Average Error: 0.0 → 0.0
Time: 40.9s
Precision: 64
Internal Precision: 128
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}}{\sqrt[3]{1 + x}}}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{\color{blue}{\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right) \cdot \sqrt[3]{1 + x}}}}\right)\]

  4. Applied associate-/r*0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\frac{\frac{1 - x}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}}{\sqrt[3]{1 + x}}}}\right)\]

  5. Final simplification0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}}{\sqrt[3]{1 + x}}}\right)\]

Reproduce

herbie shell --seed 2018365 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))

Details

Time bar (total: 36.4s)Debug log

start321.0ms

Algorithm
intervals

setup143.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize24.0ms

Local error

Found 3 expressions with local error:

0.0b
(/ (- 1 x) (+ 1 x))
0.0b
(sqrt (/ (- 1 x) (+ 1 x)))
0.0b
(atan (sqrt (/ (- 1 x) (+ 1 x))))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
3 → 65
Calls

3 calls. Slowest were:

7.0ms
(/ (- 1 x) (+ 1 x))
2.0ms
(sqrt (/ (- 1 x) (+ 1 x)))
1.0ms
(atan (sqrt (/ (- 1 x) (+ 1 x))))

series383.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

352.0ms
(sqrt (/ (- 1 x) (+ 1 x)))
27.0ms
(/ (- 1 x) (+ 1 x))
4.0ms
(atan (sqrt (/ (- 1 x) (+ 1 x))))

simplify1.6s

Counts
57 → 74
Calls

57 calls. Slowest were:

285.0ms
(- (+ (sqrt -1) (/ 1 (* x (sqrt -1)))) (+ (/ 1 (* (pow x 2) (sqrt -1))) (* 1/2 (/ 1 (* (pow x 2) (pow (sqrt -1) 3))))))
277.0ms
(/ (* (* (- 1 x) (- 1 x)) (- 1 x)) (* (* (+ 1 x) (+ 1 x)) (+ 1 x)))
269.0ms
(- (+ (sqrt -1) (/ 1 (* x (sqrt -1)))) (+ (/ 1 (* (pow x 2) (sqrt -1))) (* 1/2 (/ 1 (* (pow x 2) (pow (sqrt -1) 3))))))

prune842.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize30.0ms

Local error

Found 4 expressions with local error:

0.0b
(* (cbrt (+ 1 x)) (cbrt (+ 1 x)))
0.0b
(/ (/ (- 1 x) (* (cbrt (+ 1 x)) (cbrt (+ 1 x)))) (cbrt (+ 1 x)))
0.0b
(/ (- 1 x) (* (cbrt (+ 1 x)) (cbrt (+ 1 x))))
0.0b
(sqrt (/ (/ (- 1 x) (* (cbrt (+ 1 x)) (cbrt (+ 1 x)))) (cbrt (+ 1 x))))

rewrite36.0ms

Algorithm
rewrite-expression-head
Counts
4 → 309
Calls

4 calls. Slowest were:

11.0ms
(/ (/ (- 1 x) (* (cbrt (+ 1 x)) (cbrt (+ 1 x)))) (cbrt (+ 1 x)))
6.0ms
(sqrt (/ (/ (- 1 x) (* (cbrt (+ 1 x)) (cbrt (+ 1 x)))) (cbrt (+ 1 x))))
5.0ms
(/ (- 1 x) (* (cbrt (+ 1 x)) (cbrt (+ 1 x))))

series170.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

64.0ms
(/ (- 1 x) (* (cbrt (+ 1 x)) (cbrt (+ 1 x))))
40.0ms
(sqrt (/ (/ (- 1 x) (* (cbrt (+ 1 x)) (cbrt (+ 1 x)))) (cbrt (+ 1 x))))
40.0ms
(* (cbrt (+ 1 x)) (cbrt (+ 1 x)))
25.0ms
(/ (/ (- 1 x) (* (cbrt (+ 1 x)) (cbrt (+ 1 x)))) (cbrt (+ 1 x)))

simplify18.8s

Counts
380 → 321
Calls

380 calls. Slowest were:

747.0ms
(/ (* (* (/ (- 1 x) (* (cbrt (+ 1 x)) (cbrt (+ 1 x)))) (/ (- 1 x) (* (cbrt (+ 1 x)) (cbrt (+ 1 x))))) (/ (- 1 x) (* (cbrt (+ 1 x)) (cbrt (+ 1 x))))) (+ 1 x))
342.0ms
(/ (* (* (- 1 x) (- 1 x)) (- 1 x)) (* (* (* (cbrt (+ 1 x)) (cbrt (+ 1 x))) (* (cbrt (+ 1 x)) (cbrt (+ 1 x)))) (* (cbrt (+ 1 x)) (cbrt (+ 1 x)))))
289.0ms
(- (+ (sqrt -1) (/ 1 (* x (sqrt -1)))) (+ (/ 1 (* (pow x 2) (sqrt -1))) (* 1/2 (/ 1 (* (pow x 2) (pow (sqrt -1) 3))))))

prune5.5s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize22.0ms

Local error

Found 4 expressions with local error:

0.0b
(* (cbrt (- 1 x)) (cbrt (- 1 x)))
0.0b
(/ (* (cbrt (- 1 x)) (cbrt (- 1 x))) (/ (+ 1 x) (cbrt (- 1 x))))
0.0b
(/ (+ 1 x) (cbrt (- 1 x)))
0.0b
(cbrt (- 1 x))

rewrite33.0ms

Algorithm
rewrite-expression-head
Counts
4 → 168
Calls

4 calls. Slowest were:

15.0ms
(/ (* (cbrt (- 1 x)) (cbrt (- 1 x))) (/ (+ 1 x) (cbrt (- 1 x))))
7.0ms
(/ (+ 1 x) (cbrt (- 1 x)))
5.0ms
(* (cbrt (- 1 x)) (cbrt (- 1 x)))

series176.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

67.0ms
(/ (+ 1 x) (cbrt (- 1 x)))
54.0ms
(cbrt (- 1 x))
37.0ms
(* (cbrt (- 1 x)) (cbrt (- 1 x)))
18.0ms
(/ (* (cbrt (- 1 x)) (cbrt (- 1 x))) (/ (+ 1 x) (cbrt (- 1 x))))

simplify5.6s

Counts
189 → 180
Calls

189 calls. Slowest were:

523.0ms
(/ (* (- 1 x) (- 1 x)) (* (* (/ (+ 1 x) (cbrt (- 1 x))) (/ (+ 1 x) (cbrt (- 1 x)))) (/ (+ 1 x) (cbrt (- 1 x)))))
426.0ms
(/ (* (* (+ 1 x) (+ 1 x)) (+ 1 x)) (- 1 x))
419.0ms
(- (log (* (cbrt (- 1 x)) (cbrt (- 1 x)))) (- (log (+ 1 x)) (log (cbrt (- 1 x)))))

prune2.7s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes27.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms