Average Error: 0.2 → 0.2
Time: 6.2s
Precision: 64
Internal Precision: 128
\[\frac{a}{-\cos^{-1} a}\]
↓
\[\frac{a}{-\cos^{-1} a}\]

Error

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\frac{a}{-\cos^{-1} a}\]

  2. Final simplification0.2

    \[\leadsto \frac{a}{-\cos^{-1} a}\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (a)
  :name "Fuzzer 001"
  (/ a (- (acos a))))

Details

Time bar (total: 5.6s)Debug log

sample64.0ms

Algorithm
intervals
Results
524×(pre true 80)
268×(body nan 80)
256×(body real 80)

simplify2.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
2.0ms
(/ a (- (acos a)))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize16.0ms

Local error

Found 1 expressions with local error:

0.2b
(/ a (- (acos a)))

rewrite2.0ms

Algorithm
rewrite-expression-head
Rules
10×associate-/r*
4×add-cube-cbrt
4×*-un-lft-identity
4×add-sqr-sqrt
3×distribute-rgt-neg-in
3×distribute-lft-neg-in
1×add-log-exp
1×div-inv
1×log1p-expm1-u
1×add-exp-log
1×add-cbrt-cube
1×frac-2neg
1×neg-mul-1
1×clear-num
1×pow1
1×expm1-log1p-u
Counts
1 → 22
Calls
1 calls:
Slowest
2.0ms
(/ a (- (acos a)))

series24.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
24.0ms
(/ a (- (acos a)))

simplify68.0ms

Counts
13 → 25
Calls
13 calls:
Slowest
17.0ms
(* -1 (/ a (acos a)))
15.0ms
(* -1 (/ a (acos a)))
12.0ms
(- (acos a))
11.0ms
(* -1 (/ a (acos a)))
1.0ms
(cbrt (- (acos a)))

prune196.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.1b

localize5.0ms

Local error

Found 2 expressions with local error:

0.3b
(/ (- (acos a)) a)
0.2b
(/ 1 (/ (- (acos a)) a))

rewrite5.0ms

Algorithm
rewrite-expression-head
Rules
10×associate-/l*
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
4×associate-/r*
3×div-inv
3×add-exp-log
3×distribute-rgt-neg-in
3×pow1
3×distribute-lft-neg-in
2×add-log-exp
2×log1p-expm1-u
2×add-cbrt-cube
2×frac-2neg
2×clear-num
2×expm1-log1p-u
1×inv-pow
1×pow-flip
1×associate-/r/
1×distribute-frac-neg
1×div-sub
1×neg-mul-1
1×rec-exp
1×neg-sub0
Counts
2 → 44
Calls
2 calls:
Slowest
4.0ms
(/ (- (acos a)) a)
1.0ms
(/ 1 (/ (- (acos a)) a))

series32.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
17.0ms
(/ (- (acos a)) a)
15.0ms
(/ 1 (/ (- (acos a)) a))

simplify103.0ms

Counts
20 → 50
Calls
20 calls:
Slowest
17.0ms
(* -1 (/ a (acos a)))
17.0ms
(* -1 (/ a (acos a)))
16.0ms
(* -1 (/ a (acos a)))
15.0ms
(* -1 (/ (acos a) a))
8.0ms
(* -1 (/ (acos a) a))

prune462.0ms

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 2 expressions with local error:

0.4b
(* (- (acos a)) (/ 1 a))
0.2b
(/ 1 (* (- (acos a)) (/ 1 a)))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
10×associate-*l*
7×add-exp-log
7×pow1
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
4×associate-*r*
4×add-cbrt-cube
3×distribute-rgt-neg-in
3×distribute-lft-neg-in
2×add-log-exp
2×pow-flip
2×div-inv
2×log1p-expm1-u
2×un-div-inv
2×associate-/r/
2×associate-*r/
2×prod-exp
2×pow-prod-down
2×rec-exp
2×expm1-log1p-u
1×inv-pow
1×cbrt-unprod
1×*-commutative
1×frac-2neg
1×neg-mul-1
1×associate-/r*
1×clear-num
1×distribute-lft-neg-out
Counts
2 → 50
Calls
2 calls:
Slowest
9.0ms
(* (- (acos a)) (/ 1 a))
7.0ms
(/ 1 (* (- (acos a)) (/ 1 a)))

series55.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
28.0ms
(* (- (acos a)) (/ 1 a))
27.0ms
(/ 1 (* (- (acos a)) (/ 1 a)))

simplify566.0ms

Counts
24 → 56
Calls
24 calls:
Slowest
338.0ms
(* (* (* (- (acos a)) (- (acos a))) (- (acos a))) (* (* (/ 1 a) (/ 1 a)) (/ 1 a)))
52.0ms
(* (- (acos a)) (/ 1 a))
21.0ms
(+ (log (- (acos a))) (log (/ 1 a)))
20.0ms
(* -1 (/ (acos a) a))
16.0ms
(* -1 (/ a (acos a)))

prune373.0ms

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 2 expressions with local error:

0.4b
(/ (/ 1 (- (acos a))) (/ 1 a))
0.0b
(/ 1 (- (acos a)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
16×times-frac
15×add-cube-cbrt
15×*-un-lft-identity
15×add-sqr-sqrt
14×associate-/r*
12×div-inv
7×add-exp-log
4×associate-/l*
4×add-cbrt-cube
3×distribute-rgt-neg-in
3×pow1
3×distribute-lft-neg-in
2×add-log-exp
2×log1p-expm1-u
2×div-exp
2×frac-2neg
2×clear-num
2×rec-exp
2×expm1-log1p-u
1×associate-/l/
1×inv-pow
1×pow-flip
1×associate-/r/
1×neg-mul-1
1×cbrt-undiv
Counts
2 → 66
Calls
2 calls:
Slowest
4.0ms
(/ (/ 1 (- (acos a))) (/ 1 a))
1.0ms
(/ 1 (- (acos a)))

series27.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
16.0ms
(/ (/ 1 (- (acos a))) (/ 1 a))
11.0ms
(/ 1 (- (acos a)))

simplify1.6s

Counts
58 → 72
Calls
58 calls:
Slowest
940.0ms
(/ (* (* (/ 1 (- (acos a))) (/ 1 (- (acos a)))) (/ 1 (- (acos a)))) (* (* (/ 1 a) (/ 1 a)) (/ 1 a)))
80.0ms
(/ (/ 1 (- (acos a))) (/ 1 a))
77.0ms
(/ (/ 1 (- (acos a))) (/ 1 a))
66.0ms
(/ (/ 1 (- (acos a))) (/ 1 a))
55.0ms
(- (log (/ 1 (- (acos a)))) (log (/ 1 a)))

prune472.0ms

Pruning

8 alts after pruning (5 fresh and 3 done)

Merged error: 0b

regimes102.0ms

Accuracy

0% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch0.0ms

end0.0ms

sample1.3s

Algorithm
intervals
Results
15997×(pre true 80)
8001×(body real 80)
7996×(body nan 80)