Average Error: 14.7 → 0.3
Time: 32.9s
Precision: 64
Internal Precision: 128
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{(\left(\frac{\pi}{2}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{1}{\frac{a}{\frac{\pi}{2}}}\right))_*}{a + b}}{b - a}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.7

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

  2. Simplified9.8

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

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

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

  7. Applied associate-/l*0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019022 +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: 32.0s)Debug log

sample42.0ms

Algorithm
intervals
Results
32.0ms305×body80valid

simplify308.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
308.0ms
(* (* (/ 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: 12.8b

localize447.0ms

Local error

Found 3 expressions with local error:

12.8b
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (* (+ a b) (- b a)))
0.3b
(/ (/ 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))

series307.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
163.0ms
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (* (+ a b) (- b a)))
73.0ms
(/ (/ PI 2) a)
70.0ms
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))

simplify2.1s

Counts
32 → 67
Calls
32 calls:
Slowest
407.0ms
(* (+ (* a a) (- (* b b) (* a b))) (+ (* b b) (+ (* a a) (* b a))))
259.0ms
(- (log (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))) (log (* (+ a b) (- b a))))
189.0ms
(* (- a b) (+ (* b b) (+ (* a a) (* b a))))
176.0ms
(/ (* (* (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) (+ a b)) (+ a b)) (* (* (- b a) (- b a)) (- b a))))
161.0ms
(/ (* (* (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))))

prune775.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.1b

localize6.0ms

Local error

Found 4 expressions with local error:

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

rewrite27.0ms

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
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)
0.0ms
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))

series403.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
131.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b)) (- b a))
100.0ms
(/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (+ a b))
95.0ms
(/ (/ PI 2) a)
76.0ms
(fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))

simplify9.7s

Counts
155 → 152
Calls
155 calls:
Slowest
431.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (- (* a a) (* b b))) (* (cbrt (- b a)) (cbrt (- b a))))
327.0ms
(/ (/ (* (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)
322.0ms
(/ (/ (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)) (- (* a a) (* b b))) 1)
305.0ms
(/ (/ (* (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))))
243.0ms
(/ (/ (* (cbrt (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a))) (cbrt (fma (/ PI 2) (/ -1 b) (/ (/ PI 2) a)))) 1) (* (cbrt (- b a)) (cbrt (- b a))))

prune1.6s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

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

rewrite29.0ms

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
18.0ms
(/ (/ (* (- (/ PI a) (/ PI b)) 1/2) (+ a b)) (- b a))
9.0ms
(/ (* (- (/ PI a) (/ PI b)) 1/2) (+ a b))
0.0ms
(/ PI a)
0.0ms
(/ PI b)

series235.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
77.0ms
(/ (/ (* (- (/ PI a) (/ PI b)) 1/2) (+ a b)) (- b a))
68.0ms
(/ (* (- (/ PI a) (/ PI b)) 1/2) (+ a b))
53.0ms
(/ PI b)
36.0ms
(/ PI a)

simplify7.5s

Counts
101 → 120
Calls
101 calls:
Slowest
331.0ms
(* (- (* (/ PI a) (/ PI a)) (* (/ PI b) (/ PI b))) 1/2)
311.0ms
(/ (/ (* (- (/ PI a) (/ PI b)) 1/2) (+ (pow a 3) (pow b 3))) (sqrt (- b a)))
298.0ms
(/ (/ (* (- (/ PI a) (/ PI b)) 1/2) (+ a b)) (sqrt (- b a)))
283.0ms
(/ (/ (* (- (/ PI a) (/ PI b)) 1/2) (+ (pow a 3) (pow b 3))) 1)
271.0ms
(/ (/ (* (- (/ PI a) (/ PI b)) 1/2) (+ (pow a 3) (pow b 3))) (* (cbrt (- b a)) (cbrt (- b a))))

prune1.1s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0b

localize17.0ms

Local error

Found 4 expressions with local error:

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
14×add-cube-cbrt
14×*-un-lft-identity
14×add-sqr-sqrt
12×associate-/r*
9×times-frac
7×add-exp-log
6×add-cbrt-cube
5×div-inv
5×associate-/r/
5×pow1
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-/l*
3×frac-2neg
3×clear-num
1×inv-pow
1×pow-flip
1×fma-udef
1×flip-+
1×div-exp
1×flip3-+
1×rec-exp
1×cbrt-undiv
Counts
4 → 79
Calls
4 calls:
Slowest
6.0ms
(/ (fma (/ PI 2) (/ -1 b) (/ 1 (/ a (/ PI 2)))) (+ a b))
1.0ms
(/ 1 (/ a (/ PI 2)))
1.0ms
(/ a (/ PI 2))
0.0ms
(fma (/ PI 2) (/ -1 b) (/ 1 (/ a (/ PI 2))))

series319.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
109.0ms
(/ a (/ PI 2))
84.0ms
(/ 1 (/ a (/ PI 2)))
77.0ms
(/ (fma (/ PI 2) (/ -1 b) (/ 1 (/ a (/ PI 2)))) (+ a b))
49.0ms
(fma (/ PI 2) (/ -1 b) (/ 1 (/ a (/ PI 2))))

simplify3.6s

Counts
49 → 91
Calls
49 calls:
Slowest
231.0ms
(/ (sqrt (fma (/ PI 2) (/ -1 b) (/ 1 (/ a (/ PI 2))))) (sqrt (+ a b)))
207.0ms
(/ (* (cbrt (fma (/ PI 2) (/ -1 b) (/ 1 (/ a (/ PI 2))))) (cbrt (fma (/ PI 2) (/ -1 b) (/ 1 (/ a (/ PI 2)))))) 1)
204.0ms
(/ (cbrt (fma (/ PI 2) (/ -1 b) (/ 1 (/ a (/ PI 2))))) (cbrt (+ a b)))
196.0ms
(* (cbrt (fma (/ PI 2) (/ -1 b) (/ 1 (/ a (/ PI 2))))) (cbrt (fma (/ PI 2) (/ -1 b) (/ 1 (/ a (/ PI 2))))))
183.0ms
(/ (sqrt (fma (/ PI 2) (/ -1 b) (/ 1 (/ a (/ PI 2))))) (+ a b))

prune918.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0b

regimes221.0ms

Accuracy

0% (0.3b remaining)

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

bsearch3.0ms

end0.0ms

sample2.3s

Algorithm
intervals
Results
2.2s9588×body80valid