Average Error: 0.0 → 0.0
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \left(\sqrt{1 - \log \left(e^{\left(v \cdot v\right) \cdot 3}\right)} \cdot \frac{\sqrt{2}}{4}\right)\]

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
  2. Using strategy rm

  3. Applied add-log-exp0.0

    \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \color{blue}{\log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}}\right) \cdot \left(1 - v \cdot v\right)\]

  4. Final simplification0.0

    \[\leadsto \left(1 - v \cdot v\right) \cdot \left(\sqrt{1 - \log \left(e^{\left(v \cdot v\right) \cdot 3}\right)} \cdot \frac{\sqrt{2}}{4}\right)\]

Reproduce

herbie shell --seed 2019016 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))

Details

Time bar (total: 1.2m)Debug log

sample131.0ms

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

simplify586.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
586.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

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

rewrite44.0ms

Algorithm
rewrite-expression-head
Rules
18×sqrt-div
18×frac-times
15×flip--
15×flip3--
11×add-exp-log
11×add-cbrt-cube
11×pow1
10×associate-*r/
10×associate-*r*
10×add-sqr-sqrt
9×add-cube-cbrt
9×associate-*l/
9×*-un-lft-identity
6×sqrt-prod
5×associate-*l*
4×add-log-exp
4×cbrt-unprod
4×prod-exp
4×pow-prod-down
3×*-commutative
2×distribute-lft-in
2×sub-neg
2×distribute-rgt-in
1×div-inv
1×pow1/2
1×rem-sqrt-square
Counts
4 → 88
Calls
4 calls:
Slowest
25.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
15.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
1.0ms
(sqrt (- 1 (* 3 (* v v))))
1.0ms
(* 3 (* v v))

series213.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
97.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
62.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
35.0ms
(sqrt (- 1 (* 3 (* v v))))
19.0ms
(* 3 (* v v))

simplify13.3s

Counts
78 → 100
Calls
78 calls:
Slowest
838.0ms
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (sqrt (- 1 (* 3 (* v v)))) (sqrt (- 1 (* 3 (* v v))))) (sqrt (- 1 (* 3 (* v v))))))
487.0ms
(* (- (* v v)) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))))
466.0ms
(* (* (sqrt 2) (sqrt (- (pow 1 3) (pow (* 3 (* v v)) 3)))) (- (pow 1 3) (pow (* v v) 3)))
440.0ms
(* (* 4 (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v))))))) (+ 1 (* v v)))
437.0ms
(* (* (sqrt 2) (sqrt (- 1 (* 3 (* v v))))) (- (* 1 1) (* (* v v) (* v v))))

prune1.0s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize20.0ms

Local error

Found 4 expressions with local error:

2.1b
(log (exp (* 3 (* v v))))
0.2b
(* 3 (* v v))
0.0b
(* (/ (sqrt 2) 4) (sqrt (- 1 (log (exp (* 3 (* v v)))))))
0.0b
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (log (exp (* 3 (* v v))))))) (- 1 (* v v)))

rewrite32.0ms

Algorithm
rewrite-expression-head
Rules
18×frac-times
16×sqrt-div
14×flip--
14×flip3--
12×pow1
11×add-exp-log
11×add-cbrt-cube
10×associate-*r/
10×associate-*r*
9×add-cube-cbrt
9×associate-*l/
9×*-un-lft-identity
9×add-sqr-sqrt
5×associate-*l*
4×add-log-exp
4×cbrt-unprod
4×prod-exp
4×pow-prod-down
3×sqrt-prod
3×*-commutative
3×log-prod
2×log-pow
2×distribute-lft-in
2×sub-neg
2×distribute-rgt-in
1×div-inv
1×rem-log-exp
1×exp-prod
Counts
4 → 87
Calls
4 calls:
Slowest
21.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (log (exp (* 3 (* v v))))))) (- 1 (* v v)))
6.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (log (exp (* 3 (* v v)))))))
2.0ms
(log (exp (* 3 (* v v))))
1.0ms
(* 3 (* v v))

series233.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
115.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (log (exp (* 3 (* v v))))))) (- 1 (* v v)))
79.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (log (exp (* 3 (* v v)))))))
19.0ms
(log (exp (* 3 (* v v))))
19.0ms
(* 3 (* v v))

simplify14.0s

Counts
76 → 99
Calls
76 calls:
Slowest
1.1s
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (sqrt (- 1 (log (exp (* 3 (* v v)))))) (sqrt (- 1 (log (exp (* 3 (* v v))))))) (sqrt (- 1 (log (exp (* 3 (* v v))))))))
759.0ms
(* (- (* v v)) (* (/ (sqrt 2) 4) (sqrt (- 1 (log (exp (* 3 (* v v))))))))
476.0ms
(* (* 4 (sqrt (+ 1 (log (exp (* 3 (* v v))))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
475.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- (pow 1 3) (pow (log (exp (* 3 (* v v)))) 3)))) (- (* 1 1) (* (* v v) (* v v))))
452.0ms
(* (* (sqrt 2) (sqrt (- (pow 1 3) (pow (log (exp (* 3 (* v v)))) 3)))) (- (pow 1 3) (pow (* v v) 3)))

prune1.3s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize24.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 32 (sqrt 2))
1.0b
(/ (sqrt (- 1 (* (* v v) 3))) (/ (/ 32 (sqrt 2)) (- 1 (* (* v v) 3))))
0.2b
(* (* v v) 3)
0.2b
(* (* v v) 3)

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
192×times-frac
99×add-cube-cbrt
99×*-un-lft-identity
99×add-sqr-sqrt
57×sqrt-prod
30×div-inv
21×associate-/r*
15×associate-/r/
9×add-exp-log
9×add-cbrt-cube
8×flip--
8×flip3--
6×associate-/l*
4×add-log-exp
4×pow1
3×div-exp
3×cbrt-undiv
2×associate-/l/
2×*-commutative
2×associate-*l*
2×sqrt-div
2×frac-2neg
2×clear-num
Counts
4 → 180
Calls
4 calls:
Slowest
9.0ms
(/ (sqrt (- 1 (* (* v v) 3))) (/ (/ 32 (sqrt 2)) (- 1 (* (* v v) 3))))
2.0ms
(* (* v v) 3)
2.0ms
(* (* v v) 3)
0.0ms
(/ 32 (sqrt 2))

series126.0ms

Counts
4 → 10
Calls
4 calls:
Slowest
96.0ms
(/ (sqrt (- 1 (* (* v v) 3))) (/ (/ 32 (sqrt 2)) (- 1 (* (* v v) 3))))
19.0ms
(* (* v v) 3)
11.0ms
(* (* v v) 3)
0.0ms
(/ 32 (sqrt 2))

simplify37.4s

Counts
259 → 190
Calls
259 calls:
Slowest
638.0ms
(/ (sqrt (cbrt (- 1 (* (* v v) 3)))) (/ 1 (- 1 (* (* v v) 3))))
618.0ms
(/ (cbrt (sqrt (- 1 (* (* v v) 3)))) (/ 1 (- 1 (* (* v v) 3))))
550.0ms
(/ (cbrt (sqrt (- 1 (* (* v v) 3)))) (/ (/ 32 (sqrt 2)) (- 1 (* (* v v) 3))))
546.0ms
(/ (sqrt (- 1 (* (* v v) 3))) (/ (/ 32 (sqrt 2)) (- 1 (* (* v v) 3))))
541.0ms
(/ (sqrt (- 1 (* (* v v) 3))) (/ (/ 1 (sqrt 2)) (- 1 (* (* v v) 3))))

prune2.5s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes65.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample2.9s

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