Average Error: 14.1 → 0.3
Time: 36.7s
Precision: 64
Internal Precision: 128
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{(\left(\frac{\pi}{2}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{1}{2} \cdot \frac{\pi}{a}\right))_*}{a + b}}{b - a}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.1

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

  2. Simplified9.5

    \[\leadsto \color{blue}{\frac{(\left(\frac{\pi}{2}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{\frac{\pi}{2}}{a}\right))_*}{\left(a + b\right) \cdot \left(b - a\right)}}\]
  3. Using strategy rm

  4. Applied associate-/r*0.3

    \[\leadsto \color{blue}{\frac{\frac{(\left(\frac{\pi}{2}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{\frac{\pi}{2}}{a}\right))_*}{a + b}}{b - a}}\]

  5. Taylor expanded around inf 0.3

    \[\leadsto \frac{\frac{(\left(\frac{\pi}{2}\right) \cdot \left(\frac{-1}{b}\right) + \color{blue}{\left(\frac{1}{2} \cdot \frac{\pi}{a}\right)})_*}{a + b}}{b - a}\]

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(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: 35.9s)Debug log

sample97.0ms

Algorithm
intervals

simplify381.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

380.0ms
(* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b)))

prune12.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 7.8b

localize44.0ms

Local error

Found 3 expressions with local error:

7.9b
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (* (+ a b) (- b a)))
0.2b
(/ (/ PI 2) a)
0.2b
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
3 → 58
Calls

3 calls. Slowest were:

11.0ms
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (* (+ a b) (- b a)))
5.0ms
(/ (/ PI 2) a)
0.0ms
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))

series266.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

131.0ms
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (* (+ a b) (- b a)))
74.0ms
(/ (/ PI 2) a)
61.0ms
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))

simplify2.5s

Counts
32 → 67
Calls

32 calls. Slowest were:

545.0ms
(* (+ (* a a) (- (* b b) (* a b))) (+ b a))
359.0ms
(* (+ (* a a) (- (* b b) (* a b))) (+ (* b b) (+ (* a a) (* b a))))
307.0ms
(- (log (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))) (log (* (+ a b) (- b a))))

prune786.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.2b
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b))
0.2b
(/ (/ PI 2) a)
0.2b
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))
0.2b
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b)) (- b a))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 140
Calls

4 calls. Slowest were:

12.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b)) (- b a))
7.0ms
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b))
5.0ms
(/ (/ PI 2) a)

series402.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

128.0ms
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b))
128.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b)) (- b a))
77.0ms
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))
69.0ms
(/ (/ PI 2) a)

simplify10.1s

Counts
155 → 152
Calls

155 calls. Slowest were:

406.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (- (* a a) (* b b))) (* (cbrt (- b a)) (cbrt (- b a))))
381.0ms
(/ (/ (* (cbrt (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))) (cbrt (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)))) (* (cbrt (+ a b)) (cbrt (+ a b)))) 1)
337.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (- (* a a) (* b b))) 1)

prune1.7s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize29.0ms

Local error

Found 4 expressions with local error:

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

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 135
Calls

4 calls. Slowest were:

12.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (* 1/2 (/ PI a))) (+ a b)) (- b a))
11.0ms
(/ (fma (/ PI 2) (/ -1 b) (* 1/2 (/ PI a))) (+ a b))
0.0ms
(/ PI a)

series323.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

121.0ms
(/ (fma (/ PI 2) (/ -1 b) (* 1/2 (/ PI a))) (+ a b))
100.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (* 1/2 (/ PI a))) (+ a b)) (- b a))
54.0ms
(/ PI a)
48.0ms
(fma (/ PI 2) (/ -1 b) (* 1/2 (/ PI a)))

simplify10.7s

Counts
153 → 147
Calls

153 calls. Slowest were:

429.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (* 1/2 (/ PI a))) (- (* a a) (* b b))) (* (cbrt (- b a)) (cbrt (- b a))))
315.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (* 1/2 (/ PI a))) (- (* a a) (* b b))) 1)
310.0ms
(/ (/ (* (cbrt (fma (/ PI 2) (/ -1 b) (* 1/2 (/ PI a)))) (cbrt (fma (/ PI 2) (/ -1 b) (* 1/2 (/ PI a))))) (* (cbrt (+ a b)) (cbrt (+ a b)))) 1)

prune1.8s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 4 expressions with local error:

0.8b
(/ 1 (/ (+ a b) (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))))
0.3b
(/ (+ a b) (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)))
0.2b
(/ (/ PI 2) a)
0.2b
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 88
Calls

4 calls. Slowest were:

5.0ms
(/ (+ a b) (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)))
5.0ms
(/ (/ PI 2) a)
3.0ms
(/ 1 (/ (+ a b) (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))))

series400.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

172.0ms
(/ (+ a b) (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)))
111.0ms
(/ 1 (/ (+ a b) (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))))
60.0ms
(/ (/ PI 2) a)
56.0ms
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))

simplify2.2s

Counts
56 → 100
Calls

56 calls. Slowest were:

80.0ms
(/ (* (* (+ a b) (+ a b)) (+ a b)) (* (* (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))) (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))))
74.0ms
(/ (* (cbrt (+ a b)) (cbrt (+ a b))) (* (cbrt (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))) (cbrt (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)))))
71.0ms
(- (* 1/2 (/ PI a)) (* 1/2 (/ PI b)))

prune1.1s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0.0b

regimes228.0ms

Accuracy

0% (0.3b remaining)

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

bsearch3.0ms

end0.0ms

sample2.7s

Algorithm
intervals