Average Error: 14.2 → 0.3
Time: 35.0s
Precision: 64
Internal Precision: 128
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1}{b \cdot a} \cdot \frac{1}{\frac{a + b}{\frac{\pi}{2}}}\]

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.2

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  2. Initial simplification0.3

    \[\leadsto \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \cdot \frac{\frac{\pi}{2}}{a + b}\]

  3. Taylor expanded around -inf 0.3

    \[\leadsto \color{blue}{\frac{1}{a \cdot b}} \cdot \frac{\frac{\pi}{2}}{a + b}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.3

    \[\leadsto \frac{1}{a \cdot b} \cdot \frac{\color{blue}{1 \cdot \frac{\pi}{2}}}{a + b}\]

  6. Applied associate-/l*0.3

    \[\leadsto \frac{1}{a \cdot b} \cdot \color{blue}{\frac{1}{\frac{a + b}{\frac{\pi}{2}}}}\]

  7. Final simplification0.3

    \[\leadsto \frac{1}{b \cdot a} \cdot \frac{1}{\frac{a + b}{\frac{\pi}{2}}}\]

Reproduce

herbie shell --seed 2018365 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))

Details

Time bar (total: 29.3s)Debug log

start139.0ms

Algorithm
intervals

setup404.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize58.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ (- (/ 1 a) (/ 1 b)) (- b a))
0.3b
(/ (/ PI 2) (+ a b))
0.2b
(* (/ (- (/ 1 a) (/ 1 b)) (- b a)) (/ (/ PI 2) (+ a b)))
0.0b
(- (/ 1 a) (/ 1 b))

rewrite59.0ms

Algorithm
rewrite-expression-head
Counts
4 → 173
Calls

4 calls. Slowest were:

23.0ms
(* (/ (- (/ 1 a) (/ 1 b)) (- b a)) (/ (/ PI 2) (+ a b)))
18.0ms
(/ (- (/ 1 a) (/ 1 b)) (- b a))
7.0ms
(- (/ 1 a) (/ 1 b))

series200.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

84.0ms
(/ (/ PI 2) (+ a b))
78.0ms
(* (/ (- (/ 1 a) (/ 1 b)) (- b a)) (/ (/ PI 2) (+ a b)))
30.0ms
(/ (- (/ 1 a) (/ 1 b)) (- b a))
8.0ms
(- (/ 1 a) (/ 1 b))

simplify6.0s

Counts
176 → 185
Calls

176 calls. Slowest were:

540.0ms
(+ (- (log (- (/ 1 a) (/ 1 b))) (log (- b a))) (- (log (/ PI 2)) (log (+ a b))))
465.0ms
(/ (* (* (/ PI 2) (/ PI 2)) (/ PI 2)) (* (* (+ a b) (+ a b)) (+ a b)))
465.0ms
(+ (log (/ (- (/ 1 a) (/ 1 b)) (- b a))) (log (/ (/ PI 2) (+ a b))))

prune2.5s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0b

localize16.0ms

Local error

Found 3 expressions with local error:

0.4b
(/ 1 (* a b))
0.3b
(/ (/ PI 2) (+ a b))
0.2b
(* (/ 1 (* a b)) (/ (/ PI 2) (+ a b)))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
3 → 88
Calls

3 calls. Slowest were:

11.0ms
(* (/ 1 (* a b)) (/ (/ PI 2) (+ a b)))
6.0ms
(/ (/ PI 2) (+ a b))
1.0ms
(/ 1 (* a b))

series158.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

81.0ms
(* (/ 1 (* a b)) (/ (/ PI 2) (+ a b)))
62.0ms
(/ (/ PI 2) (+ a b))
14.0ms
(/ 1 (* a b))

simplify2.6s

Counts
74 → 97
Calls

74 calls. Slowest were:

418.0ms
(+ (log (/ 1 (* a b))) (- (log (/ PI 2)) (log (+ a b))))
347.0ms
(/ (* (* (/ PI 2) (/ PI 2)) (/ PI 2)) (* (* (+ a b) (+ a b)) (+ a b)))
279.0ms
(+ (- (log (* a b))) (- (log (/ PI 2)) (log (+ a b))))

prune1.0s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0b

localize11.0ms

Local error

Found 4 expressions with local error:

0.4b
(/ 1 (* a b))
0.3b
(/ 1 (/ (+ a b) (/ PI 2)))
0.3b
(/ (+ a b) (/ PI 2))
0.2b
(* (/ 1 (* a b)) (/ 1 (/ (+ a b) (/ PI 2))))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 111
Calls

4 calls. Slowest were:

10.0ms
(* (/ 1 (* a b)) (/ 1 (/ (+ a b) (/ PI 2))))
6.0ms
(/ (+ a b) (/ PI 2))
3.0ms
(/ 1 (/ (+ a b) (/ PI 2)))

series249.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

95.0ms
(* (/ 1 (* a b)) (/ 1 (/ (+ a b) (/ PI 2))))
72.0ms
(/ (+ a b) (/ PI 2))
69.0ms
(/ 1 (/ (+ a b) (/ PI 2)))
12.0ms
(/ 1 (* a b))

simplify5.2s

Counts
88 → 123
Calls

88 calls. Slowest were:

524.0ms
(* (* a b) (/ (+ a b) (/ PI 2)))
500.0ms
(* (* a b) (/ (+ a b) (/ PI 2)))
487.0ms
(/ (* (* (+ a b) (+ a b)) (+ a b)) (* (* (/ PI 2) (/ PI 2)) (/ PI 2)))

prune1.4s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ (- (/ 1 a) (/ 1 b)) (- b a))
0.3b
(* (/ (- (/ 1 a) (/ 1 b)) (- b a)) (/ PI 1))
0.2b
(* (* (/ (- (/ 1 a) (/ 1 b)) (- b a)) (/ PI 1)) (/ 1/2 (+ a b)))
0.0b
(/ 1/2 (+ a b))

rewrite63.0ms

Algorithm
rewrite-expression-head
Counts
4 → 148
Calls

4 calls. Slowest were:

35.0ms
(* (* (/ (- (/ 1 a) (/ 1 b)) (- b a)) (/ PI 1)) (/ 1/2 (+ a b)))
13.0ms
(* (/ (- (/ 1 a) (/ 1 b)) (- b a)) (/ PI 1))
9.0ms
(/ (- (/ 1 a) (/ 1 b)) (- b a))

series193.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

105.0ms
(* (* (/ (- (/ 1 a) (/ 1 b)) (- b a)) (/ PI 1)) (/ 1/2 (+ a b)))
32.0ms
(/ 1/2 (+ a b))
28.0ms
(/ (- (/ 1 a) (/ 1 b)) (- b a))
27.0ms
(* (/ (- (/ 1 a) (/ 1 b)) (- b a)) (/ PI 1))

simplify6.7s

Counts
145 → 160
Calls

145 calls. Slowest were:

604.0ms
(* (* (* (/ (- (/ 1 a) (/ 1 b)) (- b a)) (/ (- (/ 1 a) (/ 1 b)) (- b a))) (/ (- (/ 1 a) (/ 1 b)) (- b a))) (* (* (/ PI 1) (/ PI 1)) (/ PI 1)))
501.0ms
(* (* (/ (- (/ 1 a) (/ 1 b)) (- b a)) PI) (/ 1/2 (+ a b)))
411.0ms
(+ (+ (log (/ (- (/ 1 a) (/ 1 b)) (- b a))) (log (/ PI 1))) (log (/ 1/2 (+ a b))))

prune2.1s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

regimes101.0ms

Accuracy

0% (0.3b remaining)

Error of 0.3b against oracle of 0.0b and baseline of 0.3b

bsearch2.0ms