Average Error: 14.3 → 0.3
Time: 21.5s
Precision: 64
Internal Precision: 128
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{a}}{b} \cdot \frac{\frac{\pi}{2}}{a + b}\]

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.3

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

  2. Simplified0.3

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

  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 associate-/r*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019010 
(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: 21.0s)Debug log

sample100.0ms

Algorithm
intervals

simplify334.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
334.0ms
(* (* (/ 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: 0.2b

localize32.0ms

Local error

Found 4 expressions with local error:

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

rewrite69.0ms

Algorithm
rewrite-expression-head
Rules
78×*-un-lft-identity
72×times-frac
62×add-sqr-sqrt
46×add-cube-cbrt
43×div-inv
32×distribute-lft-out--
30×associate-*l*
20×add-exp-log
20×add-cbrt-cube
18×associate-*r*
12×associate-/l*
8×difference-of-squares
8×associate-/r/
6×add-log-exp
6×div-exp
6×associate-/r*
6×cbrt-undiv
6×pow1
4×associate-/l/
4×flip--
4×cbrt-unprod
4×prod-exp
4×flip3--
2×flip-+
2×frac-sub
2×flip3-+
2×frac-2neg
2×clear-num
1×*-commutative
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×div-sub
1×diff-log
1×sub-neg
1×frac-times
Counts
4 → 173
Calls
4 calls:
Slowest
30.0ms
(* (/ (- (/ 1 a) (/ 1 b)) (- b a)) (/ (/ PI 2) (+ a b)))
18.0ms
(/ (- (/ 1 a) (/ 1 b)) (- b a))
9.0ms
(/ (/ PI 2) (+ a b))
7.0ms
(- (/ 1 a) (/ 1 b))

series226.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
110.0ms
(* (/ (- (/ 1 a) (/ 1 b)) (- b a)) (/ (/ PI 2) (+ a b)))
74.0ms
(/ (/ PI 2) (+ a b))
27.0ms
(/ (- (/ 1 a) (/ 1 b)) (- b a))
14.0ms
(- (/ 1 a) (/ 1 b))

simplify5.4s

Counts
176 → 185
Calls
176 calls:
Slowest
489.0ms
(+ (- (log (- (/ 1 a) (/ 1 b))) (log (- b a))) (- (log (/ PI 2)) (log (+ a b))))
396.0ms
(/ (* (* (/ PI 2) (/ PI 2)) (/ PI 2)) (* (* (+ a b) (+ a b)) (+ a b)))
336.0ms
(+ (log (/ (- (/ 1 a) (/ 1 b)) (- b a))) (- (log (/ PI 2)) (log (+ a b))))
281.0ms
(* (/ (* (* (- (/ 1 a) (/ 1 b)) (- (/ 1 a) (/ 1 b))) (- (/ 1 a) (/ 1 b))) (* (* (- b a) (- b a)) (- b a))) (/ (* (* (/ PI 2) (/ PI 2)) (/ PI 2)) (* (* (+ a b) (+ a b)) (+ a b))))
264.0ms
(+ (- (log (- (/ 1 a) (/ 1 b))) (log (- b a))) (log (/ (/ PI 2) (+ a b))))

prune1.8s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0b

localize8.0ms

Local error

Found 3 expressions with local error:

0.6b
(/ 1 (* a b))
0.3b
(/ (/ PI 2) (+ a b))
0.2b
(* (/ 1 (* a b)) (/ (/ PI 2) (+ a b)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
24×times-frac
21×add-cube-cbrt
21×*-un-lft-identity
21×add-sqr-sqrt
18×associate-*r*
16×add-exp-log
11×div-inv
10×add-cbrt-cube
6×pow1
4×associate-/l*
4×associate-/r/
4×prod-exp
4×associate-*l*
4×associate-/r*
3×add-log-exp
3×div-exp
3×rec-exp
2×flip-+
2×cbrt-unprod
2×flip3-+
2×frac-2neg
2×clear-num
2×cbrt-undiv
1×associate-/l/
1×inv-pow
1×pow-flip
1×*-commutative
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×frac-times
Counts
3 → 88
Calls
3 calls:
Slowest
9.0ms
(* (/ 1 (* a b)) (/ (/ PI 2) (+ a b)))
3.0ms
(/ (/ PI 2) (+ a b))
1.0ms
(/ 1 (* a b))

series147.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
74.0ms
(* (/ 1 (* a b)) (/ (/ PI 2) (+ a b)))
61.0ms
(/ (/ PI 2) (+ a b))
12.0ms
(/ 1 (* a b))

simplify2.7s

Counts
74 → 97
Calls
74 calls:
Slowest
441.0ms
(/ (* (* (/ PI 2) (/ PI 2)) (/ PI 2)) (* (* (+ a b) (+ a b)) (+ a b)))
411.0ms
(+ (log (/ 1 (* a b))) (- (log (/ PI 2)) (log (+ a b))))
324.0ms
(+ (- (log (* a b))) (- (log (/ PI 2)) (log (+ a b))))
257.0ms
(+ (log (/ 1 (* a b))) (log (/ (/ PI 2) (+ a b))))
199.0ms
(* (* (* (/ 1 (* a b)) (/ 1 (* a b))) (/ 1 (* a b))) (/ (* (* (/ PI 2) (/ PI 2)) (/ PI 2)) (* (* (+ a b) (+ a b)) (+ a b))))

prune764.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0b

localize12.0ms

Local error

Found 3 expressions with local error:

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

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
24×times-frac
22×add-cube-cbrt
22×*-un-lft-identity
22×add-sqr-sqrt
18×associate-*r*
12×div-inv
10×add-exp-log
10×add-cbrt-cube
8×associate-/l*
5×pow1
4×associate-/r/
4×associate-*l*
3×add-log-exp
3×associate-/r*
2×associate-/l/
2×flip-+
2×cbrt-unprod
2×prod-exp
2×div-exp
2×flip3-+
2×frac-2neg
2×clear-num
2×cbrt-undiv
1×*-commutative
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×frac-times
Counts
3 → 87
Calls
3 calls:
Slowest
7.0ms
(* (/ (/ 1 a) b) (/ (/ PI 2) (+ a b)))
7.0ms
(/ (/ PI 2) (+ a b))
5.0ms
(/ (/ 1 a) b)

series142.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
80.0ms
(* (/ (/ 1 a) b) (/ (/ PI 2) (+ a b)))
52.0ms
(/ (/ PI 2) (+ a b))
10.0ms
(/ (/ 1 a) b)

simplify1.8s

Counts
74 → 96
Calls
74 calls:
Slowest
351.0ms
(/ (* (* (/ PI 2) (/ PI 2)) (/ PI 2)) (* (* (+ a b) (+ a b)) (+ a b)))
289.0ms
(+ (log (/ (/ 1 a) b)) (log (/ (/ PI 2) (+ a b))))
221.0ms
(* (/ 1 a) (/ (/ PI 2) (+ a b)))
218.0ms
(* (* (* (/ (/ 1 a) b) (/ (/ 1 a) b)) (/ (/ 1 a) b)) (/ (* (* (/ PI 2) (/ PI 2)) (/ PI 2)) (* (* (+ a b) (+ a b)) (+ a b))))
90.0ms
(* (* (* (/ (/ 1 a) b) (/ (/ 1 a) b)) (/ (/ 1 a) b)) (* (* (/ (/ PI 2) (+ a b)) (/ (/ PI 2) (+ a b))) (/ (/ PI 2) (+ a b))))

prune789.0ms

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0b

localize8.0ms

Local error

Found 2 expressions with local error:

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
54×times-frac
27×add-cube-cbrt
27×*-un-lft-identity
27×add-sqr-sqrt
22×associate-/l*
14×div-inv
9×add-exp-log
9×add-cbrt-cube
6×associate-/r/
4×div-exp
4×associate-/r*
4×cbrt-undiv
3×flip-+
3×flip3-+
2×associate-/l/
2×add-log-exp
2×frac-2neg
2×clear-num
2×pow1
Counts
2 → 86
Calls
2 calls:
Slowest
6.0ms
(/ (/ (/ PI 2) (+ b a)) (* a b))
3.0ms
(/ (/ PI 2) (+ b a))

series158.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
90.0ms
(/ (/ (/ PI 2) (+ b a)) (* a b))
68.0ms
(/ (/ PI 2) (+ b a))

simplify3.1s

Counts
97 → 92
Calls
97 calls:
Slowest
437.0ms
(- (- (log (/ PI 2)) (log (+ b a))) (log (* a b)))
340.0ms
(/ (* (* (/ PI 2) (/ PI 2)) (/ PI 2)) (* (* (+ b a) (+ b a)) (+ b a)))
304.0ms
(/ (/ PI 2) (- (* b b) (* a a)))
239.0ms
(/ (/ (/ PI 2) (- (* b b) (* a a))) a)
166.0ms
(- (log (/ (/ PI 2) (+ b a))) (log (* a b)))

prune773.0ms

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0b

regimes86.0ms

Accuracy

0% (0.3b remaining)

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

bsearch1.0ms

end0.0ms

sample2.5s

Algorithm
intervals