Average Error: 0.6 → 0.6
Time: 26.2s
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\log_* (1 + (e^{\frac{(\left(-5 \cdot v\right) \cdot v + 1)_*}{(v \cdot v + -1)_*}} - 1)^*)\right)\]

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. 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 log1p-expm1-u0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))

Details

Time bar (total: 25.5s)Debug log

sample233.0ms

Algorithm
intervals

simplify533.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

533.0ms
(acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize32.0ms

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))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
2 → 39
Calls

2 calls. Slowest were:

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

series52.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

47.0ms
(/ (fma (* -5 v) v 1) (fma v v -1))
5.0ms
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))

simplify2.9s

Counts
31 → 45
Calls

31 calls. Slowest were:

521.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))
499.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))
402.0ms
(/ (* (* (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)))

prune459.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.3b

localize18.0ms

Local error

Found 3 expressions with local error:

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

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
3 → 50
Calls

3 calls. Slowest were:

7.0ms
(/ (fma (* -5 v) v 1) (fma v v -1))
1.0ms
(acos (log1p (expm1 (/ (fma (* -5 v) v 1) (fma v v -1)))))
0.0ms
(log1p (expm1 (/ (fma (* -5 v) v 1) (fma v v -1))))

series183.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

145.0ms
(log1p (expm1 (/ (fma (* -5 v) v 1) (fma v v -1))))
33.0ms
(/ (fma (* -5 v) v 1) (fma v v -1))
5.0ms
(acos (log1p (expm1 (/ (fma (* -5 v) v 1) (fma v v -1)))))

simplify4.0s

Counts
35 → 59
Calls

35 calls. Slowest were:

604.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))
554.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))
554.0ms
(- (+ (* 4 (/ 1 (pow v 4))) (+ 5 (* 4 (/ 1 (pow v 2))))))

prune696.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.3b

localize13.0ms

Local error

Found 4 expressions with local error:

1.0b
(* (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))
1.0b
(sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
1.0b
(sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
0.3b
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

10.0ms
(* (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))
2.0ms
(sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
2.0ms
(sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))

series42.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

20.0ms
(sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
14.0ms
(sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
5.0ms
(* (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))
2.0ms
(acos (/ (fma (* -5 v) v 1) (fma v v -1)))

simplify2.9s

Counts
42 → 82
Calls

42 calls. Slowest were:

101.0ms
(sqrt (* (cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (cbrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))))
100.0ms
(sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))
98.0ms
(sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))

prune921.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.3b

localize18.0ms

Local error

Found 4 expressions with local error:

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

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 75
Calls

4 calls. Slowest were:

23.0ms
(* (* (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))) (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))))
1.0ms
(sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
1.0ms
(sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))

series43.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

14.0ms
(sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
13.0ms
(sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
12.0ms
(sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))
4.0ms
(* (* (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))) (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))))

simplify3.3s

Counts
50 → 87
Calls

50 calls. Slowest were:

168.0ms
(* (* (* (* (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))) (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))) (* (* (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))) (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))) (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))))) (* (* (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))) (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))) (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))))
163.0ms
(* (* (* (* (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))) (* (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))))) (* (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))))) (* (* (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))) (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))) (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))))
111.0ms
(* (* (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))) (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1)))))) (sqrt (sqrt (acos (/ (fma (* -5 v) v 1) (fma v v -1))))))

prune937.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.3b

regimes41.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample8.1s

Algorithm
intervals