Average Error: 0.5 → 0.1
Time: 4.1m
Precision: 64
Internal Precision: 128
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi}}{\sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_* \cdot 2}}}{\left(1 - v \cdot v\right) \cdot t}\]

Error

Bits error versus v

Bits error versus t

Derivation

  1. Initial program 0.5

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  2. Initial simplification0.3

    \[\leadsto \frac{\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi}}{\sqrt{2 \cdot (\left(v \cdot v\right) \cdot -3 + 1)_*} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)}\]
  3. Using strategy rm

  4. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi}}{\sqrt{2 \cdot (\left(v \cdot v\right) \cdot -3 + 1)_*}}}{t \cdot \left(1 - v \cdot v\right)}}\]

  5. Final simplification0.1

    \[\leadsto \frac{\frac{\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{\pi}}{\sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_* \cdot 2}}}{\left(1 - v \cdot v\right) \cdot t}\]

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))

Details

Time bar (total: 4.0m)Debug log

start228.0ms

Algorithm
intervals

setup154.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.3b

localize33.0ms

Local error

Found 4 expressions with local error:

0.6b
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))
0.4b
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))
0.0b
(sqrt (* 2 (fma (* v v) -3 1)))
0.0b
(* t (- 1 (* v v)))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 86
Calls

4 calls. Slowest were:

17.0ms
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))
7.0ms
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))
1.0ms
(* t (- 1 (* v v)))

series647.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

346.0ms
(/ (/ (fma (* v v) -5 1) PI) (* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v)))))
198.0ms
(* (sqrt (* 2 (fma (* v v) -3 1))) (* t (- 1 (* v v))))
70.0ms
(sqrt (* 2 (fma (* v v) -3 1)))
33.0ms
(* t (- 1 (* v v)))

simplify6.8s

Counts
58 → 98
Calls

58 calls. Slowest were:

471.0ms
(- (* t (sqrt 2)) (* 5/2 (* t (* (sqrt 2) (pow v 2)))))
399.0ms
(- (/ 1 (* t (* (sqrt 2) PI))) (+ (* 53/8 (/ (pow v 4) (* t (* (sqrt 2) PI)))) (* 5/2 (/ (pow v 2) (* t (* (sqrt 2) PI))))))
392.0ms
(- (log (/ (fma (* v v) -5 1) PI)) (+ (log (sqrt (* 2 (fma (* v v) -3 1)))) (log (* t (- 1 (* v v))))))

prune1.7s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0b

localize11.0ms

Local error

Found 4 expressions with local error:

0.1b
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (* t (- 1 (* v v))))
0.0b
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))
0.0b
(sqrt (* 2 (fma (* v v) -3 1)))
0.0b
(* t (- 1 (* v v)))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
4 → 129
Calls

4 calls. Slowest were:

16.0ms
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (* t (- 1 (* v v))))
9.0ms
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))
3.0ms
(* t (- 1 (* v v)))

series981.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

716.0ms
(/ (/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1)))) (* t (- 1 (* v v))))
184.0ms
(/ (/ (fma (* v v) -5 1) PI) (sqrt (* 2 (fma (* v v) -3 1))))
46.0ms
(sqrt (* 2 (fma (* v v) -3 1)))
35.0ms
(* t (- 1 (* v v)))

simplify10.0s

Counts
131 → 141
Calls

131 calls. Slowest were:

427.0ms
(- (/ 1 (* t (* (sqrt 2) PI))) (+ (* 53/8 (/ (pow v 4) (* t (* (sqrt 2) PI)))) (* 5/2 (/ (pow v 2) (* t (* (sqrt 2) PI))))))
310.0ms
(- (+ (* (sqrt 2) (* v (sqrt -3))) (* 1/2 (/ (sqrt 2) (* v (sqrt -3))))) (* 1/8 (/ (sqrt 2) (* (pow v 3) (pow (sqrt -3) 3)))))
275.0ms
(+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v))))

prune2.6s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize72.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ (fma (* v v) -5 1) (sqrt 2))
0.6b
(/ (/ (fma (* v v) -5 1) (sqrt 2)) (/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (fma (* v v) -3 1)))))
0.4b
(/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (fma (* v v) -3 1))))
0.0b
(/ (/ 1 PI) (sqrt (fma (* v v) -3 1)))

rewrite86.0ms

Algorithm
rewrite-expression-head
Counts
4 → 655
Calls

4 calls. Slowest were:

19.0ms
(/ (/ (fma (* v v) -5 1) (sqrt 2)) (/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (fma (* v v) -3 1)))))
13.0ms
(/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (fma (* v v) -3 1))))
7.0ms
(/ (/ 1 PI) (sqrt (fma (* v v) -3 1)))

series686.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

360.0ms
(/ (/ (fma (* v v) -5 1) (sqrt 2)) (/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (fma (* v v) -3 1)))))
157.0ms
(/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (fma (* v v) -3 1))))
93.0ms
(/ (fma (* v v) -5 1) (sqrt 2))
76.0ms
(/ (/ 1 PI) (sqrt (fma (* v v) -3 1)))

simplify2.8m

Counts
1103 → 667
Calls

1103 calls. Slowest were:

738.0ms
(/ (* (cbrt (/ (fma (* v v) -5 1) (sqrt 2))) (cbrt (/ (fma (* v v) -5 1) (sqrt 2)))) (/ t (/ 1 (* (cbrt (sqrt (fma (* v v) -3 1))) (cbrt (sqrt (fma (* v v) -3 1)))))))
639.0ms
(/ (/ (cbrt (fma (* v v) -5 1)) (cbrt (sqrt 2))) (/ (- 1 (* v v)) (/ (sqrt (/ 1 PI)) (sqrt (fma (* v v) -3 1)))))
635.0ms
(/ (/ (* (cbrt (fma (* v v) -5 1)) (cbrt (fma (* v v) -5 1))) (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (/ t (/ 1 (sqrt 1))))

prune9.2s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.6b
(/ (fma v (* v -5) 1) (/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (* 2 (fma (* v v) -3 1))))))
0.1b
(/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (* 2 (fma (* v v) -3 1)))))
0.0b
(/ (/ 1 PI) (sqrt (* 2 (fma (* v v) -3 1))))
0.0b
(sqrt (* 2 (fma (* v v) -3 1)))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 232
Calls

4 calls. Slowest were:

10.0ms
(/ (fma v (* v -5) 1) (/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (* 2 (fma (* v v) -3 1))))))
6.0ms
(/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (* 2 (fma (* v v) -3 1)))))
3.0ms
(/ (/ 1 PI) (sqrt (* 2 (fma (* v v) -3 1))))

series559.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

242.0ms
(/ (fma v (* v -5) 1) (/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (* 2 (fma (* v v) -3 1))))))
161.0ms
(/ (* t (- 1 (* v v))) (/ (/ 1 PI) (sqrt (* 2 (fma (* v v) -3 1)))))
113.0ms
(/ (/ 1 PI) (sqrt (* 2 (fma (* v v) -3 1))))
43.0ms
(sqrt (* 2 (fma (* v v) -3 1)))

simplify35.2s

Counts
304 → 244
Calls

304 calls. Slowest were:

569.0ms
(/ (* (* (/ 1 PI) (/ 1 PI)) (/ 1 PI)) (* (* (sqrt (* 2 (fma (* v v) -3 1))) (sqrt (* 2 (fma (* v v) -3 1)))) (sqrt (* 2 (fma (* v v) -3 1)))))
562.0ms
(/ (sqrt (fma v (* v -5) 1)) (/ (- 1 (* v v)) (/ (cbrt (/ 1 PI)) (sqrt (* 2 (fma (* v v) -3 1))))))
451.0ms
(/ (- 1 (* v v)) (/ (/ 1 PI) (cbrt (sqrt (* 2 (fma (* v v) -3 1))))))

prune3.0s

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

regimes117.0ms

Accuracy

0% (0.1b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.1b

bsearch2.0ms