Average Error: 1.9 → 1.9
Time: 50.2s
Precision: 64
Internal Precision: 128
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{x \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}{y}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.9

    \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]

  2. Taylor expanded around inf 1.9

    \[\leadsto \frac{x \cdot e^{\left(y \cdot \log z + \color{blue}{\left(1.0 \cdot \log \left(\frac{1}{a}\right) - t \cdot \log \left(\frac{1}{a}\right)\right)}\right) - b}}{y}\]

  3. Simplified1.9

    \[\leadsto \frac{x \cdot e^{\left(y \cdot \log z + \color{blue}{\left(t - 1.0\right) \cdot \log a}\right) - b}}{y}\]

  4. Final simplification1.9

    \[\leadsto \frac{x \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}{y}\]

Reproduce

herbie shell --seed 2019022 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))

Details

Time bar (total: 48.9s)Debug log

sample505.0ms

Algorithm
intervals
Results
290.0ms1488×body80nan
132.0ms493×body80valid

simplify364.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
364.0ms
(/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)

prune34.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.5b

localize64.0ms

Local error

Found 4 expressions with local error:

1.2b
(/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)
0.9b
(exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
0.2b
(* y (log z))
0.1b
(* (- t 1.0) (log a))

rewrite74.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×add-exp-log
6×associate-*r*
6×add-cbrt-cube
6×pow1
5×add-log-exp
3×exp-sum
3×associate-*l*
3×exp-prod
2×*-commutative
2×associate-*l/
2×exp-diff
1×associate-/l/
1×rem-exp-log
1×associate-/l*
1×flip--
1×div-inv
1×cbrt-unprod
1×associate-*r/
1×associate--l+
1×prod-exp
1×pow-prod-down
1×flip3--
1×frac-2neg
1×sub-neg
1×clear-num
Counts
4 → 57
Calls
4 calls:
Slowest
42.0ms
(exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
21.0ms
(/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)
8.0ms
(* (- t 1.0) (log a))
2.0ms
(* y (log z))

series427.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
211.0ms
(/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)
129.0ms
(exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
52.0ms
(* (- t 1.0) (log a))
35.0ms
(* y (log z))

simplify2.8s

Counts
34 → 69
Calls
34 calls:
Slowest
302.0ms
(+ (* t (* (log a) (exp (- (* 1.0 (log a)))))) (+ (exp (- (* 1.0 (log a)))) (* (log z) (* (exp (- (* 1.0 (log a)))) y))))
279.0ms
(- (+ (* t (log -1)) (* 1.0 (log (/ -1 a)))) (+ (* t (log (/ -1 a))) (* 1.0 (log -1))))
220.0ms
(* (* (* (- t 1.0) (- t 1.0)) (- t 1.0)) (* (* (log a) (log a)) (log a)))
170.0ms
(+ (/ (* x (exp (- (* 1.0 (log a))))) y) (+ (* x (* (log z) (exp (- (* 1.0 (log a)))))) (/ (* t (* x (* (log a) (exp (- (* 1.0 (log a))))))) y)))
168.0ms
(exp (* (cbrt (- (+ (* y (log z)) (* (- t 1.0) (log a))) b)) (cbrt (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))))

prune1.2s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.0b

localize3.0ms

Local error

Found 4 expressions with local error:

1.2b
(/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)
0.9b
(exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
0.2b
(* y (log z))
0.1b
(* (- t 1.0) (log a))

rewrite72.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×add-exp-log
6×associate-*r*
6×add-cbrt-cube
6×pow1
5×add-log-exp
3×exp-sum
3×associate-*l*
3×exp-prod
2×*-commutative
2×associate-*l/
2×exp-diff
1×associate-/l/
1×rem-exp-log
1×associate-/l*
1×flip--
1×div-inv
1×cbrt-unprod
1×associate-*r/
1×associate--l+
1×prod-exp
1×pow-prod-down
1×flip3--
1×frac-2neg
1×sub-neg
1×clear-num
Counts
4 → 57
Calls
4 calls:
Slowest
40.0ms
(exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
21.0ms
(/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)
8.0ms
(* (- t 1.0) (log a))
2.0ms
(* y (log z))

series440.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
219.0ms
(/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)
130.0ms
(exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
46.0ms
(* (- t 1.0) (log a))
44.0ms
(* y (log z))

simplify2.7s

Counts
34 → 69
Calls
34 calls:
Slowest
385.0ms
(+ (* t (* (log a) (exp (- (* 1.0 (log a)))))) (+ (exp (- (* 1.0 (log a)))) (* (log z) (* (exp (- (* 1.0 (log a)))) y))))
196.0ms
(- (+ (* t (log -1)) (* 1.0 (log (/ -1 a)))) (+ (* t (log (/ -1 a))) (* 1.0 (log -1))))
189.0ms
(/ (* x (exp (- (* 1.0 (log (/ 1 a))) (+ (* (log (/ 1 z)) y) (+ b (* t (log (/ 1 a)))))))) y)
182.0ms
(* (* (* (- t 1.0) (- t 1.0)) (- t 1.0)) (* (* (log a) (log a)) (log a)))
177.0ms
(exp (* (cbrt (- (+ (* y (log z)) (* (- t 1.0) (log a))) b)) (cbrt (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))))

prune1.1s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize32.0ms

Local error

Found 4 expressions with local error:

1.2b
(/ (* x (* (* (pow z y) (exp (- b))) (* (pow a t) (pow a (- 1.0))))) y)
0.5b
(* (* (pow z y) (exp (- b))) (* (pow a t) (pow a (- 1.0))))
0.2b
(* (pow z y) (exp (- b)))
0.0b
(* x (* (* (pow z y) (exp (- b))) (* (pow a t) (pow a (- 1.0)))))

rewrite108.0ms

Algorithm
rewrite-expression-head
Rules
89×associate-*r/
39×prod-exp
30×add-exp-log
27×frac-times
26×exp-neg
25×un-div-inv
25×neg-sub0
24×pow-neg
18×add-cbrt-cube
18×pow-to-exp
15×associate-/l/
13×exp-diff
12×pow-sub
9×cbrt-unprod
9×associate-*l/
6×add-cube-cbrt
6×*-un-lft-identity
6×pow1
6×add-sqr-sqrt
5×associate-*r*
4×add-log-exp
4×associate-*l*
3×*-commutative
1×associate-/l*
1×div-inv
1×pow-prod-down
1×frac-2neg
1×clear-num
Counts
4 → 115
Calls
4 calls:
Slowest
39.0ms
(/ (* x (* (* (pow z y) (exp (- b))) (* (pow a t) (pow a (- 1.0))))) y)
38.0ms
(* x (* (* (pow z y) (exp (- b))) (* (pow a t) (pow a (- 1.0)))))
18.0ms
(* (* (pow z y) (exp (- b))) (* (pow a t) (pow a (- 1.0))))
8.0ms
(* (pow z y) (exp (- b)))

series1.2s

Counts
4 → 12
Calls
4 calls:
Slowest
504.0ms
(/ (* x (* (* (pow z y) (exp (- b))) (* (pow a t) (pow a (- 1.0))))) y)
310.0ms
(* x (* (* (pow z y) (exp (- b))) (* (pow a t) (pow a (- 1.0)))))
274.0ms
(* (* (pow z y) (exp (- b))) (* (pow a t) (pow a (- 1.0))))
89.0ms
(* (pow z y) (exp (- b)))

simplify8.8s

Counts
100 → 127
Calls
100 calls:
Slowest
683.0ms
(+ (+ (* (log z) y) (- b)) (+ (log (pow a t)) (* (log a) (- 1.0))))
401.0ms
(+ (log (* (pow z y) (exp (- b)))) (+ (log (pow a t)) (* (log a) (- 1.0))))
392.0ms
(+ (log (* (pow z y) (exp (- b)))) (+ (* (log a) t) (* (log a) (- 1.0))))
333.0ms
(* (* (* (pow z y) (pow z y)) (pow z y)) (* (* (exp (- b)) (exp (- b))) (exp (- b))))
308.0ms
(* x (* (* (pow z y) (exp (- b))) (* (pow a t) 1)))

prune2.0s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize31.0ms

Local error

Found 4 expressions with local error:

5.3b
(/ (/ (* (pow z y) x) (/ (exp b) (pow a t))) (* y (pow a 1.0)))
0.7b
(/ (exp b) (pow a t))
0.5b
(/ (* (pow z y) x) (/ (exp b) (pow a t)))
0.0b
(* (pow z y) x)

rewrite37.0ms

Algorithm
rewrite-expression-head
Rules
102×times-frac
43×add-cube-cbrt
43×*-un-lft-identity
43×add-sqr-sqrt
22×associate-/l*
20×add-exp-log
20×add-cbrt-cube
17×associate-/r*
16×div-exp
10×cbrt-undiv
9×div-inv
4×add-log-exp
4×pow1
3×associate-/r/
3×associate-*l*
3×frac-2neg
3×clear-num
3×pow-to-exp
1×associate-/l/
1×*-commutative
Counts
4 → 137
Calls
4 calls:
Slowest
15.0ms
(/ (/ (* (pow z y) x) (/ (exp b) (pow a t))) (* y (pow a 1.0)))
11.0ms
(/ (* (pow z y) x) (/ (exp b) (pow a t)))
5.0ms
(/ (exp b) (pow a t))
3.0ms
(* (pow z y) x)

series918.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
451.0ms
(/ (/ (* (pow z y) x) (/ (exp b) (pow a t))) (* y (pow a 1.0)))
270.0ms
(/ (* (pow z y) x) (/ (exp b) (pow a t)))
105.0ms
(* (pow z y) x)
92.0ms
(/ (exp b) (pow a t))

simplify7.1s

Counts
148 → 149
Calls
148 calls:
Slowest
398.0ms
(/ (exp b) (exp (* t (- (log -1) (log (/ -1 a))))))
376.0ms
(/ (* (exp (* -1 (* (log (/ 1 z)) y))) (* x (exp (* -1 (* t (log (/ 1 a))))))) (exp b))
335.0ms
(/ (/ (* (pow z y) x) (/ (exp b) (pow a t))) (pow a 1.0))
305.0ms
(/ (* (* (/ (* (pow z y) x) (/ (exp b) (pow a t))) (/ (* (pow z y) x) (/ (exp b) (pow a t)))) (/ (* (pow z y) x) (/ (exp b) (pow a t)))) (* (* (* y (pow a 1.0)) (* y (pow a 1.0))) (* y (pow a 1.0))))
273.0ms
(/ (/ (* (* (* (pow z y) x) (* (pow z y) x)) (* (pow z y) x)) (* (* (/ (exp b) (pow a t)) (/ (exp b) (pow a t))) (/ (exp b) (pow a t)))) (* (* (* y (pow a 1.0)) (* y (pow a 1.0))) (* y (pow a 1.0))))

prune2.4s

Pruning

6 alts after pruning (3 fresh and 3 done)

Merged error: 0b

regimes1.0s

Accuracy

0% (1.9b remaining)

Error of 1.9b against oracle of 0.0b and baseline of 1.9b

bsearch2.0ms

end0.0ms

sample15.6s

Algorithm
intervals
Results
9.5s45466×body80nan
3.5s15051×body80valid