Average Error: 58.6 → 3.4
Time: 39.5s
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
Target14.1
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. 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 3.4

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

  4. Final simplification3.4

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

Reproduce

herbie shell --seed 2019016 +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: 39.1s)Debug log

sample612.0ms

Algorithm
intervals
Results
328×(pre true 80)
128×(body real 1280)
72×(body real 2560)
70×(body exit 10240)
29×(body real 640)
17×(body real 80)
6×(body real 320)
6×(body real 160)

simplify145.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
145.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

prune34.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 27.5b

localize47.0ms

Local error

Found 4 expressions with local error:

15.7b
(/ eps (expm1 (* eps a)))
12.2b
(/ (expm1 (* (+ a b) eps)) (expm1 (* eps b)))
3.2b
(* (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b))) (/ eps (expm1 (* eps a))))
0.0b
(expm1 (* (+ a b) eps))

rewrite11.0ms

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
6.0ms
(* (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b))) (/ eps (expm1 (* eps a))))
4.0ms
(/ (expm1 (* (+ a b) eps)) (expm1 (* eps b)))
1.0ms
(/ eps (expm1 (* eps a)))
0.0ms
(expm1 (* (+ a b) eps))

series592.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
443.0ms
(* (/ (expm1 (* (+ a b) eps)) (expm1 (* eps b))) (/ eps (expm1 (* eps a))))
71.0ms
(/ (expm1 (* (+ a b) eps)) (expm1 (* eps b)))
41.0ms
(/ eps (expm1 (* eps a)))
36.0ms
(expm1 (* (+ a b) eps))

simplify2.6s

Counts
67 → 103
Calls
67 calls:
Slowest
375.0ms
(* (* (* (/ (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)))))
250.0ms
(+ (- (log (expm1 (* (+ a b) eps))) (log (expm1 (* eps b)))) (log (/ eps (expm1 (* eps a)))))
207.0ms
(/ (* (- (exp (* (+ a b) eps)) 1) eps) (- (+ (* (exp (* a eps)) (exp (* eps b))) 1) (+ (exp (* eps b)) (exp (* a eps)))))
196.0ms
(/ (* (- (exp (* (+ a b) eps)) 1) eps) (- (+ (* (exp (* a eps)) (exp (* eps b))) 1) (+ (exp (* eps b)) (exp (* a eps)))))
174.0ms
(+ (* 1/2 (* (pow a 2) (pow eps 2))) (+ (* eps b) (* a eps)))

prune1.4s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize16.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Rules
Counts
0 → 0
Calls
0 calls:
Slowest

series0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

simplify0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

prune16.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize18.0ms

Local error

Found 4 expressions with local error:

15.7b
(/ eps (expm1 (* eps a)))
15.1b
(/ (* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a)))) (expm1 (* eps b)))
3.3b
(* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a))))
0.0b
(expm1 (* (+ a b) eps))

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
11×add-exp-log
11×add-cbrt-cube
9×add-cube-cbrt
9×*-un-lft-identity
9×add-sqr-sqrt
6×associate-/r*
6×pow1
5×log1p-expm1-u
4×add-log-exp
4×associate-*r*
4×expm1-log1p-u
3×div-inv
3×associate-*l*
3×times-frac
2×cbrt-unprod
2×associate-*r/
2×prod-exp
2×div-exp
2×frac-2neg
2×clear-num
2×cbrt-undiv
1×associate-/l/
1×expm1-udef
1×associate-/l*
1×*-commutative
1×pow-prod-down
1×expm1-log1p
Counts
4 → 71
Calls
4 calls:
Slowest
13.0ms
(/ (* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a)))) (expm1 (* eps b)))
7.0ms
(* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a))))
1.0ms
(/ eps (expm1 (* eps a)))
1.0ms
(expm1 (* (+ a b) eps))

series750.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
519.0ms
(/ (* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a)))) (expm1 (* eps b)))
116.0ms
(* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a))))
62.0ms
(expm1 (* (+ a b) eps))
52.0ms
(/ eps (expm1 (* eps a)))

simplify2.1s

Counts
41 → 83
Calls
41 calls:
Slowest
285.0ms
(- (+ (log (expm1 (* (+ a b) eps))) (log (/ eps (expm1 (* eps a))))) (log (expm1 (* eps b))))
262.0ms
(+ (* 1/2 (* (pow a 2) (pow eps 2))) (+ (* eps b) (* a eps)))
165.0ms
(- (log (* (expm1 (* (+ a b) eps)) (/ eps (expm1 (* eps a))))) (log (expm1 (* eps b))))
155.0ms
(* (* (* (expm1 (* (+ a b) eps)) (expm1 (* (+ a b) eps))) (expm1 (* (+ a b) eps))) (* (* (/ eps (expm1 (* eps a))) (/ eps (expm1 (* eps a)))) (/ eps (expm1 (* eps a)))))
152.0ms
(/ (* (- (exp (* (+ a b) eps)) 1) eps) (- (+ (* (exp (* a eps)) (exp (* eps b))) 1) (+ (exp (* eps b)) (exp (* a eps)))))

prune894.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize6.0ms

Local error

Found 4 expressions with local error:

20.9b
(/ (expm1 (* (+ a b) eps)) (/ (expm1 (* eps b)) (/ eps (expm1 (* eps a)))))
15.7b
(/ eps (expm1 (* eps a)))
4.6b
(/ (expm1 (* eps b)) (/ eps (expm1 (* eps a))))
0.0b
(expm1 (* (+ a b) eps))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
111×times-frac
64×add-cube-cbrt
64×*-un-lft-identity
64×add-sqr-sqrt
24×associate-/r*
23×div-inv
11×add-exp-log
11×add-cbrt-cube
6×associate-/l*
6×associate-/r/
5×log1p-expm1-u
4×add-log-exp
4×div-exp
4×cbrt-undiv
4×pow1
4×expm1-log1p-u
3×expm1-udef
3×frac-2neg
3×clear-num
2×div-sub
1×expm1-log1p
Counts
4 → 150
Calls
4 calls:
Slowest
5.0ms
(/ (expm1 (* (+ a b) eps)) (/ (expm1 (* eps b)) (/ eps (expm1 (* eps a)))))
3.0ms
(/ (expm1 (* eps b)) (/ eps (expm1 (* eps a))))
0.0ms
(/ eps (expm1 (* eps a)))
0.0ms
(expm1 (* (+ a b) eps))

series578.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
430.0ms
(/ (expm1 (* (+ a b) eps)) (/ (expm1 (* eps b)) (/ eps (expm1 (* eps a)))))
71.0ms
(/ (expm1 (* eps b)) (/ eps (expm1 (* eps a))))
40.0ms
(expm1 (* (+ a b) eps))
37.0ms
(/ eps (expm1 (* eps a)))

simplify7.1s

Counts
177 → 162
Calls
177 calls:
Slowest
486.0ms
(* -1 (/ (- (+ (exp (* a eps)) (exp (* eps b))) (+ (* (exp (* a eps)) (exp (* eps b))) 1)) eps))
325.0ms
(/ (sqrt (expm1 (* (+ a b) eps))) (cbrt (/ (expm1 (* eps b)) (/ eps (expm1 (* eps a))))))
263.0ms
(/ (* (cbrt (expm1 (* (+ a b) eps))) (cbrt (expm1 (* (+ a b) eps)))) (/ (* (cbrt (expm1 (* eps b))) (cbrt (expm1 (* eps b)))) (* (cbrt (/ eps (expm1 (* eps a)))) (cbrt (/ eps (expm1 (* eps a)))))))
255.0ms
(+ (* 1/2 (* a (* (pow eps 2) (pow b 2)))) (+ (* 1/2 (* (pow a 2) (* (pow eps 2) b))) (* a (* eps b))))
252.0ms
(+ (* 1/2 (* (pow a 2) (pow eps 2))) (+ (* eps b) (* a eps)))

prune2.0s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes84.0ms

Accuracy

0% (3.3b remaining)

Error of 3.4b against oracle of 0.0b and baseline of 3.4b

bsearch1.0ms

end0.0ms

sample20.1s

Algorithm
intervals
Results
10025×(pre true 80)
3466×(body real 1280)
2584×(body real 2560)
2021×(body exit 10240)
1053×(body real 640)
449×(body real 80)
327×(body real 320)
125×(body real 160)