Average Error: 14.0 → 0.3
Time: 41.3s
Precision: 64
Internal Precision: 128
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\left(\frac{1}{a} \cdot \pi - \frac{\pi}{b}\right) \cdot \frac{1}{2}}{b + a}}{b - a}\]

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 14.0

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

  2. Simplified9.4

    \[\leadsto \color{blue}{\frac{(\left(\frac{\pi}{2}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{\frac{\pi}{2}}{a}\right))_*}{\left(a + b\right) \cdot \left(b - a\right)}}\]
  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 0 0.3

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

  6. Simplified0.3

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

  8. Applied div-inv0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019016 +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: 40.6s)Debug log

sample76.0ms

Algorithm
intervals
Results
305×(pre true 80)
305×(body real 80)

simplify347.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
346.0ms
(* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b)))

prune28.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 8.0b

localize54.0ms

Local error

Found 3 expressions with local error:

8.0b
(/ (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))

rewrite20.0ms

Algorithm
rewrite-expression-head
Rules
8×associate-/r/
8×add-exp-log
8×add-cbrt-cube
7×associate-/l*
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
4×frac-times
3×add-log-exp
3×flip--
3×div-inv
3×log1p-expm1-u
3×flip-+
3×flip3--
3×times-frac
3×flip3-+
3×pow1
3×expm1-log1p-u
2×associate-*r/
2×associate-*l/
2×div-exp
2×frac-2neg
2×clear-num
2×cbrt-undiv
1×associate-/l/
1×fma-udef
1×cbrt-unprod
1×prod-exp
1×associate-/r*
Counts
3 → 58
Calls
3 calls:
Slowest
13.0ms
(/ (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
133.0ms
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (* (+ a b) (- b a)))
73.0ms
(/ (/ PI 2) a)
59.0ms
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))

simplify2.1s

Counts
32 → 67
Calls
32 calls:
Slowest
339.0ms
(* (+ (* a a) (- (* b b) (* a b))) (+ (* b b) (+ (* a a) (* b a))))
282.0ms
(- (log (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))) (log (* (+ a b) (- b a))))
217.0ms
(* (- a b) (+ (* b b) (+ (* a a) (* b a))))
174.0ms
(/ (* (* (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))) (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))) (* (* (* (+ a b) (- b a)) (* (+ a b) (- b a))) (* (+ a b) (- b a))))
151.0ms
(* (+ (* a a) (- (* b b) (* a b))) (+ b a))

prune760.0ms

Pruning

9 alts after pruning (9 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))

rewrite29.0ms

Algorithm
rewrite-expression-head
Rules
90×times-frac
57×add-cube-cbrt
57×*-un-lft-identity
57×add-sqr-sqrt
22×associate-/l*
12×associate-/r/
11×add-exp-log
11×add-cbrt-cube
8×div-inv
6×associate-/r*
5×flip-+
5×flip3-+
4×add-log-exp
4×log1p-expm1-u
4×div-exp
4×cbrt-undiv
4×pow1
4×expm1-log1p-u
3×frac-2neg
3×clear-num
2×associate-/l/
1×flip--
1×fma-udef
1×flip3--
Counts
4 → 140
Calls
4 calls:
Slowest
13.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b)) (- b a))
8.0ms
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b))
5.0ms
(/ (/ PI 2) a)
0.0ms
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))

series363.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
135.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b)) (- b a))
103.0ms
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b))
71.0ms
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))
53.0ms
(/ (/ PI 2) a)

simplify10.2s

Counts
155 → 152
Calls
155 calls:
Slowest
412.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (- (* a a) (* b b))) (* (cbrt (- b a)) (cbrt (- b a))))
354.0ms
(/ (/ (* (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))))
346.0ms
(/ (/ (* (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)
344.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (- (* a a) (* b b))) 1)
289.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (- (* a a) (* b b))) (sqrt (- b a)))

prune2.0s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0b

localize19.0ms

Local error

Found 4 expressions with local error:

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

rewrite41.0ms

Algorithm
rewrite-expression-head
Rules
42×times-frac
24×add-cube-cbrt
24×*-un-lft-identity
24×add-sqr-sqrt
12×associate-/r/
11×add-exp-log
11×add-cbrt-cube
10×associate-/l*
8×div-inv
6×associate-/r*
5×flip-+
5×flip3-+
4×associate-/l/
4×add-log-exp
4×log1p-expm1-u
4×div-exp
4×frac-2neg
4×clear-num
4×cbrt-undiv
4×pow1
4×expm1-log1p-u
3×associate-*l/
2×flip--
2×flip3--
1×frac-sub
Counts
4 → 108
Calls
4 calls:
Slowest
22.0ms
(/ (/ (* (- (/ PI a) (/ PI b)) 1/2) (+ a b)) (- b a))
15.0ms
(/ (* (- (/ PI a) (/ PI b)) 1/2) (+ a b))
0.0ms
(/ PI a)
0.0ms
(/ PI b)

series262.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
91.0ms
(/ (* (- (/ PI a) (/ PI b)) 1/2) (+ a b))
87.0ms
(/ (/ (* (- (/ PI a) (/ PI b)) 1/2) (+ a b)) (- b a))
48.0ms
(/ PI b)
37.0ms
(/ PI a)

simplify8.1s

Counts
101 → 120
Calls
101 calls:
Slowest
367.0ms
(/ (/ (* (- (/ PI a) (/ PI b)) 1/2) (+ a b)) (sqrt (- b a)))
355.0ms
(/ (/ (* (- (/ PI a) (/ PI b)) 1/2) (+ a b)) (cbrt (- b a)))
353.0ms
(/ (/ (* (- (/ PI a) (/ PI b)) 1/2) (+ (pow a 3) (pow b 3))) (sqrt (- b a)))
334.0ms
(* (- (* (/ PI a) (/ PI a)) (* (/ PI b) (/ PI b))) 1/2)
321.0ms
(/ (/ (* (- (/ PI a) (/ PI b)) 1/2) (+ (pow a 3) (pow b 3))) (* (cbrt (- b a)) (cbrt (- b a))))

prune1.2s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0b

localize31.0ms

Local error

Found 4 expressions with local error:

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

rewrite20.0ms

Algorithm
rewrite-expression-head
Rules
42×times-frac
25×add-cube-cbrt
25×*-un-lft-identity
25×add-sqr-sqrt
12×associate-/r/
11×add-exp-log
11×add-cbrt-cube
10×associate-/l*
8×div-inv
6×associate-/r*
5×associate-/l/
5×flip-+
5×flip3-+
4×add-log-exp
4×log1p-expm1-u
4×associate-*l/
4×associate-*r*
4×div-exp
4×cbrt-undiv
4×pow1
4×expm1-log1p-u
3×frac-2neg
3×clear-num
2×flip--
2×frac-sub
2×un-div-inv
2×associate-*r/
2×flip3--
1×*-commutative
Counts
4 → 113
Calls
4 calls:
Slowest
10.0ms
(/ (/ (* (- (* PI (/ 1 a)) (/ PI b)) 1/2) (+ a b)) (- b a))
8.0ms
(/ (* (- (* PI (/ 1 a)) (/ PI b)) 1/2) (+ a b))
1.0ms
(* PI (/ 1 a))
0.0ms
(/ PI b)

series245.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
92.0ms
(/ (/ (* (- (* PI (/ 1 a)) (/ PI b)) 1/2) (+ a b)) (- b a))
67.0ms
(/ (* (- (* PI (/ 1 a)) (/ PI b)) 1/2) (+ a b))
48.0ms
(* PI (/ 1 a))
38.0ms
(/ PI b)

simplify10.7s

Counts
106 → 125
Calls
106 calls:
Slowest
584.0ms
(/ (sqrt (/ (* (- (* PI (/ 1 a)) (/ PI b)) 1/2) (+ a b))) (cbrt (- b a)))
331.0ms
(* (- (pow (* PI (/ 1 a)) 3) (pow (/ PI b) 3)) 1/2)
311.0ms
(- (- (log (* (- (* PI (/ 1 a)) (/ PI b)) 1/2)) (log (+ a b))) (log (- b a)))
294.0ms
(/ (/ (* (- (* PI (/ 1 a)) (/ PI b)) 1/2) (+ a b)) (- b a))
285.0ms
(/ (/ (* (- (* PI (/ 1 a)) (/ PI b)) 1/2) (+ (pow a 3) (pow b 3))) 1)

prune1.2s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0b

regimes214.0ms

Accuracy

0% (0.3b remaining)

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

bsearch4.0ms

end0.0ms

sample2.1s

Algorithm
intervals
Results
9556×(pre true 80)
9556×(body real 80)