Average Error: 58.8 → 3.2
Time: 1.3m
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
Target14.1
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 56.4

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

  3. Simplified55.2

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

  4. Taylor expanded around 0 3.2

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

  5. Final simplification3.2

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

Reproduce

herbie shell --seed 1137770271 
(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: 1.2m)Debug log

sample1.3s

Algorithm
intervals
Results
27×447.0ms(body real 640)
81×282.0ms(body real 2560)
76×246.0ms(body exit 10240)
120×242.0ms(body real 1280)
332×18.0ms(pre true 80)
12×12.0ms(body real 320)
13×4.0ms(body real 80)
3×2.0ms(body real 160)

simplify311.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
311.0ms
(/ (* 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: 58.8b

localize98.0ms

Local error

Found 4 expressions with local error:

5.9b
(- (exp (* (+ a b) eps)) 1)
3.5b
(- (exp (* a eps)) 1)
3.4b
(- (exp (* b eps)) 1)
0.5b
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))

rewrite90.0ms

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
45.0ms
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))
22.0ms
(- (exp (* (+ a b) eps)) 1)
11.0ms
(- (exp (* b eps)) 1)
11.0ms
(- (exp (* a eps)) 1)

series247.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
76.0ms
(* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))
72.0ms
(- (exp (* (+ a b) eps)) 1)
50.0ms
(- (exp (* a eps)) 1)
49.0ms
(- (exp (* b eps)) 1)

simplify6.4s

Counts
41 → 79
Calls
41 calls:
Slowest
817.0ms
(* (* (* (- (exp (* a eps)) 1) (- (exp (* a eps)) 1)) (- (exp (* a eps)) 1)) (* (* (- (exp (* b eps)) 1) (- (exp (* b eps)) 1)) (- (exp (* b eps)) 1)))
733.0ms
(* (+ (exp (* a eps)) 1) (+ (* (exp (* b eps)) (exp (* b eps))) (+ (* 1 1) (* (exp (* b eps)) 1))))
476.0ms
(+ (* eps b) (+ (* 1/2 (* (pow eps 2) (pow b 2))) (* 1/6 (* (pow eps 3) (pow b 3)))))
472.0ms
(+ (* 1/2 (* (pow a 2) (pow eps 2))) (+ (* 1/6 (* (pow a 3) (pow eps 3))) (* a eps)))
423.0ms
(* (- (* (exp (* a eps)) (exp (* a eps))) (* 1 1)) (- (exp (* b eps)) 1))

prune2.3s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 47.3b

localize89.0ms

Local error

Found 4 expressions with local error:

5.9b
(- (exp (* (+ a b) eps)) 1)
3.8b
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))))
3.5b
(- (exp (* a eps)) 1)
1.2b
(* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))

rewrite135.0ms

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
54.0ms
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))))
45.0ms
(* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))
17.0ms
(- (exp (* (+ a b) eps)) 1)
17.0ms
(- (exp (* a eps)) 1)

series2.8s

Counts
4 → 12
Calls
4 calls:
Slowest
2.4s
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2)))))
275.0ms
(* (- (exp (* a eps)) 1) (+ (* eps b) (* (* (* eps b) (* eps b)) (+ (* b (* 1/6 eps)) 1/2))))
75.0ms
(- (exp (* (+ a b) eps)) 1)
57.0ms
(- (exp (* a eps)) 1)

simplify14.8s

Counts
54 → 91
Calls
54 calls:
Slowest
898.0ms
(/ (* (* (* 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))))))
728.0ms
(- (+ (/ (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))))))))
708.0ms
(- (+ (/ (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))))))))
659.0ms
(* (+ (* (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))))))
603.0ms
(* (* (* (- (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)))))

prune2.5s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize23.0ms

Local error

Found 1 expressions with local error:

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

rewrite9.0ms

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))

simplify57.0ms

Counts
10 → 19
Calls
10 calls:
Slowest
11.0ms
(* (exp (/ 1 a)) (exp (/ 1 b)))
8.0ms
(+ (/ 1 a) (/ 1 b))
6.0ms
(+ (/ 1 a) (/ 1 b))
5.0ms
(+ (/ 1 b) (/ 1 a))
5.0ms
(+ (/ 1 b) (/ 1 a))

prune311.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (cbrt (+ (/ 1 a) (/ 1 b))) (cbrt (+ (/ 1 a) (/ 1 b))))
0.5b
(cbrt (+ (/ 1 a) (/ 1 b)))
0.5b
(cbrt (+ (/ 1 a) (/ 1 b)))
0.5b
(cbrt (+ (/ 1 a) (/ 1 b)))

rewrite47.0ms

Algorithm
rewrite-expression-head
Rules
35×cbrt-prod
33×cbrt-div
31×*-un-lft-identity
20×distribute-lft-out
20×div-inv
11×flip-+
11×add-cube-cbrt
11×frac-add
11×flip3-+
11×add-sqr-sqrt
10×associate-*l*
10×associate-*r*
9×pow1
9×frac-times
7×pow1/3
6×add-exp-log
4×add-log-exp
4×add-cbrt-cube
3×associate-*r/
3×associate-*l/
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 105
Calls
4 calls:
Slowest
28.0ms
(* (cbrt (+ (/ 1 a) (/ 1 b))) (cbrt (+ (/ 1 a) (/ 1 b))))
5.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))
5.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))
5.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))

series597.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
171.0ms
(* (cbrt (+ (/ 1 a) (/ 1 b))) (cbrt (+ (/ 1 a) (/ 1 b))))
145.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))
141.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))
140.0ms
(cbrt (+ (/ 1 a) (/ 1 b)))

simplify11.0s

Counts
94 → 117
Calls
94 calls:
Slowest
850.0ms
(- (* (cbrt -1) (pow (/ -1 b) 1/3)) (+ (* 1/9 (* (/ (cbrt -1) (pow a 2)) (pow (* (pow b 5) -1) 1/3))) (* 1/3 (* (/ (cbrt -1) a) (pow (pow b 2) 1/3)))))
726.0ms
(- (+ (* 2/3 (* (/ 1 a) (pow b 1/3))) (pow (/ 1 b) 2/3)) (* 1/9 (* (/ 1 (pow a 2)) (pow (pow b 4) 1/3))))
621.0ms
(- (+ (pow (/ 1 b) 1/3) (* 1/3 (* (/ 1 a) (pow (pow b 2) 1/3)))) (* 1/9 (* (/ 1 (pow a 2)) (pow (pow b 5) 1/3))))
584.0ms
(- (+ (pow (/ 1 b) 1/3) (* 1/3 (* (/ 1 a) (pow (pow b 2) 1/3)))) (* 1/9 (* (/ 1 (pow a 2)) (pow (pow b 5) 1/3))))
582.0ms
(- (+ (pow (/ 1 b) 1/3) (* 1/3 (* (/ 1 a) (pow (pow b 2) 1/3)))) (* 1/9 (* (/ 1 (pow a 2)) (pow (pow b 5) 1/3))))

prune2.2s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

regimes350.0ms

Accuracy

0% (3.1b remaining)

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

bsearch2.0ms

end0.0ms

sample28.4s

Algorithm
intervals
Results
2591×10.0s(body real 2560)
3457×7.8s(body real 1280)
2096×7.1s(body exit 10240)
1065×1.6s(body real 640)
350×766.0ms(body real 320)
10103×546.0ms(pre true 80)
428×164.0ms(body real 80)
116×78.0ms(body real 160)