Average Error: 1.0 → 0.0
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\sqrt[3]{\frac{\frac{\frac{\frac{\frac{64}{27}}{\pi}}{\pi \cdot \pi}}{1 - v \cdot v}}{\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)}}}{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}\]

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. Simplified0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied add-cbrt-cube0.0

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

  6. Applied cbrt-undiv0.0

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

  7. Simplified0.0

    \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{\frac{\frac{\frac{\frac{64}{27}}{\pi}}{\pi \cdot \pi}}{1 - v \cdot v}}{\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)}}}}{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}\]

  8. Final simplification0.0

    \[\leadsto \frac{\sqrt[3]{\frac{\frac{\frac{\frac{\frac{64}{27}}{\pi}}{\pi \cdot \pi}}{1 - v \cdot v}}{\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)}}}{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}\]

Reproduce

herbie shell --seed 2019010 +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: 1.3m)Debug log

sample160.0ms

Algorithm
intervals

simplify131.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
131.0ms
(/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v))))))

prune19.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize21.0ms

Local error

Found 3 expressions with local error:

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

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
204×times-frac
106×add-cube-cbrt
106×*-un-lft-identity
106×add-sqr-sqrt
57×sqrt-prod
35×div-inv
22×associate-/l*
16×associate-/r/
10×add-exp-log
10×add-cbrt-cube
9×associate-/r*
8×flip--
8×flip3--
4×div-exp
4×cbrt-undiv
3×add-log-exp
3×log1p-expm1-u
3×frac-2neg
3×clear-num
3×pow1
3×expm1-log1p-u
2×associate-/l/
Counts
3 → 197
Calls
3 calls:
Slowest
8.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (fma -6 (* v v) 2)))
3.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
0.0ms
(/ 4/3 PI)

series296.0ms

Counts
3 → 7
Calls
3 calls:
Slowest
214.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (fma -6 (* v v) 2)))
82.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
0.0ms
(/ 4/3 PI)

simplify24.1s

Counts
284 → 204
Calls
284 calls:
Slowest
779.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
749.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
525.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (fma -6 (* v v) 2)))
463.0ms
(/ (/ (/ 4/3 PI) (- (pow 1 3) (pow (* v v) 3))) 1)
444.0ms
(/ (/ (/ 4/3 PI) (- (* 1 1) (* (* v v) (* v v)))) 1)

prune2.5s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize21.0ms

Local error

Found 4 expressions with local error:

1.0b
(cbrt (/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v)))))
0.0b
(* (- 1 (* v v)) (- 1 (* v v)))
0.0b
(/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v))))
0.0b
(/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v)))

rewrite56.0ms

Algorithm
rewrite-expression-head
Rules
208×times-frac
69×add-cube-cbrt
69×*-un-lft-identity
69×add-sqr-sqrt
42×cbrt-prod
38×associate-/l*
32×div-inv
32×add-exp-log
32×add-cbrt-cube
24×associate-/r/
22×flip--
22×flip3--
15×div-exp
15×cbrt-undiv
12×frac-times
9×pow1
6×associate-*r/
6×associate-*l/
4×add-log-exp
4×log1p-expm1-u
4×cbrt-unprod
4×prod-exp
4×associate-/r*
4×expm1-log1p-u
3×associate-*l*
3×associate-*r*
2×associate-/l/
2×distribute-lft-in
2×frac-2neg
2×sub-neg
2×clear-num
2×distribute-rgt-in
1×pow-prod-up
1×*-commutative
1×pow-prod-down
1×pow1/3
1×pow-plus
1×cbrt-div
1×pow2
Counts
4 → 235
Calls
4 calls:
Slowest
21.0ms
(/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v))))
18.0ms
(/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v)))
5.0ms
(* (- 1 (* v v)) (- 1 (* v v)))
5.0ms
(cbrt (/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v)))))

series458.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
195.0ms
(/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v))))
131.0ms
(/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v)))
108.0ms
(cbrt (/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v)))))
24.0ms
(* (- 1 (* v v)) (- 1 (* v v)))

simplify42.4s

Counts
258 → 247
Calls
258 calls:
Slowest
1.4s
(* (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
1.3s
(* (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
1.1s
(/ (/ (sqrt (/ (/ 64/27 PI) (* PI PI))) (- 1 (* v v))) (- 1 (* v v)))
1.0s
(/ (/ 1 (- 1 (* v v))) (- 1 (* v v)))
958.0ms
(/ (/ (cbrt (/ (/ 64/27 PI) (* PI PI))) (- 1 (* v v))) (- 1 (* v v)))

prune3.7s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes75.0ms

Accuracy

0% (0.0b remaining)

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

bsearch4.0ms

end0.0ms

sample3.2s

Algorithm
intervals