Average Error: 1.0 → 0.0
Time: 4.2m
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 2019016 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))

Details

Time bar (total: 4.2m)Debug log

sample170.0ms

Algorithm
intervals
Results
524×(pre true 80)
268×(body nan 80)
256×(body real 80)

simplify145.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
145.0ms
(/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v))))))

prune13.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize25.0ms

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))))

rewrite25.0ms

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
13.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (+ 2 (* (* v v) -6))))
4.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
2.0ms
(* (* v v) -6)
0.0ms
(/ 4/3 PI)

series337.0ms

Counts
4 → 10
Calls
4 calls:
Slowest
238.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (+ 2 (* (* v v) -6))))
84.0ms
(/ (/ 4/3 PI) (- 1 (* v v)))
14.0ms
(* (* v v) -6)
0.0ms
(/ 4/3 PI)

simplify20.2s

Counts
289 → 212
Calls
289 calls:
Slowest
566.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
497.0ms
(- (+ (* 4/3 (/ 1 (* (pow v 2) PI))) (+ (* 4/3 (/ 1 (* (pow v 4) PI))) (* 4/3 (/ 1 (* (pow v 6) PI))))))
455.0ms
(/ (/ (/ 4/3 PI) (- (pow 1 3) (pow (* v v) 3))) (sqrt (sqrt (+ 2 (* (* v v) -6)))))
347.0ms
(/ (/ (/ 4/3 PI) (- (pow 1 3) (pow (* v v) 3))) (sqrt (* (cbrt (+ 2 (* (* v v) -6))) (cbrt (+ 2 (* (* v v) -6))))))
334.0ms
(/ (/ (/ 4/3 PI) (- (* 1 1) (* (* v v) (* v v)))) (* (cbrt (sqrt (+ 2 (* (* v v) -6)))) (cbrt (sqrt (+ 2 (* (* v v) -6))))))

prune2.1s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize39.0ms

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
(/ (cbrt (/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v))))) (sqrt (+ 2 (* (* v v) -6))))

rewrite59.0ms

Algorithm
rewrite-expression-head
Rules
798×times-frac
336×cbrt-prod
233×add-cube-cbrt
233×*-un-lft-identity
233×add-sqr-sqrt
138×sqrt-prod
82×associate-/r/
65×div-inv
62×flip--
62×flip3--
45×associate-/l*
36×frac-times
18×associate-*r/
18×associate-*l/
9×pow1
8×add-exp-log
7×add-cbrt-cube
6×associate-/r*
4×add-log-exp
4×associate-*l*
3×associate-*r*
2×*-commutative
2×sqrt-div
2×distribute-lft-in
2×sub-neg
2×cbrt-div
2×distribute-rgt-in
1×associate-/l/
1×flip-+
1×cbrt-unprod
1×pow-prod-up
1×prod-exp
1×pow-prod-down
1×div-exp
1×pow1/3
1×pow-plus
1×flip3-+
1×frac-2neg
1×clear-num
1×cbrt-undiv
1×pow2
Counts
4 → 428
Calls
4 calls:
Slowest
26.0ms
(/ (cbrt (/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v))))) (sqrt (+ 2 (* (* v v) -6))))
8.0ms
(cbrt (/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v)))))
8.0ms
(* (- 1 (* v v)) (- 1 (* v v)))
4.0ms
(* (* v v) -6)

series346.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
201.0ms
(/ (cbrt (/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v))))) (sqrt (+ 2 (* (* v v) -6))))
105.0ms
(cbrt (/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v)))))
21.0ms
(* (- 1 (* v v)) (- 1 (* v v)))
18.0ms
(* (* v v) -6)

simplify2.0m

Counts
677 → 440
Calls
677 calls:
Slowest
959.0ms
(/ (cbrt (sqrt (/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v)))))) (sqrt (+ 2 (* (* v v) -6))))
780.0ms
(* (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
685.0ms
(/ (cbrt (sqrt (/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v)))))) (sqrt (sqrt (+ 2 (* (* v v) -6)))))
596.0ms
(/ (cbrt (sqrt (/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v)))))) (sqrt (sqrt (+ 2 (* (* v v) -6)))))
592.0ms
(/ (cbrt (cbrt (/ (/ (/ (/ 64/27 PI) (* PI PI)) (- 1 (* v v))) (* (- 1 (* v v)) (- 1 (* v v)))))) (sqrt (sqrt (+ 2 (* (* v v) -6)))))

prune4.4s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

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

rewrite28.0ms

Algorithm
rewrite-expression-head
Rules
1092×times-frac
466×add-cube-cbrt
466×*-un-lft-identity
466×add-sqr-sqrt
183×sqrt-prod
122×div-inv
91×associate-/r/
27×associate-/r*
25×flip--
25×flip3--
24×associate-/l*
21×flip-+
21×flip3-+
18×add-exp-log
18×add-cbrt-cube
9×div-exp
9×cbrt-undiv
4×add-log-exp
4×pow1
3×associate-/l/
3×frac-2neg
3×clear-num
2×sqrt-div
1×*-commutative
1×associate-*l*
Counts
4 → 556
Calls
4 calls:
Slowest
8.0ms
(/ (/ (/ 4/3 PI) (- 1 (pow v 4))) (/ (sqrt (+ 2 (* (* v v) -6))) (+ 1 (* v v))))
4.0ms
(/ (sqrt (+ 2 (* (* v v) -6))) (+ 1 (* v v)))
2.0ms
(* (* v v) -6)
0.0ms
(/ 4/3 PI)

series232.0ms

Counts
4 → 10
Calls
4 calls:
Slowest
162.0ms
(/ (/ (/ 4/3 PI) (- 1 (pow v 4))) (/ (sqrt (+ 2 (* (* v v) -6))) (+ 1 (* v v))))
53.0ms
(/ (sqrt (+ 2 (* (* v v) -6))) (+ 1 (* v v)))
17.0ms
(* (* v v) -6)
0.0ms
(/ 4/3 PI)

simplify1.6m

Counts
975 → 566
Calls
975 calls:
Slowest
559.0ms
(/ (/ 4/3 (sqrt (- 1 (pow v 4)))) (/ (sqrt 1) (* (cbrt (+ 1 (* v v))) (cbrt (+ 1 (* v v))))))
498.0ms
(/ (/ (* (cbrt (/ 4/3 PI)) (cbrt (/ 4/3 PI))) (sqrt (- 1 (pow v 4)))) (/ 1 (* (cbrt (+ 1 (* v v))) (cbrt (+ 1 (* v v))))))
405.0ms
(/ (/ (/ 4/3 PI) (- (pow 1 3) (pow (pow v 4) 3))) (/ (sqrt (* (cbrt (+ 2 (* (* v v) -6))) (cbrt (+ 2 (* (* v v) -6))))) (sqrt (+ 1 (* v v)))))
400.0ms
(/ (/ (* (cbrt (/ 4/3 PI)) (cbrt (/ 4/3 PI))) (sqrt (- 1 (pow v 4)))) (/ (sqrt (* (cbrt (+ 2 (* (* v v) -6))) (cbrt (+ 2 (* (* v v) -6))))) 1))
390.0ms
(/ (/ 1 (* (cbrt (- 1 (pow v 4))) (cbrt (- 1 (pow v 4))))) (/ (sqrt (+ 2 (* (* v v) -6))) (+ (pow 1 3) (pow (* v v) 3))))

prune5.4s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes37.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample2.0s

Algorithm
intervals
Results
16013×(pre true 80)
8012×(body nan 80)
8001×(body real 80)