Average Error: 1.0 → 0.0
Time: 54.7s
Precision: 64
Internal Precision: 128
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3}}{\pi}}{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*} \cdot \left(1 - v \cdot v\right)}\]

Error

Bits error versus v

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  2. Initial simplification0.0

    \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}\]
  3. Using strategy rm

  4. Applied associate-/l/0.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{4}{3}}{\pi}}{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*} \cdot \left(1 - v \cdot v\right)}}\]

  5. Final simplification0.0

    \[\leadsto \frac{\frac{\frac{4}{3}}{\pi}}{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*} \cdot \left(1 - v \cdot v\right)}\]

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))

Details

Time bar (total: 49.1s)Debug log

start166.0ms

Algorithm
intervals

setup170.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize34.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 4/3 PI)
0.0b
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (fma -6 (* v v) 2)))
0.0b
(/ (/ 4/3 PI) (- 1 (* v v)))
0.0b
(sqrt (fma -6 (* v v) 2))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 211
Calls

4 calls. Slowest were:

8.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
7.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (fma -6 (* v v) 2)))
1.0ms
(sqrt (fma -6 (* v v) 2))

series360.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

240.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (fma -6 (* v v) 2)))
77.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
42.0ms
(sqrt (fma -6 (* v v) 2))
0.0ms
(/ 4/3 PI)

simplify26.5s

Counts
291 → 221
Calls

291 calls. Slowest were:

840.0ms
(/ (/ (/ 4/3 PI) (- (* 1 1) (* (* v v) (* v v)))) (sqrt 1))
830.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
820.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))

prune3.4s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize10.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 4/3 PI)
0.0b
(sqrt (fma -6 (* v v) 2))
0.0b
(* (sqrt (fma -6 (* v v) 2)) (- 1 (* v v)))
0.0b
(/ (/ 4/3 PI) (* (sqrt (fma -6 (* v v) 2)) (- 1 (* v v))))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

11.0ms
(/ (/ 4/3 PI) (* (sqrt (fma -6 (* v v) 2)) (- 1 (* v v))))
9.0ms
(* (sqrt (fma -6 (* v v) 2)) (- 1 (* v v)))
2.0ms
(sqrt (fma -6 (* v v) 2))

series328.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

223.0ms
(/ (/ 4/3 PI) (* (sqrt (fma -6 (* v v) 2)) (- 1 (* v v))))
63.0ms
(* (sqrt (fma -6 (* v v) 2)) (- 1 (* v v)))
42.0ms
(sqrt (fma -6 (* v v) 2))
0.0ms
(/ 4/3 PI)

simplify4.1s

Counts
48 → 92
Calls

48 calls. Slowest were:

378.0ms
(- (sqrt 2) (+ (* 9/2 (/ (pow v 4) (pow (sqrt 2) 3))) (* 3 (/ (pow v 2) (sqrt 2)))))
376.0ms
(* (* (* (sqrt (fma -6 (* v v) 2)) (sqrt (fma -6 (* v v) 2))) (sqrt (fma -6 (* v v) 2))) (* (* (- 1 (* v v)) (- 1 (* v v))) (- 1 (* v v))))
341.0ms
(- (+ (/ 1 (* v (sqrt -6))) (+ (* v (sqrt -6)) (* 1/2 (/ 1 (* v (pow (sqrt -6) 3)))))) (+ (/ v (sqrt -6)) (* (pow v 3) (sqrt -6))))

prune1.3s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize26.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 4/3 PI)
1.0b
(log (/ 4/3 PI))
1.0b
(log (* (sqrt (fma -6 (* v v) 2)) (- 1 (* v v))))
1.0b
(exp (- (log (/ 4/3 PI)) (log (* (sqrt (fma -6 (* v v) 2)) (- 1 (* v v))))))

rewrite67.0ms

Algorithm
rewrite-expression-head
Counts
4 → 94
Calls

4 calls. Slowest were:

54.0ms
(exp (- (log (/ 4/3 PI)) (log (* (sqrt (fma -6 (* v v) 2)) (- 1 (* v v))))))
9.0ms
(log (* (sqrt (fma -6 (* v v) 2)) (- 1 (* v v))))
1.0ms
(log (/ 4/3 PI))

series591.0ms

Counts
4 → 8
Calls

4 calls. Slowest were:

433.0ms
(exp (- (log (/ 4/3 PI)) (log (* (sqrt (fma -6 (* v v) 2)) (- 1 (* v v))))))
158.0ms
(log (* (sqrt (fma -6 (* v v) 2)) (- 1 (* v v))))
0.0ms
(/ 4/3 PI)
0.0ms
(log (/ 4/3 PI))

simplify10.1s

Counts
60 → 102
Calls

60 calls. Slowest were:

544.0ms
(- (+ (log (* -1 (sqrt -6))) (/ 1 (* (pow v 2) (pow (sqrt -6) 2)))) (+ (* 3 (log (/ -1 v))) (+ (* 1/2 (/ 1 (pow v 4))) (+ (/ 1 (* (pow v 4) (pow (sqrt -6) 4))) (/ 1 (pow v 2))))))
497.0ms
(exp (fma (* (cbrt (log (/ 4/3 PI))) (cbrt (log (/ 4/3 PI)))) (cbrt (log (/ 4/3 PI))) (- (* (log (* (sqrt (fma -6 (* v v) 2)) (- 1 (* v v)))) 1))))
466.0ms
(- (+ (log (* -1 (sqrt -6))) (/ 1 (* (pow v 2) (pow (sqrt -6) 2)))) (+ (* 3 (log (/ 1 v))) (+ (* 1/2 (/ 1 (pow v 4))) (+ (/ 1 (* (pow v 4) (pow (sqrt -6) 4))) (/ 1 (pow v 2))))))

prune1.8s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes48.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms