Average Error: 0.6 → 0.6
Time: 32.2s
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\log \left((e^{\log_* (1 + \cos^{-1} \left(\frac{(\left(-5 \cdot v\right) \cdot v + 1)_*}{(v \cdot v + -1)_*}\right))} - 1)^*\right)}\]

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. Initial simplification0.6

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

  4. Applied add-exp-log0.6

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

  6. Applied expm1-log1p-u0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))

Details

Time bar (total: 18.8s)Debug log

start254.0ms

Algorithm
intervals

setup725.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.8b

localize18.0ms

Local error

Found 2 expressions with local error:

0.8b
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))
0.0b
(/ (fma (* -5 v) v 1) (fma v v -1))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
2 → 39
Calls

2 calls. Slowest were:

4.0ms
(/ (fma (* -5 v) v 1) (fma v v -1))
1.0ms
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))

series29.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

26.0ms
(/ (fma (* -5 v) v 1) (fma v v -1))
3.0ms
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))

simplify3.0s

Counts
31 → 45
Calls

31 calls. Slowest were:

673.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))
538.0ms
(- (+ (* 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)))

prune549.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.8b

localize14.0ms

Local error

Found 4 expressions with local error:

0.8b
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))
0.0b
(exp (log (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))
0.0b
(/ (fma (* -5 v) v 1) (fma v v -1))
0.0b
(log (acos (/ (fma (* -5 v) v 1) (fma v v -1))))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

6.0ms
(/ (fma (* -5 v) v 1) (fma v v -1))
4.0ms
(exp (log (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))
2.0ms
(log (acos (/ (fma (* -5 v) v 1) (fma v v -1))))

series166.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

110.0ms
(log (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
46.0ms
(/ (fma (* -5 v) v 1) (fma v v -1))
5.0ms
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))
5.0ms
(exp (log (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))

simplify4.4s

Counts
49 → 82
Calls

49 calls. Slowest were:

659.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))
609.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))
361.0ms
(/ (* (* (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)))

prune1.3s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.8b

localize1.0ms

Local error

Found 2 expressions with local error:

0.8b
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))
0.0b
(/ (fma (* -5 v) v 1) (fma v v -1))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
2 → 39
Calls

2 calls. Slowest were:

3.0ms
(/ (fma (* -5 v) v 1) (fma v v -1))
0.0ms
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))

series32.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

30.0ms
(/ (fma (* -5 v) v 1) (fma v v -1))
3.0ms
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))

simplify3.0s

Counts
31 → 45
Calls

31 calls. Slowest were:

622.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))
576.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))
361.0ms
(/ (* (* (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)))

prune591.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.8b

localize19.0ms

Local error

Found 4 expressions with local error:

0.8b
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))
0.8b
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))
0.0b
(* (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1))))) (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1))))))
0.0b
(exp (* (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1))))) (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

10.0ms
(* (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1))))) (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1))))))
8.0ms
(exp (* (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1))))) (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))))
1.0ms
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))

series125.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

111.0ms
(* (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1))))) (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1))))))
6.0ms
(exp (* (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1))))) (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))))
4.0ms
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))
4.0ms
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))

simplify3.2s

Counts
39 → 78
Calls

39 calls. Slowest were:

132.0ms
(* (* (* (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1))))) (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))) (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))) (* (* (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1))))) (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))) (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))))
113.0ms
(* (log (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (log (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))
112.0ms
(sqrt (sqrt (log (acos (/ (fma (* -5 v) v 1) (fma v v -1))))))

prune1.2s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.8b

regimes51.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms