Average Error: 1.9 → 1.8
Time: 47.0s
Precision: 64
Internal Precision: 128
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{x}{\frac{y}{e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}}\]

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. Using strategy rm

  3. Applied associate-/l*1.8

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

  4. Final simplification1.8

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

Reproduce

herbie shell --seed 2019022 +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: 45.6s)Debug log

sample437.0ms

Algorithm
intervals
Results
249.0ms1488×body80nan
112.0ms493×body80valid

simplify347.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
347.0ms
(/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)

prune21.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.8b

localize38.0ms

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
4×log1p-expm1-u
4×expm1-log1p-u
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 → 65
Calls
4 calls:
Slowest
42.0ms
(exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
15.0ms
(* (- t 1.0) (log a))
12.0ms
(/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)
2.0ms
(* y (log z))

series480.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
217.0ms
(/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)
157.0ms
(exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
74.0ms
(* (- t 1.0) (log a))
31.0ms
(* y (log z))

simplify2.9s

Counts
34 → 77
Calls
34 calls:
Slowest
365.0ms
(- (+ (* t (log -1)) (* 1.0 (log (/ -1 a)))) (+ (* t (log (/ -1 a))) (* 1.0 (log -1))))
255.0ms
(+ (* t (* (log a) (exp (- (* 1.0 (log a)))))) (+ (exp (- (* 1.0 (log a)))) (* (log z) (* (exp (- (* 1.0 (log a)))) y))))
187.0ms
(exp (- (* 1.0 (log (/ 1 a))) (+ (* (log (/ 1 z)) y) (+ b (* t (log (/ 1 a)))))))
172.0ms
(- (* 1.0 (log (/ 1 a))) (* t (log (/ 1 a))))
171.0ms
(/ (* x (exp (- (* 1.0 (log (/ 1 a))) (+ (* (log (/ 1 z)) y) (+ b (* t (log (/ 1 a)))))))) y)

prune1.5s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize7.0ms

Local error

Found 4 expressions with local error:

0.9b
(exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
0.4b
(/ x (/ y (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))))
0.2b
(* y (log z))
0.1b
(* (- t 1.0) (log a))

rewrite45.0ms

Algorithm
rewrite-expression-head
Rules
9×add-cube-cbrt
9×*-un-lft-identity
9×add-sqr-sqrt
6×add-exp-log
6×associate-*r*
6×add-cbrt-cube
6×pow1
5×add-log-exp
5×associate-/r*
4×log1p-expm1-u
4×expm1-log1p-u
3×exp-sum
3×associate-*l*
3×exp-prod
2×div-inv
2×*-commutative
2×associate-/r/
2×associate-*l/
2×exp-diff
1×rem-exp-log
1×flip--
1×cbrt-unprod
1×associate--l+
1×prod-exp
1×pow-prod-down
1×flip3--
1×frac-2neg
1×sub-neg
1×clear-num
Counts
4 → 69
Calls
4 calls:
Slowest
32.0ms
(exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
5.0ms
(/ x (/ y (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))))
4.0ms
(* (- t 1.0) (log a))
2.0ms
(* y (log z))

series408.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
208.0ms
(/ x (/ y (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))))
125.0ms
(exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
45.0ms
(* (- t 1.0) (log a))
30.0ms
(* y (log z))

simplify3.9s

Counts
39 → 81
Calls
39 calls:
Slowest
304.0ms
(- (+ (* t (log -1)) (* 1.0 (log (/ -1 a)))) (+ (* t (log (/ -1 a))) (* 1.0 (log -1))))
290.0ms
(exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))
245.0ms
(+ (* t (* (log a) (exp (- (* 1.0 (log a)))))) (+ (exp (- (* 1.0 (log a)))) (* (log z) (* (exp (- (* 1.0 (log a)))) y))))
233.0ms
(/ 1 (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b)))
227.0ms
(* (* (* (- t 1.0) (- t 1.0)) (- t 1.0)) (* (* (log a) (log a)) (log a)))

prune1.2s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize53.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b))))
0.4b
(/ x (/ y (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b))))))
0.1b
(fma t (log a) (- b))
0.0b
(/ y (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b)))))

rewrite26.0ms

Algorithm
rewrite-expression-head
Rules
12×associate-*l/
9×add-exp-log
9×prod-exp
9×add-cbrt-cube
7×associate-/r*
6×add-cube-cbrt
6×*-un-lft-identity
6×pow1
6×add-sqr-sqrt
5×associate-/r/
4×add-log-exp
4×log1p-expm1-u
4×associate-*r*
4×pow-to-exp
4×expm1-log1p-u
3×pow-sub
3×div-inv
3×cbrt-unprod
3×pow-neg
3×neg-sub0
2×fma-udef
2×frac-2neg
2×clear-num
1×*-commutative
1×pow-prod-down
1×exp-sum
1×associate-*l*
Counts
4 → 69
Calls
4 calls:
Slowest
17.0ms
(* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b))))
4.0ms
(/ x (/ y (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b))))))
4.0ms
(/ y (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b)))))
0.0ms
(fma t (log a) (- b))

series958.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
360.0ms
(/ x (/ y (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b))))))
279.0ms
(/ y (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b)))))
263.0ms
(* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b))))
56.0ms
(fma t (log a) (- b))

simplify5.2s

Counts
37 → 81
Calls
37 calls:
Slowest
401.0ms
(* (* (* (* (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)))))
382.0ms
(+ (+ (* (log a) (- 1.0)) (* (log z) y)) (fma t (log a) (- b)))
380.0ms
(+ (+ (* (log a) (- 1.0)) (log (pow z y))) (fma t (log a) (- b)))
321.0ms
(+ (+ (log (pow a (- 1.0))) (log (pow z y))) (fma t (log a) (- b)))
317.0ms
(+ (+ (log (pow a (- 1.0))) (* (log z) y)) (fma t (log a) (- b)))

prune1.2s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

localize25.0ms

Local error

Found 4 expressions with local error:

0.9b
(cbrt (/ x (/ y (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b)))))))
0.9b
(cbrt (/ x (/ y (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b)))))))
0.9b
(cbrt (/ x (/ y (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b)))))))
0.5b
(* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b))))

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
15×cbrt-prod
9×add-exp-log
9×prod-exp
9×add-cbrt-cube
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×associate-*l/
4×associate-*r*
4×pow-to-exp
4×expm1-log1p-u
3×div-inv
3×cbrt-unprod
3×associate-/r/
3×pow1/3
3×cbrt-div
1×pow-sub
1×fma-udef
1×*-commutative
1×pow-prod-down
1×exp-sum
1×associate-*l*
1×pow-neg
1×neg-sub0
Counts
4 → 73
Calls
4 calls:
Slowest
17.0ms
(* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b))))
1.0ms
(cbrt (/ x (/ y (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b)))))))
1.0ms
(cbrt (/ x (/ y (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b)))))))
1.0ms
(cbrt (/ x (/ y (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b)))))))

series2.1s

Counts
4 → 12
Calls
4 calls:
Slowest
670.0ms
(cbrt (/ x (/ y (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b)))))))
560.0ms
(cbrt (/ x (/ y (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b)))))))
555.0ms
(cbrt (/ x (/ y (* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b)))))))
268.0ms
(* (* (pow a (- 1.0)) (pow z y)) (exp (fma t (log a) (- b))))

simplify6.1s

Counts
45 → 85
Calls
45 calls:
Slowest
438.0ms
(+ (+ (* (log a) (- 1.0)) (* (log z) y)) (fma t (log a) (- b)))
430.0ms
(+ (+ (* (log a) (- 1.0)) (log (pow z y))) (fma t (log a) (- b)))
350.0ms
(* (* (* (* (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)))))
328.0ms
(+ (+ (log (pow a (- 1.0))) (log (pow z y))) (fma t (log a) (- b)))
321.0ms
(+ (+ (log (pow a (- 1.0))) (* (log z) y)) (fma t (log a) (- b)))

prune2.0s

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0b

regimes969.0ms

Accuracy

0% (1.8b remaining)

Error of 1.8b against oracle of 0.0b and baseline of 1.8b

bsearch3.0ms

end0.0ms

sample15.7s

Algorithm
intervals
Results
9.0s45466×body80nan
4.5s15051×body80valid