Average Error: 13.9 → 0.3
Time: 31.5s
Precision: 64
Internal Precision: 128
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{(\left(\frac{\pi}{2}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{\frac{\pi}{2}}{a}\right))_*}{a + b}}{b - a}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 13.9

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

  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019004 +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: 30.2s)Debug log

sample103.0ms

Algorithm
intervals

simplify351.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

351.0ms
(* (* (/ 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: 10.8b

localize42.0ms

Local error

Found 3 expressions with local error:

10.8b
(/ (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))

series221.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

89.0ms
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (* (+ a b) (- b a)))
72.0ms
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))
59.0ms
(/ (/ PI 2) a)

simplify2.0s

Counts
32 → 67
Calls

32 calls. Slowest were:

340.0ms
(* (+ (* a a) (- (* b b) (* a b))) (+ (* b b) (+ (* a a) (* b a))))
283.0ms
(- (log (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))) (log (* (+ a b) (- b a))))
189.0ms
(* (+ (* a a) (- (* b b) (* a b))) (+ b a))

prune736.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 4 expressions with local error:

0.2b
(/ (/ PI 2) a)
0.2b
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b))
0.2b
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))
0.1b
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b)) (- b a))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 140
Calls

4 calls. Slowest were:

6.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b)) (- b a))
4.0ms
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b))
2.0ms
(/ (/ PI 2) a)

series364.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

113.0ms
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b))
108.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b)) (- b a))
77.0ms
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))
66.0ms
(/ (/ PI 2) a)

simplify9.4s

Counts
155 → 152
Calls

155 calls. Slowest were:

492.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (- (* a a) (* b b))) (* (cbrt (- b a)) (cbrt (- b a))))
283.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (- (* a a) (* b b))) 1)
257.0ms
(/ (/ (* (cbrt (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))) (cbrt (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)))) (* (cbrt (+ a b)) (cbrt (+ a b)))) (* (cbrt (- b a)) (cbrt (- b a))))

prune1.6s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0b

localize20.0ms

Local error

Found 4 expressions with local error:

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

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
4 → 123
Calls

4 calls. Slowest were:

17.0ms
(/ (- (/ PI (* a 2)) (/ PI (* 2 b))) (- b a))
11.0ms
(* (/ 1 (+ a b)) (/ (- (/ PI (* a 2)) (/ PI (* 2 b))) (- b a)))
0.0ms
(/ PI (* a 2))

series242.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

79.0ms
(* (/ 1 (+ a b)) (/ (- (/ PI (* a 2)) (/ PI (* 2 b))) (- b a)))
55.0ms
(/ (- (/ PI (* a 2)) (/ PI (* 2 b))) (- b a))
55.0ms
(/ PI (* 2 b))
52.0ms
(/ PI (* a 2))

simplify5.4s

Counts
101 → 135
Calls

101 calls. Slowest were:

447.0ms
(* (* (* (/ 1 (+ a b)) (/ 1 (+ a b))) (/ 1 (+ a b))) (* (* (/ (- (/ PI (* a 2)) (/ PI (* 2 b))) (- b a)) (/ (- (/ PI (* a 2)) (/ PI (* 2 b))) (- b a))) (/ (- (/ PI (* a 2)) (/ PI (* 2 b))) (- b a))))
337.0ms
(- (pow (/ PI (* a 2)) 3) (pow (/ PI (* 2 b)) 3))
250.0ms
(/ (* (* (- (/ PI (* a 2)) (/ PI (* 2 b))) (- (/ PI (* a 2)) (/ PI (* 2 b)))) (- (/ PI (* a 2)) (/ PI (* 2 b)))) (* (* (- b a) (- b a)) (- b a)))

prune1.3s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0b

localize16.0ms

Local error

Found 4 expressions with local error:

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

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

9.0ms
(/ (- (* PI (/ 1 (* a 2))) (/ PI (* 2 b))) (- b a))
1.0ms
(* PI (/ 1 (* a 2)))
1.0ms
(/ 1 (* a 2))

series260.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

91.0ms
(/ (- (* PI (/ 1 (* a 2))) (/ PI (* 2 b))) (- b a))
83.0ms
(/ PI (* 2 b))
74.0ms
(* PI (/ 1 (* a 2)))
12.0ms
(/ 1 (* a 2))

simplify4.8s

Counts
57 → 97
Calls

57 calls. Slowest were:

464.0ms
(- (pow (* PI (/ 1 (* a 2))) 3) (pow (/ PI (* 2 b)) 3))
407.0ms
(/ (- (* PI (/ 1 (* a 2))) (/ PI (* 2 b))) (cbrt (- b a)))
386.0ms
(/ (- (* PI (/ 1 (* a 2))) (/ PI (* 2 b))) (sqrt (- b a)))

prune998.0ms

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0b

regimes121.0ms

Accuracy

0% (0.2b remaining)

Error of 0.3b against oracle of 0.1b and baseline of 0.3b

bsearch1.0ms

end0.0ms

sample2.0s

Algorithm
intervals