Average Error: 0.5 → 0.5
Time: 27.6s
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\]

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  2. Using strategy rm

  3. Applied acos-asin0.5

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]

  4. Final simplification0.5

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\]

Reproduce

herbie shell --seed 2019010 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))

Details

Time bar (total: 26.9s)Debug log

sample227.0ms

Algorithm
intervals

simplify459.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
459.0ms
(acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1)))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.6b

localize19.0ms

Local error

Found 3 expressions with local error:

0.5b
(acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1)))
0.2b
(* 5 (* v v))
0.0b
(/ (- 1 (* 5 (* v v))) (- (* v v) 1))

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
12×add-cube-cbrt
12×times-frac
12×*-un-lft-identity
12×add-sqr-sqrt
5×add-exp-log
5×add-cbrt-cube
4×difference-of-sqr-1
4×associate-/r*
3×add-log-exp
3×associate-/l*
3×pow1
2×associate-/l/
2×flip--
2×associate-/r/
2×flip3--
1×div-inv
1×*-commutative
1×associate-*r*
1×div-exp
1×div-sub
1×frac-2neg
1×clear-num
1×cbrt-undiv
1×acos-asin
Counts
3 → 53
Calls
3 calls:
Slowest
4.0ms
(/ (- 1 (* 5 (* v v))) (- (* v v) 1))
1.0ms
(* 5 (* v v))
1.0ms
(acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1)))

series39.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
24.0ms
(/ (- 1 (* 5 (* v v))) (- (* v v) 1))
12.0ms
(* 5 (* v v))
4.0ms
(acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1)))

simplify5.0s

Counts
46 → 62
Calls
46 calls:
Slowest
731.0ms
(/ (* (* (- 1 (* 5 (* v v))) (- 1 (* 5 (* v v)))) (- 1 (* 5 (* v v)))) (* (* (- (* v v) 1) (- (* v v) 1)) (- (* v v) 1)))
497.0ms
(/ (sqrt (- 1 (* 5 (* v v)))) (cbrt (- (* v v) 1)))
410.0ms
(/ (- 1 (* 5 (* v v))) (- (* v v) 1))
206.0ms
(/ (- 1 (* 5 (* v v))) (cbrt (- (* v v) 1)))
201.0ms
(/ (- 1 (* 5 (* v v))) (sqrt (- (* v v) 1)))

prune454.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.5b

localize9.0ms

Local error

Found 4 expressions with local error:

0.6b
(asin (/ (- 1 (* 5 (* v v))) (- (* v v) 1)))
0.2b
(* 5 (* v v))
0.0b
(- (/ PI 2) (asin (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))
0.0b
(/ (- 1 (* 5 (* v v))) (- (* v v) 1))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
15×*-un-lft-identity
15×add-sqr-sqrt
13×add-cube-cbrt
12×times-frac
6×add-log-exp
6×add-exp-log
6×add-cbrt-cube
4×difference-of-sqr-1
4×associate-/r*
4×pow1
3×associate-/l*
3×flip--
3×flip3--
2×associate-/l/
2×associate-/r/
2×asin-acos
1×difference-of-squares
1×associate--r-
1×distribute-lft-out--
1×div-inv
1×*-commutative
1×associate-*r*
1×div-exp
1×div-sub
1×diff-log
1×frac-2neg
1×sub-neg
1×clear-num
1×cbrt-undiv
Counts
4 → 68
Calls
4 calls:
Slowest
4.0ms
(/ (- 1 (* 5 (* v v))) (- (* v v) 1))
4.0ms
(- (/ PI 2) (asin (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))
1.0ms
(asin (/ (- 1 (* 5 (* v v))) (- (* v v) 1)))
1.0ms
(* 5 (* v v))

series88.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
43.0ms
(- (/ PI 2) (asin (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))
23.0ms
(/ (- 1 (* 5 (* v v))) (- (* v v) 1))
15.0ms
(* 5 (* v v))
6.0ms
(asin (/ (- 1 (* 5 (* v v))) (- (* v v) 1)))

simplify7.9s

Counts
54 → 80
Calls
54 calls:
Slowest
729.0ms
(/ (* (* (- 1 (* 5 (* v v))) (- 1 (* 5 (* v v)))) (- 1 (* 5 (* v v)))) (* (* (- (* v v) 1) (- (* v v) 1)) (- (* v v) 1)))
539.0ms
(- (/ PI 2) (asin (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))
492.0ms
(+ (sqrt (/ PI 2)) (sqrt (asin (/ (- 1 (* 5 (* v v))) (- (* v v) 1)))))
488.0ms
(acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1)))
486.0ms
(- (sqrt (/ PI 2)) (sqrt (asin (/ (- 1 (* 5 (* v v))) (- (* v v) 1)))))

prune731.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.5b

localize14.0ms

Local error

Found 3 expressions with local error:

0.6b
(acos (+ -1 (* (+ (* v v) (pow v 4)) 4)))
0.0b
(* (+ (* v v) (pow v 4)) 4)
0.0b
(+ (* v v) (pow v 4))

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
5×add-log-exp
4×add-cube-cbrt
4×*-un-lft-identity
4×add-sqr-sqrt
3×add-exp-log
3×associate-*l*
3×add-cbrt-cube
3×pow1
2×flip-+
2×associate-*l/
2×flip3-+
1×*-commutative
1×sum-log
1×+-commutative
1×acos-asin
Counts
3 → 32
Calls
3 calls:
Slowest
5.0ms
(* (+ (* v v) (pow v 4)) 4)
2.0ms
(+ (* v v) (pow v 4))
1.0ms
(acos (+ -1 (* (+ (* v v) (pow v 4)) 4)))

series100.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
49.0ms
(+ (* v v) (pow v 4))
47.0ms
(* (+ (* v v) (pow v 4)) 4)
4.0ms
(acos (+ -1 (* (+ (* v v) (pow v 4)) 4)))

simplify933.0ms

Counts
14 → 41
Calls
14 calls:
Slowest
476.0ms
(* (- (* (* v v) (* v v)) (* (pow v 4) (pow v 4))) 4)
256.0ms
(* (+ (pow (* v v) 3) (pow (pow v 4) 3)) 4)
58.0ms
(acos (- (+ (* 4 (pow v 4)) (* 4 (pow v 2))) 1))
39.0ms
(acos (- (+ (* 4 (pow v 4)) (* 4 (pow v 2))) 1))
30.0ms
(acos (- (+ (* 4 (pow v 4)) (* 4 (pow v 2))) 1))

prune396.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.5b

localize7.0ms

Local error

Found 4 expressions with local error:

0.6b
(acos (+ -1 (* (+ (* v v) (pow v 4)) 4)))
0.0b
(exp (log (acos (+ -1 (* (+ (* v v) (pow v 4)) 4)))))
0.0b
(* (+ (* v v) (pow v 4)) 4)
0.0b
(log (acos (+ -1 (* (+ (* v v) (pow v 4)) 4))))

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×log-prod
6×pow1
5×add-exp-log
4×add-log-exp
4×add-cbrt-cube
4×exp-prod
3×exp-sum
3×associate-*l*
2×log-pow
2×associate-*l/
1×rem-exp-log
1×flip-+
1×*-commutative
1×rem-log-exp
1×flip3-+
1×acos-asin
Counts
4 → 48
Calls
4 calls:
Slowest
5.0ms
(* (+ (* v v) (pow v 4)) 4)
2.0ms
(exp (log (acos (+ -1 (* (+ (* v v) (pow v 4)) 4)))))
1.0ms
(log (acos (+ -1 (* (+ (* v v) (pow v 4)) 4))))
1.0ms
(acos (+ -1 (* (+ (* v v) (pow v 4)) 4)))

series197.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
129.0ms
(log (acos (+ -1 (* (+ (* v v) (pow v 4)) 4))))
59.0ms
(* (+ (* v v) (pow v 4)) 4)
4.0ms
(acos (+ -1 (* (+ (* v v) (pow v 4)) 4)))
4.0ms
(exp (log (acos (+ -1 (* (+ (* v v) (pow v 4)) 4)))))

simplify1.5s

Counts
28 → 60
Calls
28 calls:
Slowest
525.0ms
(* (- (* (* v v) (* v v)) (* (pow v 4) (pow v 4))) 4)
286.0ms
(* (+ (pow (* v v) 3) (pow (pow v 4) 3)) 4)
61.0ms
(acos (- (+ (* 4 (pow v 4)) (* 4 (pow v 2))) 1))
59.0ms
(acos (- (+ (* 4 (pow v 4)) (* 4 (pow v 2))) 1))
46.0ms
(log (acos (- (+ (* 4 (pow v 4)) (* 4 (pow v 2))) 1)))

prune609.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.5b

regimes145.0ms

Accuracy

0% (0.0b remaining)

Error of 0.5b against oracle of 0.5b and baseline of 0.5b

bsearch3.0ms

end0.0ms

sample8.1s

Algorithm
intervals