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

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 *-un-lft-identity0.0

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

  5. Applied sqrt-prod0.0

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

  6. Applied add-sqr-sqrt0.0

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

  7. Applied div-inv0.0

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

  8. Applied times-frac0.0

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

  9. Applied times-frac0.0

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

  10. Simplified0.0

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

  12. Applied add-cube-cbrt0.0

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

  13. Applied associate-/r*0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2019022 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))

Details

Time bar (total: 59.9s)Debug log

sample86.0ms

Algorithm
intervals
Results
36.0ms256×body80valid
32.0ms246×body80nan

simplify143.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
143.0ms
(/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v))))))

prune24.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize21.0ms

Local error

Found 3 expressions with local error:

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

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
192×times-frac
97×add-cube-cbrt
97×*-un-lft-identity
97×add-sqr-sqrt
57×sqrt-prod
30×div-inv
18×associate-/l*
16×associate-/r/
8×add-exp-log
8×add-cbrt-cube
7×flip--
7×flip3--
6×associate-/r*
3×add-log-exp
3×div-exp
3×cbrt-undiv
3×pow1
2×sqrt-div
2×frac-2neg
2×clear-num
1×associate-/l/
1×flip-+
1×*-commutative
1×associate-*l*
1×flip3-+
Counts
3 → 168
Calls
3 calls:
Slowest
11.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (+ 2 (* (* v v) -6))))
3.0ms
(* (* v v) -6)
0.0ms
(/ 4/3 PI)

series249.0ms

Counts
3 → 7
Calls
3 calls:
Slowest
231.0ms
(/ (/ (/ 4/3 PI) (- 1 (* v v))) (sqrt (+ 2 (* (* v v) -6))))
18.0ms
(* (* v v) -6)
0.0ms
(/ 4/3 PI)

simplify17.0s

Counts
253 → 175
Calls
253 calls:
Slowest
487.0ms
(/ (/ (* (cbrt (/ 4/3 PI)) (cbrt (/ 4/3 PI))) (* (cbrt (- 1 (* v v))) (cbrt (- 1 (* v v))))) 1)
335.0ms
(sqrt (+ (* 2 2) (- (* (* (* v v) -6) (* (* v v) -6)) (* 2 (* (* v v) -6)))))
335.0ms
(/ (/ (/ 4/3 PI) (- (pow 1 3) (pow (* v v) 3))) (sqrt (sqrt (+ 2 (* (* v v) -6)))))
297.0ms
(/ (/ (/ 4/3 PI) (- (* 1 1) (* (* v v) (* v v)))) (sqrt 1))
292.0ms
(/ (/ (/ 4/3 PI) (- (pow 1 3) (pow (* v v) 3))) (sqrt (* (cbrt (+ 2 (* (* v v) -6))) (cbrt (+ 2 (* (* v v) -6))))))

prune1.7s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize31.0ms

Local error

Found 4 expressions with local error:

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

rewrite41.0ms

Algorithm
rewrite-expression-head
Rules
672×times-frac
342×sqrt-prod
325×add-cube-cbrt
325×*-un-lft-identity
325×add-sqr-sqrt
186×associate-*r*
95×div-inv
34×associate-/r/
34×sqrt-div
30×associate-/l*
18×add-exp-log
18×add-cbrt-cube
15×flip--
15×flip3--
12×associate-/r*
7×associate-*l*
6×div-exp
6×cbrt-undiv
6×pow1
4×add-log-exp
3×cbrt-unprod
3×prod-exp
2×flip-+
2×*-commutative
2×flip3-+
2×frac-2neg
2×clear-num
1×associate-/l/
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×frac-times
Counts
4 → 470
Calls
4 calls:
Slowest
17.0ms
(* (/ 4/3 (sqrt (- 1 (* v v)))) (/ (/ (/ 1 PI) (sqrt (- 1 (* v v)))) (sqrt (+ 2 (* (* v v) -6)))))
7.0ms
(/ (/ (/ 1 PI) (sqrt (- 1 (* v v)))) (sqrt (+ 2 (* (* v v) -6))))
5.0ms
(* (* v v) -6)
1.0ms
(/ 4/3 (sqrt (- 1 (* v v))))

series456.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
230.0ms
(* (/ 4/3 (sqrt (- 1 (* v v)))) (/ (/ (/ 1 PI) (sqrt (- 1 (* v v)))) (sqrt (+ 2 (* (* v v) -6)))))
153.0ms
(/ (/ (/ 1 PI) (sqrt (- 1 (* v v)))) (sqrt (+ 2 (* (* v v) -6))))
48.0ms
(/ 4/3 (sqrt (- 1 (* v v))))
25.0ms
(* (* v v) -6)

simplify27.6s

Counts
623 → 482
Calls
623 calls:
Slowest
609.0ms
(+ (log (/ 4/3 (sqrt (- 1 (* v v))))) (- (log (/ (/ 1 PI) (sqrt (- 1 (* v v))))) (log (sqrt (+ 2 (* (* v v) -6))))))
547.0ms
(/ (* (* (/ (/ 1 PI) (sqrt (- 1 (* v v)))) (/ (/ 1 PI) (sqrt (- 1 (* v v))))) (/ (/ 1 PI) (sqrt (- 1 (* v v))))) (* (* (sqrt (+ 2 (* (* v v) -6))) (sqrt (+ 2 (* (* v v) -6)))) (sqrt (+ 2 (* (* v v) -6)))))
434.0ms
(* (/ 4/3 (sqrt (- 1 (* v v)))) (/ (/ (/ 1 PI) (sqrt (- 1 (* v v)))) (sqrt (+ 2 (* (* v v) -6)))))
332.0ms
(- (+ (* 1/6 (/ 1 (* (pow v 5) (pow (sqrt -1) 3)))) (+ (* 2/3 (/ 1 (* (pow v 5) (sqrt -1)))) (* 2/3 (/ 1 (* (pow v 3) (sqrt -1)))))) (* 4/3 (/ (sqrt -1) v)))
331.0ms
(sqrt (+ (* 2 2) (- (* (* (* v v) -6) (* (* v v) -6)) (* 2 (* (* v v) -6)))))

prune6.1s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize29.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (* v v) -6)
0.0b
(cbrt (sqrt (- 1 (* v v))))
0.0b
(cbrt (sqrt (- 1 (* v v))))
0.0b
(cbrt (sqrt (- 1 (* v v))))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
18×cbrt-prod
10×add-cube-cbrt
10×*-un-lft-identity
10×add-sqr-sqrt
9×sqrt-prod
6×sqrt-div
6×cbrt-div
4×add-log-exp
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×flip--
3×pow1/3
3×flip3--
1×*-commutative
1×associate-*l*
Counts
4 → 57
Calls
4 calls:
Slowest
7.0ms
(* (* v v) -6)
1.0ms
(cbrt (sqrt (- 1 (* v v))))
1.0ms
(cbrt (sqrt (- 1 (* v v))))
1.0ms
(cbrt (sqrt (- 1 (* v v))))

series359.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
122.0ms
(cbrt (sqrt (- 1 (* v v))))
114.0ms
(cbrt (sqrt (- 1 (* v v))))
111.0ms
(cbrt (sqrt (- 1 (* v v))))
12.0ms
(* (* v v) -6)

simplify1.9s

Counts
36 → 69
Calls
36 calls:
Slowest
272.0ms
(- (exp (* 1/6 (- (log -1) (* 2 (log (/ -1 v)))))) (+ (* 1/6 (/ (exp (* 1/6 (- (log -1) (* 2 (log (/ -1 v)))))) (pow v 2))) (* 5/72 (/ (exp (* 1/6 (- (log -1) (* 2 (log (/ -1 v)))))) (pow v 4)))))
216.0ms
(- (exp (* 1/6 (- (log -1) (* 2 (log (/ 1 v)))))) (+ (* 1/6 (/ (exp (* 1/6 (- (log -1) (* 2 (log (/ 1 v)))))) (pow v 2))) (* 5/72 (/ (exp (* 1/6 (- (log -1) (* 2 (log (/ 1 v)))))) (pow v 4)))))
209.0ms
(- (exp (* 1/6 (- (log -1) (* 2 (log (/ -1 v)))))) (+ (* 1/6 (/ (exp (* 1/6 (- (log -1) (* 2 (log (/ -1 v)))))) (pow v 2))) (* 5/72 (/ (exp (* 1/6 (- (log -1) (* 2 (log (/ -1 v)))))) (pow v 4)))))
203.0ms
(- (exp (* 1/6 (- (log -1) (* 2 (log (/ 1 v)))))) (+ (* 1/6 (/ (exp (* 1/6 (- (log -1) (* 2 (log (/ 1 v)))))) (pow v 2))) (* 5/72 (/ (exp (* 1/6 (- (log -1) (* 2 (log (/ 1 v)))))) (pow v 4)))))
201.0ms
(- (exp (* 1/6 (- (log -1) (* 2 (log (/ -1 v)))))) (+ (* 1/6 (/ (exp (* 1/6 (- (log -1) (* 2 (log (/ -1 v)))))) (pow v 2))) (* 5/72 (/ (exp (* 1/6 (- (log -1) (* 2 (log (/ -1 v)))))) (pow v 4)))))

prune1.1s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes57.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample2.9s

Algorithm
intervals
Results
1.5s8000×body80valid
1.1s7999×body80nan