (exp (/ 1 a)) (exp (/ 1 b)))
8.0ms
(+ (/ 1 a) (/ 1 b))
8.0ms
(+ (* 1 b) (* a 1))
5.0ms
(+ (/ 1 a) (/ 1 b))
5.0ms
(+ (/ 1 a) (/ 1 b))

prune230.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 1 expressions with local error:

12.7b
(/ (+ b a) (* a b))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
3×associate-/l*
3×add-cube-cbrt
3×add-exp-log
3×add-cbrt-cube
3×times-frac
3×*-un-lft-identity
3×add-sqr-sqrt
2×associate-/l/
1×add-log-exp
1×div-inv
1×log1p-expm1-u
1×flip-+
1×div-exp
1×flip3-+
1×frac-2neg
1×associate-/r*
1×clear-num
1×cbrt-undiv
1×pow1
1×expm1-log1p-u
Counts
1 → 23
Calls
1 calls:
Slowest
6.0ms
(/ (+ b a) (* a b))

series38.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
38.0ms
(/ (+ b a) (* a b))

simplify194.0ms

Counts
15 → 26
Calls
15 calls:
Slowest
112.0ms
(/ (* (* (+ b a) (+ b a)) (+ b a)) (* (* (* a b) (* a b)) (* a b)))
30.0ms
(- (log (+ b a)) (log (* a b)))
13.0ms
(- (* b b) (* a a))
9.0ms
(/ (* (cbrt (+ b a)) (cbrt (+ b a))) a)
5.0ms
(+ (pow b 3) (pow a 3))

prune193.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize8.0ms

Local error

Found 2 expressions with local error:

8.3b
(* (/ 1 a) (/ (+ b a) b))
0.0b
(/ (+ b a) b)

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-exp-log
4×associate-*l*
4×associate-*r*
4×add-cbrt-cube
4×pow1
3×associate-/l*
3×div-inv
2×associate-/l/
2×add-log-exp
2×log1p-expm1-u
2×expm1-log1p-u
1×flip-+
1×cbrt-unprod
1×*-commutative
1×associate-*r/
1×prod-exp
1×associate-*l/
1×pow-prod-down
1×flip3-+
1×frac-2neg
1×clear-num
1×frac-times
Counts
2 → 41
Calls
2 calls:
Slowest
7.0ms
(* (/ 1 a) (/ (+ b a) b))
4.0ms
(/ (+ b a) b)

series36.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
24.0ms
(* (/ 1 a) (/ (+ b a) b))
11.0ms
(/ (+ b a) b)

simplify904.0ms

Counts
22 → 47
Calls
22 calls:
Slowest
576.0ms
(* (* (* (/ 1 a) (/ 1 a)) (/ 1 a)) (* (* (/ (+ b a) b) (/ (+ b a) b)) (/ (+ b a) b)))
118.0ms
(* (/ 1 a) (/ (+ b a) b))
99.0ms
(+ (log (/ 1 a)) (log (/ (+ b a) b)))
47.0ms
(* 1 (/ (+ b a) b))
13.0ms
(- (* b b) (* a a))

prune336.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes163.0ms

Accuracy

0% (3.3b remaining)

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

bsearch2.0ms

end0.0ms

sample18.0s

Algorithm
intervals
Results
6.5s2573×body2560valid
5.0s3435×body1280valid
4.5s1983×body10240exit
860.0ms1093×body640valid
527.0ms9988×pre80true
186.0ms337×body320valid
73.0ms441×body80valid
45.0ms126×body160valid