Average Error: 1.9 → 1.9
Time: 54.1s
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 0 1.9

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

  3. 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 2018362 +o rules:numerics
(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: 26.4s)Debug log

start1.0s

Algorithm
intervals

setup406.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.9b

localize50.0ms

Local error

Found 4 expressions with local error:

0.8b
(exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
0.7b
(/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)
0.3b
(* y (log z))
0.1b
(* (- t 1.0) (log a))

rewrite44.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

26.0ms
(exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
12.0ms
(/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)
4.0ms
(* (- t 1.0) (log a))

series455.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

204.0ms
(/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)
159.0ms
(exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
55.0ms
(* (- t 1.0) (log a))
38.0ms
(* y (log z))

simplify3.5s

Counts
34 → 77
Calls

34 calls. Slowest were:

409.0ms
(- (+ (* t (log -1)) (* 1.0 (log (/ -1 a)))) (+ (* t (log (/ -1 a))) (* 1.0 (log -1))))
344.0ms
(+ (* t (* (log a) (exp (- (* 1.0 (log a)))))) (+ (exp (- (* 1.0 (log a)))) (* (log z) (* (exp (- (* 1.0 (log a)))) y))))
246.0ms
(* (* (* (- t 1.0) (- t 1.0)) (- t 1.0)) (* (* (log a) (log a)) (log a)))

prune1.4s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.1b

localize33.0ms

Local error

Found 4 expressions with local error:

0.8b
(exp (- (+ (* (log z) y) (* (- t 1.0) (log a))) b))
0.7b
(/ (* x (exp (- (+ (* (log z) y) (* (- t 1.0) (log a))) b))) y)
0.3b
(* (log z) y)
0.1b
(* (- t 1.0) (log a))

rewrite105.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

64.0ms
(exp (- (+ (* (log z) y) (* (- t 1.0) (log a))) b))
29.0ms
(/ (* x (exp (- (+ (* (log z) y) (* (- t 1.0) (log a))) b))) y)
7.0ms
(* (- t 1.0) (log a))

series482.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

207.0ms
(/ (* x (exp (- (+ (* (log z) y) (* (- t 1.0) (log a))) b))) y)
156.0ms
(exp (- (+ (* (log z) y) (* (- t 1.0) (log a))) b))
62.0ms
(* (log z) y)
57.0ms
(* (- t 1.0) (log a))

simplify3.2s

Counts
33 → 77
Calls

33 calls. Slowest were:

293.0ms
(+ (* t (* (log a) (exp (- (* 1.0 (log a)))))) (+ (exp (- (* 1.0 (log a)))) (* (log z) (* (exp (- (* 1.0 (log a)))) y))))
284.0ms
(- (+ (* t (log -1)) (* 1.0 (log (/ -1 a)))) (+ (* t (log (/ -1 a))) (* 1.0 (log -1))))
235.0ms
(* (* (* (- t 1.0) (- t 1.0)) (- t 1.0)) (* (* (log a) (log a)) (log a)))

prune1.3s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.1b

localize45.0ms

Local error

Found 4 expressions with local error:

0.7b
(/ (* x (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b))))) y)
0.3b
(* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b))))
0.2b
(fma t (log a) (- b))
0.1b
(* x (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b)))))

rewrite84.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

34.0ms
(* x (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b)))))
25.0ms
(/ (* x (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b))))) y)
24.0ms
(* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b))))

series1.2s

Counts
4 → 12
Calls

4 calls. Slowest were:

408.0ms
(/ (* x (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b))))) y)
362.0ms
(* x (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b)))))
324.0ms
(* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b))))
73.0ms
(fma t (log a) (- b))

simplify5.3s

Counts
32 → 75
Calls

32 calls. Slowest were:

473.0ms
(* (* (* (* (pow a (- 1.0)) (pow a (- 1.0))) (pow a (- 1.0))) (* (* (pow z y) (pow z y)) (pow z y))) (* (* (exp (fma t (log a) (- b))) (exp (fma t (log a) (- b)))) (exp (fma t (log a) (- b)))))
454.0ms
(+ (+ (* (log a) (- 1.0)) (* (log z) y)) (fma t (log a) (- b)))
389.0ms
(+ (+ (log (pow a (- 1.0))) (log (pow z y))) (fma t (log a) (- b)))

prune1.6s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize33.0ms

Local error

Found 4 expressions with local error:

0.7b
(/ (/ (* x (* (* 1 (pow z y)) (exp (fma t (log a) (- b))))) (pow a 1.0)) y)
0.2b
(* (* 1 (pow z y)) (exp (fma t (log a) (- b))))
0.2b
(fma t (log a) (- b))
0.0b
(exp (fma t (log a) (- b)))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 62
Calls

4 calls. Slowest were:

12.0ms
(/ (/ (* x (* (* 1 (pow z y)) (exp (fma t (log a) (- b))))) (pow a 1.0)) y)
9.0ms
(* (* 1 (pow z y)) (exp (fma t (log a) (- b))))
1.0ms
(exp (fma t (log a) (- b)))

series766.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

525.0ms
(/ (/ (* x (* (* 1 (pow z y)) (exp (fma t (log a) (- b))))) (pow a 1.0)) y)
125.0ms
(* (* 1 (pow z y)) (exp (fma t (log a) (- b))))
58.0ms
(exp (fma t (log a) (- b)))
58.0ms
(fma t (log a) (- b))

simplify3.1s

Counts
31 → 74
Calls

31 calls. Slowest were:

351.0ms
(* (exp (* -1 (* (log (/ 1 z)) y))) (exp (- (+ b (* t (log (/ 1 a)))))))
323.0ms
(exp (- (* t (log -1)) (+ (* t (log (/ -1 a))) b)))
313.0ms
(/ (* (exp (* -1 (* (log (/ 1 z)) y))) (* x (exp (- (+ b (* t (log (/ 1 a)))))))) (* a y))

prune1.5s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

regimes742.0ms

Accuracy

0% (1.9b remaining)

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

bsearch4.0ms