Average Error: 58.7 → 3.3
Time: 39.8s
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. Taylor expanded around 0 56.4

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

  3. Simplified54.9

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

  4. Taylor expanded around 0 3.3

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

  5. Final simplification3.3

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

Reproduce

herbie shell --seed 2019022 
(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: 39.2s)Debug log

sample552.0ms

Algorithm
intervals
Results
209.0ms85×body2560valid
144.0ms81×body10240exit
138.0ms123×body1280valid
19.0ms25×body640valid
12.0ms338×pre80true
5.0ms10×body320valid
2.0ms13×body80valid
0.0ms1×body160valid

simplify185.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
185.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 59.1b

localize53.0ms

Local error

Found 4 expressions with local error:

4.9b
(- (exp (* (+ a b) eps)) 1)
3.0b
(- (exp (* b eps)) 1)
2.7b
(- (exp (* a eps)) 1)
0.2b
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))

rewrite40.0ms

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
24.0ms
(* (- (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)

series429.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
325.0ms
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))
38.0ms
(- (exp (* (+ a b) eps)) 1)
36.0ms
(- (exp (* a eps)) 1)
29.0ms
(- (exp (* b eps)) 1)

simplify4.1s

Counts
41 → 79
Calls
41 calls:
Slowest
652.0ms
(* (* (* (- (exp (* a eps)) 1) (- (exp (* a eps)) 1)) (- (exp (* a eps)) 1)) (* (* (- (exp (* b eps)) 1) (- (exp (* b eps)) 1)) (- (exp (* b eps)) 1)))
482.0ms
(* (+ (exp (* a eps)) 1) (+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1))))
326.0ms
(+ (* eps b) (+ (* 1/2 (* (pow eps 2) (pow b 2))) (* 1/6 (* (pow eps 3) (pow b 3)))))
274.0ms
(* (- (* (exp (* a eps)) (exp (* a eps))) (* 1 1)) (- (exp (* b eps)) 1))
261.0ms
(* (- (pow (exp (* a eps)) 3) (pow 1 3)) (- (* (exp (* b eps)) (exp (* b eps))) (* 1 1)))

prune1000.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 48.0b

localize35.0ms

Local error

Found 4 expressions with local error:

4.9b
(- (exp (* (+ a b) eps)) 1)
3.2b
(/ (* 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)
2.0b
(* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))

rewrite58.0ms

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-*l*
3×associate-*r*
3×difference-of-sqr-1
2×associate-/l/
2×cbrt-unprod
2×prod-exp
2×div-exp
2×sub-neg
2×cbrt-undiv
1×associate-/l*
1×div-inv
1×*-commutative
1×pow-prod-down
1×times-frac
1×distribute-lft-in
1×frac-2neg
1×associate-/r*
1×clear-num
1×distribute-rgt-in
Counts
4 → 79
Calls
4 calls:
Slowest
26.0ms
(* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))
17.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))))
8.0ms
(- (exp (* a eps)) 1)
6.0ms
(- (exp (* (+ a b) eps)) 1)

series1.7s

Counts
4 → 12
Calls
4 calls:
Slowest
1.5s
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))))
122.0ms
(* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))
41.0ms
(- (exp (* (+ a b) eps)) 1)
26.0ms
(- (exp (* a eps)) 1)

simplify9.2s

Counts
54 → 91
Calls
54 calls:
Slowest
543.0ms
(- (+ (/ (exp (* (+ a b) eps)) (* (- (* 1/6 (exp (* a eps))) 1/6) (* (pow b 3) (pow eps 2)))) (+ (* 1/2 (/ (exp (* a eps)) (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (* (pow b 4) (pow eps 3))))) (* 1/2 (/ (exp (* (+ a b) eps)) (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (* (pow b 4) (pow eps 3))))))) (+ (* 1/2 (/ (* (exp (* (+ a b) eps)) (exp (* a eps))) (* (pow eps 3) (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (pow b 4))))) (+ (/ 1 (* (- (* 1/6 (exp (* a eps))) 1/6) (* (pow eps 2) (pow b 3)))) (* 1/2 (/ 1 (* (pow (- (* 1/6 (exp (* a eps))) 1/6) 2) (* (pow eps 3) (pow b 4))))))))
493.0ms
(/ (* (* (* 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))))))
417.0ms
(* (* (* (- (exp (* a eps)) 1) (- (exp (* a eps)) 1)) (- (exp (* a eps)) 1)) (* (* (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))))
406.0ms
(* (+ (* (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))))))
396.0ms
(* (+ (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))))))

prune1.5s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize7.0ms

Local error

Found 1 expressions with local error:

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

rewrite3.0ms

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))

series7.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
7.0ms
(+ (/ 1 a) (/ 1 b))

simplify31.0ms

Counts
10 → 19
Calls
10 calls:
Slowest
4.0ms
(* (exp (/ 1 a)) (exp (/ 1 b)))
4.0ms
(+ (/ 1 a) (/ 1 b))
4.0ms
(+ (/ 1 a) (/ 1 b))
4.0ms
(+ (/ 1 a) (/ 1 b))
4.0ms
(+ (/ 1 a) (/ 1 b))

prune181.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 1 expressions with local error:

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

rewrite3.0ms

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

series19.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
19.0ms
(/ (+ b a) (* a b))

simplify149.0ms

Counts
15 → 24
Calls
15 calls:
Slowest
87.0ms
(/ (* (* (+ b a) (+ b a)) (+ b a)) (* (* (* a b) (* a b)) (* a b)))
16.0ms
(- (log (+ b a)) (log (* a b)))
13.0ms
(- (* b b) (* a a))
5.0ms
(/ (* (cbrt (+ b a)) (cbrt (+ b a))) a)
4.0ms
(+ (/ 1 a) (/ 1 b))

prune127.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

regimes97.0ms

Accuracy

0% (3.2b remaining)

Error of 3.3b against oracle of 0.1b and baseline of 3.3b

bsearch2.0ms

end0.0ms

sample19.6s

Algorithm
intervals
Results
6.0s2573×body2560valid
5.8s3435×body1280valid
5.2s1983×body10240exit
1.5s1093×body640valid
480.0ms9988×pre80true
255.0ms337×body320valid
76.0ms441×body80valid
48.0ms126×body160valid