Average Error: 1.0 → 0.0
Time: 1.0m
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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied add-cbrt-cube0.0

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

  6. Applied cbrt-undiv0.0

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

  7. Simplified0.0

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

  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 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))

Details

Time bar (total: 1.0m)Debug log

sample119.0ms

Algorithm
intervals

simplify89.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
89.0ms
(/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v))))))

prune15.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize31.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 4/3 PI)
0.2b
(* (* v v) -6)
0.0b
(/ (/ 4/3 PI) (- 1 (* v v)))
0.0b
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (+ 2 (* (* v v) -6))))

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
204×times-frac
107×add-cube-cbrt
107×*-un-lft-identity
107×add-sqr-sqrt
57×sqrt-prod
35×div-inv
22×associate-/l*
18×associate-/r/
11×add-exp-log
11×add-cbrt-cube
9×associate-/r*
8×flip--
8×flip3--
4×add-log-exp
4×div-exp
4×cbrt-undiv
4×pow1
3×frac-2neg
3×clear-num
2×associate-/l/
2×sqrt-div
1×flip-+
1×*-commutative
1×associate-*l*
1×flip3-+
Counts
4 → 202
Calls
4 calls:
Slowest
6.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (+ 2 (* (* v v) -6))))
3.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
2.0ms
(* (* v v) -6)
0.0ms
(/ 4/3 PI)

series297.0ms

Counts
4 → 10
Calls
4 calls:
Slowest
201.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (+ 2 (* (* v v) -6))))
79.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
17.0ms
(* (* v v) -6)
0.0ms
(/ 4/3 PI)

simplify20.2s

Counts
289 → 212
Calls
289 calls:
Slowest
588.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
582.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
410.0ms
(/ (/ (/ 4/3 PI) (- (pow 1 3) (pow (* v v) 3))) (sqrt (sqrt (+ 2 (* (* v v) -6)))))
389.0ms
(/ (/ (/ 4/3 PI) (- (pow 1 3) (pow (* v v) 3))) (sqrt (sqrt (+ 2 (* (* v v) -6)))))
361.0ms
(/ (/ (/ 4/3 PI) (- (pow 1 3) (pow (* v v) 3))) (sqrt (* (cbrt (+ 2 (* (* v v) -6))) (cbrt (+ 2 (* (* v v) -6))))))

prune2.2s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize44.0ms

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)))))
0.2b
(* (* v v) -6)
0.0b
(* (- 1 (* v v)) (- 1 (* v v)))
0.0b
(/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v))))

rewrite30.0ms

Algorithm
rewrite-expression-head
Rules
168×times-frac
52×add-cube-cbrt
52×*-un-lft-identity
52×add-sqr-sqrt
42×cbrt-prod
30×associate-/l*
27×add-exp-log
27×add-cbrt-cube
23×div-inv
22×associate-/r/
21×flip--
21×flip3--
12×div-exp
12×cbrt-undiv
12×frac-times
9×pow1
6×associate-*r/
6×associate-*l/
4×add-log-exp
4×cbrt-unprod
4×prod-exp
4×associate-*l*
3×associate-*r*
2×*-commutative
2×distribute-lft-in
2×sub-neg
2×distribute-rgt-in
1×associate-/l/
1×pow-prod-up
1×pow-prod-down
1×pow1/3
1×pow-plus
1×frac-2neg
1×associate-/r*
1×clear-num
1×cbrt-div
1×pow2
Counts
4 → 184
Calls
4 calls:
Slowest
11.0ms
(/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v))))
8.0ms
(cbrt (/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v)))))
4.0ms
(* (- 1 (* v v)) (- 1 (* v v)))
2.0ms
(* (* v v) -6)

series292.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
158.0ms
(/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v))))
103.0ms
(cbrt (/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v)))))
20.0ms
(* (- 1 (* v v)) (- 1 (* v v)))
11.0ms
(* (* v v) -6)

simplify31.0s

Counts
193 → 196
Calls
193 calls:
Slowest
1.2s
(* (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
940.0ms
(cbrt (sqrt (/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v))))))
736.0ms
(* (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
584.0ms
(* (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))) (+ 1 (* v v)))
571.0ms
(/ (/ (/ 64/27 PI) (* PI PI)) (- (pow 1 3) (pow (* v v) 3)))

prune2.8s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes71.0ms

Accuracy

0% (0.0b remaining)

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

bsearch4.0ms

end0.0ms

sample3.1s

Algorithm
intervals