Average Error: 0.6 → 0.6
Time: 28.4s
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\log \left(e^{\frac{(\left(-5 \cdot v\right) \cdot v + 1)_*}{(v \cdot v + -1)_*}}\right)\right)\]

Error

Bits error versus v

Derivation

  1. Initial program 0.6

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]

  2. Simplified0.6

    \[\leadsto \color{blue}{\cos^{-1} \left(\frac{(\left(-5 \cdot v\right) \cdot v + 1)_*}{(v \cdot v + -1)_*}\right)}\]
  3. Using strategy rm

  4. Applied add-log-exp0.6

    \[\leadsto \cos^{-1} \color{blue}{\left(\log \left(e^{\frac{(\left(-5 \cdot v\right) \cdot v + 1)_*}{(v \cdot v + -1)_*}}\right)\right)}\]

  5. Final simplification0.6

    \[\leadsto \cos^{-1} \left(\log \left(e^{\frac{(\left(-5 \cdot v\right) \cdot v + 1)_*}{(v \cdot v + -1)_*}}\right)\right)\]

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))

Details

Time bar (total: 27.8s)Debug log

sample284.0ms

Algorithm
intervals

simplify632.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

632.0ms
(acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1)))

prune16.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize35.0ms

Local error

Found 2 expressions with local error:

0.3b
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))
0.0b
(/ (fma (* -5 v) v 1) (fma v v -1))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 39
Calls

2 calls. Slowest were:

6.0ms
(/ (fma (* -5 v) v 1) (fma v v -1))
1.0ms
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))

series40.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

33.0ms
(/ (fma (* -5 v) v 1) (fma v v -1))
7.0ms
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))

simplify3.1s

Counts
31 → 45
Calls

31 calls. Slowest were:

647.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))
613.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))
404.0ms
(/ (* (* (fma (* -5 v) v 1) (fma (* -5 v) v 1)) (fma (* -5 v) v 1)) (* (* (fma v v -1) (fma v v -1)) (fma v v -1)))

prune371.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.3b

localize15.0ms

Local error

Found 4 expressions with local error:

0.3b
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))
0.0b
(expm1 (log1p (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))
0.0b
(/ (fma (* -5 v) v 1) (fma v v -1))
0.0b
(log1p (acos (/ (fma (* -5 v) v 1) (fma v v -1))))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

6.0ms
(/ (fma (* -5 v) v 1) (fma v v -1))
1.0ms
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))
1.0ms
(log1p (acos (/ (fma (* -5 v) v 1) (fma v v -1))))

series346.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

196.0ms
(expm1 (log1p (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))
113.0ms
(log1p (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
32.0ms
(/ (fma (* -5 v) v 1) (fma v v -1))
5.0ms
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))

simplify3.7s

Counts
39 → 73
Calls

39 calls. Slowest were:

673.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))
539.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))
423.0ms
(/ (* (* (fma (* -5 v) v 1) (fma (* -5 v) v 1)) (fma (* -5 v) v 1)) (* (* (fma v v -1) (fma v v -1)) (fma v v -1)))

prune793.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.3b

localize11.0ms

Local error

Found 3 expressions with local error:

0.3b
(acos (log (exp (/ (fma (* -5 v) v 1) (fma v v -1)))))
0.0b
(log (exp (/ (fma (* -5 v) v 1) (fma v v -1))))
0.0b
(/ (fma (* -5 v) v 1) (fma v v -1))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
3 → 66
Calls

3 calls. Slowest were:

8.0ms
(log (exp (/ (fma (* -5 v) v 1) (fma v v -1))))
7.0ms
(/ (fma (* -5 v) v 1) (fma v v -1))
2.0ms
(acos (log (exp (/ (fma (* -5 v) v 1) (fma v v -1)))))

series98.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

50.0ms
(/ (fma (* -5 v) v 1) (fma v v -1))
44.0ms
(log (exp (/ (fma (* -5 v) v 1) (fma v v -1))))
5.0ms
(acos (log (exp (/ (fma (* -5 v) v 1) (fma v v -1)))))

simplify5.4s

Counts
51 → 75
Calls

51 calls. Slowest were:

589.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))
581.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))
569.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))

prune726.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.3b

localize15.0ms

Local error

Found 4 expressions with local error:

1.0b
(* (cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))
1.0b
(cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
1.0b
(cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
1.0b
(cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

5.0ms
(* (cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))
1.0ms
(cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
1.0ms
(cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))

series863.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

233.0ms
(cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
232.0ms
(* (cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))
220.0ms
(cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
177.0ms
(cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))

simplify2.9s

Counts
41 → 81
Calls

41 calls. Slowest were:

111.0ms
(* (cbrt (cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))) (cbrt (cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))))
109.0ms
(cbrt (* (cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))))
106.0ms
(cbrt (cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))

prune695.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.3b

regimes42.0ms

Accuracy

0% (0.0b remaining)

Error of 0.6b against oracle of 0.6b and baseline of 0.6b

bsearch1.0ms

end0.0ms

sample7.7s

Algorithm
intervals