Average Error: 58.9 → 3.1
Time: 37.3s
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

Original58.9
Target14.1
Herbie3.1
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.9

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

  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. Simplified55.0

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

  4. Taylor expanded around 0 3.1

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

  5. Final simplification3.1

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

Reproduce

herbie shell --seed 2019021 
(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: 36.7s)Debug log

sample531.0ms

Algorithm
intervals
Results
180.0ms102×body2560valid
136.0ms104×body1280valid
124.0ms72×body10240exit
18.0ms29×body640valid
11.0ms330×pre80true
5.0ms7×body320valid
2.0ms12×body80valid
2.0ms4×body160valid

simplify173.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
173.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 59.0b

localize37.0ms

Local error

Found 4 expressions with local error:

5.5b
(- (exp (* (+ a b) eps)) 1)
3.6b
(- (exp (* a eps)) 1)
2.7b
(- (exp (* b eps)) 1)
0.2b
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))

rewrite34.0ms

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
18.0ms
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))
6.0ms
(- (exp (* (+ a b) eps)) 1)
4.0ms
(- (exp (* b eps)) 1)
4.0ms
(- (exp (* a eps)) 1)

series155.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
48.0ms
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))
39.0ms
(- (exp (* (+ a b) eps)) 1)
38.0ms
(- (exp (* b eps)) 1)
30.0ms
(- (exp (* a eps)) 1)

simplify3.7s

Counts
41 → 79
Calls
41 calls:
Slowest
504.0ms
(* (* (* (- (exp (* a eps)) 1) (- (exp (* a eps)) 1)) (- (exp (* a eps)) 1)) (* (* (- (exp (* b eps)) 1) (- (exp (* b eps)) 1)) (- (exp (* b eps)) 1)))
389.0ms
(* (+ (exp (* a eps)) 1) (+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1))))
294.0ms
(+ (* eps b) (+ (* 1/2 (* (pow eps 2) (pow b 2))) (* 1/6 (* (pow eps 3) (pow b 3)))))
244.0ms
(* (- (pow (exp (* a eps)) 3) (pow 1 3)) (- (* (exp (* b eps)) (exp (* b eps))) (* 1 1)))
240.0ms
(+ (* 1/2 (* (pow a 2) (pow eps 2))) (+ (* 1/6 (* (pow a 3) (pow eps 3))) (* a eps)))

prune1.0s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 48.4b

localize43.0ms

Local error

Found 4 expressions with local error:

5.5b
(- (exp (* (+ a b) eps)) 1)
2.7b
(- (exp (* b eps)) 1)
2.5b
(* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1))
2.1b
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1)))

rewrite107.0ms

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
49.0ms
(* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1))
40.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1)))
10.0ms
(- (exp (* (+ a b) eps)) 1)
7.0ms
(- (exp (* b eps)) 1)

series761.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
618.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1)))
80.0ms
(* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- (exp (* b eps)) 1))
35.0ms
(- (exp (* (+ a b) eps)) 1)
27.0ms
(- (exp (* b eps)) 1)

simplify9.0s

Counts
56 → 93
Calls
56 calls:
Slowest
410.0ms
(/ (* (* (* 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))))
381.0ms
(* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- 1))
374.0ms
(* (+ (* (* (* a eps) (* a eps)) (+ (* eps (* 1/6 a)) 1/2)) (* a eps)) (- 1))
371.0ms
(+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1)))
352.0ms
(- (+ (/ (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)))))))

prune1.2s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 1 expressions with local error:

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

rewrite4.0ms

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

series8.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
8.0ms
(+ (/ 1 a) (/ 1 b))

simplify30.0ms

Counts
10 → 19
Calls
10 calls:
Slowest
8.0ms
(* (exp (/ 1 a)) (exp (/ 1 b)))
3.0ms
(+ (/ 1 a) (/ 1 b))
3.0ms
(+ (/ 1 a) (/ 1 b))
2.0ms
(+ (/ 1 b) (/ 1 a))
2.0ms
(+ (/ 1 a) (/ 1 b))

prune113.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 1 expressions with local error:

15.5b
(/ (+ 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×flip-+
1×div-exp
1×flip3-+
1×frac-2neg
1×associate-/r*
1×clear-num
1×cbrt-undiv
1×pow1
Counts
1 → 21
Calls
1 calls:
Slowest
5.0ms
(/ (+ b a) (* a b))

series20.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
20.0ms
(/ (+ b a) (* a b))

simplify147.0ms

Counts
15 → 24
Calls
15 calls:
Slowest
65.0ms
(/ (* (* (+ b a) (+ b a)) (+ b a)) (* (* (* a b) (* a b)) (* a b)))
27.0ms
(- (log (+ b a)) (log (* a b)))
16.0ms
(- (* b b) (* a a))
6.0ms
(/ (* (cbrt (+ b a)) (cbrt (+ b a))) a)
5.0ms
(+ (/ 1 a) (/ 1 b))

prune196.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

regimes146.0ms

Accuracy

0% (2.9b remaining)

Error of 3.1b against oracle of 0.2b and baseline of 3.1b

bsearch1.0ms

end0.0ms

sample19.3s

Algorithm
intervals
Results
6.0s2688×body2560valid
5.8s3366×body1280valid
5.1s2014×body10240exit
1.1s1073×body640valid
794.0ms10018×pre80true
194.0ms334×body320valid
76.0ms129×body160valid
74.0ms414×body80valid