Average Error: 0.0 → 0.0
Time: 37.3s
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[e^{\log \left(1 - v \cdot v\right) + \log \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right)}\]

Error

Bits error versus v

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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-exp-log0.0

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

  4. Applied add-exp-log0.0

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

  5. Applied prod-exp0.0

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

  6. Final simplification0.0

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

Runtime

Time bar (total: 37.3s)Debug logProfile

start244.0ms

Algorithm
intervals

setup332.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize45.0ms

Local error

Found 4 expressions with local error:

0.1b
(* 3 (* v v))
0.0b
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
0.0b
(sqrt (- 1 (* 3 (* v v))))
0.0b
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 96
Calls

4 calls. Slowest were:

15.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
6.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
1.0ms
(sqrt (- 1 (* 3 (* v v))))

series235.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

103.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
81.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
35.0ms
(sqrt (- 1 (* 3 (* v v))))

simplify15.7s

Counts
78 → 108
Calls

78 calls. Slowest were:

1.2s
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (sqrt (- 1 (* 3 (* v v)))) (sqrt (- 1 (* 3 (* v v))))) (sqrt (- 1 (* 3 (* v v))))))
722.0ms
(* (* 4 (sqrt (+ 1 (* 3 (* v v))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))
637.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- (* 1 1) (* (* 3 (* v v)) (* 3 (* v v)))))) (- 1 (* v v)))

prune1.8s

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

1.9b
(log (- 1 (* v v)))
0.1b
(* 3 (* v v))
0.0b
(+ (log (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))) (log (- 1 (* v v))))
0.0b
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))

rewrite47.0ms

Algorithm
rewrite-expression-head
Counts
4 → 92
Calls

4 calls. Slowest were:

31.0ms
(+ (log (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))) (log (- 1 (* v v))))
11.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
1.0ms
(log (- 1 (* v v)))

series508.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

281.0ms
(+ (log (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))) (log (- 1 (* v v))))
112.0ms
(log (- 1 (* v v)))
96.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))

simplify8.2s

Counts
59 → 104
Calls

59 calls. Slowest were:

1.3s
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (sqrt (- 1 (* 3 (* v v)))) (sqrt (- 1 (* 3 (* v v))))) (sqrt (- 1 (* 3 (* v v))))))
589.0ms
(- (log -1) (+ (* 2 (log (/ 1 v))) (+ (* 1/2 (/ 1 (pow v 4))) (/ 1 (pow v 2)))))
396.0ms
(* 4 (sqrt (+ (* 1 1) (+ (* (* 3 (* v v)) (* 3 (* v v))) (* 1 (* 3 (* v v)))))))

prune2.0s

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

herbie shell --seed 2018357 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))