Average Error: 0.5 → 0.5
Time: 22.0s
Precision: 64
Internal Precision: 128
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{(a1 \cdot a1 + \left(a2 \cdot a2\right))_*}{\sqrt{2}} \cdot \cos th\]

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Derivation

  1. Initial program 0.5

    \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

  2. Simplified0.5

    \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot (a1 \cdot a1 + \left(a2 \cdot a2\right))_*}\]
  3. Using strategy rm

  4. Applied associate-*l/0.5

    \[\leadsto \color{blue}{\frac{\cos th \cdot (a1 \cdot a1 + \left(a2 \cdot a2\right))_*}{\sqrt{2}}}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity0.5

    \[\leadsto \frac{\cos th \cdot (a1 \cdot a1 + \left(a2 \cdot a2\right))_*}{\color{blue}{1 \cdot \sqrt{2}}}\]

  7. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{\cos th}{1} \cdot \frac{(a1 \cdot a1 + \left(a2 \cdot a2\right))_*}{\sqrt{2}}}\]

  8. Simplified0.5

    \[\leadsto \color{blue}{\cos th} \cdot \frac{(a1 \cdot a1 + \left(a2 \cdot a2\right))_*}{\sqrt{2}}\]

  9. Final simplification0.5

    \[\leadsto \frac{(a1 \cdot a1 + \left(a2 \cdot a2\right))_*}{\sqrt{2}} \cdot \cos th\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))

Details

Time bar (total: 20.9s)Debug log

sample218.0ms

Algorithm
intervals
Results
77.0ms77×body1280valid
59.0ms271×body80valid
40.0ms58×body640valid
19.0ms40×body320valid
5.0ms15×body160valid

simplify92.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
91.0ms
(+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize31.0ms

Local error

Found 3 expressions with local error:

0.7b
(/ (cos th) (sqrt 2))
0.4b
(* (/ (cos th) (sqrt 2)) (fma a1 a1 (* a2 a2)))
0.0b
(fma a1 a1 (* a2 a2))

rewrite18.0ms

Algorithm
rewrite-expression-head
Rules
19×add-cube-cbrt
19×*-un-lft-identity
19×add-sqr-sqrt
18×times-frac
13×associate-*l*
10×add-exp-log
10×add-cbrt-cube
5×pow1
3×add-log-exp
3×associate-/l*
3×log1p-expm1-u
3×fma-udef
3×associate-*r*
3×associate-/r*
3×expm1-log1p-u
2×div-inv
2×cbrt-unprod
2×prod-exp
2×div-exp
2×cbrt-undiv
1×*-commutative
1×associate-*l/
1×pow-prod-down
1×distribute-lft-in
1×frac-2neg
1×clear-num
1×distribute-rgt-in
Counts
3 → 73
Calls
3 calls:
Slowest
10.0ms
(* (/ (cos th) (sqrt 2)) (fma a1 a1 (* a2 a2)))
6.0ms
(/ (cos th) (sqrt 2))
0.0ms
(fma a1 a1 (* a2 a2))

series205.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
147.0ms
(* (/ (cos th) (sqrt 2)) (fma a1 a1 (* a2 a2)))
40.0ms
(/ (cos th) (sqrt 2))
18.0ms
(fma a1 a1 (* a2 a2))

simplify1.2s

Counts
57 → 82
Calls
57 calls:
Slowest
261.0ms
(/ (* (* (cos th) (cos th)) (cos th)) (* (* (sqrt 2) (sqrt 2)) (sqrt 2)))
168.0ms
(- (+ (/ 1 (sqrt 2)) (* 1/24 (/ (pow th 4) (sqrt 2)))) (* 1/2 (/ (pow th 2) (sqrt 2))))
111.0ms
(* (/ (* (* (cos th) (cos th)) (cos th)) (* (* (sqrt 2) (sqrt 2)) (sqrt 2))) (* (* (fma a1 a1 (* a2 a2)) (fma a1 a1 (* a2 a2))) (fma a1 a1 (* a2 a2))))
64.0ms
(* (* (* (/ (cos th) (sqrt 2)) (/ (cos th) (sqrt 2))) (/ (cos th) (sqrt 2))) (* (* (fma a1 a1 (* a2 a2)) (fma a1 a1 (* a2 a2))) (fma a1 a1 (* a2 a2))))
62.0ms
(+ (/ (* (pow a1 2) (cos th)) (sqrt 2)) (/ (* (cos th) (pow a2 2)) (sqrt 2)))

prune818.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 3 expressions with local error:

0.4b
(/ (* (cos th) (fma a1 a1 (* a2 a2))) (sqrt 2))
0.1b
(* (cos th) (fma a1 a1 (* a2 a2)))
0.0b
(fma a1 a1 (* a2 a2))

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
10×add-exp-log
10×add-cbrt-cube
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
5×pow1
3×add-log-exp
3×log1p-expm1-u
3×fma-udef
3×associate-*l*
3×associate-*r*
3×times-frac
3×associate-/r*
3×expm1-log1p-u
2×cbrt-unprod
2×prod-exp
2×div-exp
2×cbrt-undiv
1×associate-/l*
1×div-inv
1×*-commutative
1×pow-prod-down
1×distribute-lft-in
1×frac-2neg
1×clear-num
1×distribute-rgt-in
Counts
3 → 54
Calls
3 calls:
Slowest
14.0ms
(/ (* (cos th) (fma a1 a1 (* a2 a2))) (sqrt 2))
4.0ms
(* (cos th) (fma a1 a1 (* a2 a2)))
0.0ms
(fma a1 a1 (* a2 a2))

series305.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
154.0ms
(/ (* (cos th) (fma a1 a1 (* a2 a2))) (sqrt 2))
120.0ms
(* (cos th) (fma a1 a1 (* a2 a2)))
31.0ms
(fma a1 a1 (* a2 a2))

simplify1.1s

Counts
33 → 63
Calls
33 calls:
Slowest
471.0ms
(* (* (* (cos th) (cos th)) (cos th)) (* (* (fma a1 a1 (* a2 a2)) (fma a1 a1 (* a2 a2))) (fma a1 a1 (* a2 a2))))
99.0ms
(+ (/ (* (pow a1 2) (cos th)) (sqrt 2)) (/ (* (cos th) (pow a2 2)) (sqrt 2)))
84.0ms
(/ (* (* (* (cos th) (cos th)) (cos th)) (* (* (fma a1 a1 (* a2 a2)) (fma a1 a1 (* a2 a2))) (fma a1 a1 (* a2 a2)))) (* (* (sqrt 2) (sqrt 2)) (sqrt 2)))
77.0ms
(+ (/ (* (pow a1 2) (cos th)) (sqrt 2)) (/ (* (cos th) (pow a2 2)) (sqrt 2)))
66.0ms
(- (log (* (cos th) (fma a1 a1 (* a2 a2)))) (log (sqrt 2)))

prune843.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

localize5.0ms

Local error

Found 3 expressions with local error:

0.4b
(/ (fma a1 a1 (* a2 a2)) (sqrt 2))
0.1b
(* (cos th) (/ (fma a1 a1 (* a2 a2)) (sqrt 2)))
0.0b
(fma a1 a1 (* a2 a2))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
19×add-cube-cbrt
19×*-un-lft-identity
19×add-sqr-sqrt
18×times-frac
13×associate-*r*
10×add-exp-log
10×add-cbrt-cube
5×pow1
3×add-log-exp
3×associate-/l*
3×log1p-expm1-u
3×associate-*l*
3×associate-/r*
3×expm1-log1p-u
2×div-inv
2×cbrt-unprod
2×prod-exp
2×div-exp
2×cbrt-undiv
1×fma-udef
1×*-commutative
1×associate-*r/
1×pow-prod-down
1×frac-2neg
1×clear-num
Counts
3 → 71
Calls
3 calls:
Slowest
6.0ms
(* (cos th) (/ (fma a1 a1 (* a2 a2)) (sqrt 2)))
3.0ms
(/ (fma a1 a1 (* a2 a2)) (sqrt 2))
0.0ms
(fma a1 a1 (* a2 a2))

series295.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
197.0ms
(* (cos th) (/ (fma a1 a1 (* a2 a2)) (sqrt 2)))
79.0ms
(/ (fma a1 a1 (* a2 a2)) (sqrt 2))
19.0ms
(fma a1 a1 (* a2 a2))

simplify1.4s

Counts
55 → 80
Calls
55 calls:
Slowest
365.0ms
(* (* (* (cos th) (cos th)) (cos th)) (* (* (/ (fma a1 a1 (* a2 a2)) (sqrt 2)) (/ (fma a1 a1 (* a2 a2)) (sqrt 2))) (/ (fma a1 a1 (* a2 a2)) (sqrt 2))))
129.0ms
(/ (* (* (fma a1 a1 (* a2 a2)) (fma a1 a1 (* a2 a2))) (fma a1 a1 (* a2 a2))) (* (* (sqrt 2) (sqrt 2)) (sqrt 2)))
95.0ms
(+ (/ (* (pow a1 2) (cos th)) (sqrt 2)) (/ (* (cos th) (pow a2 2)) (sqrt 2)))
78.0ms
(* (* (* (cos th) (cos th)) (cos th)) (/ (* (* (fma a1 a1 (* a2 a2)) (fma a1 a1 (* a2 a2))) (fma a1 a1 (* a2 a2))) (* (* (sqrt 2) (sqrt 2)) (sqrt 2))))
74.0ms
(+ (/ (* (pow a1 2) (cos th)) (sqrt 2)) (/ (* (cos th) (pow a2 2)) (sqrt 2)))

prune908.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

localize25.0ms

Local error

Found 4 expressions with local error:

5.7b
(sqrt (fma a1 a1 (* a2 a2)))
0.4b
(/ (* (* (cos th) (sqrt (fma a1 a1 (* a2 a2)))) (hypot a1 a2)) (sqrt 2))
0.1b
(* (* (cos th) (sqrt (fma a1 a1 (* a2 a2)))) (hypot a1 a2))
0.1b
(* (cos th) (sqrt (fma a1 a1 (* a2 a2))))

rewrite60.0ms

Algorithm
rewrite-expression-head
Rules
20×add-exp-log
20×add-cbrt-cube
12×add-sqr-sqrt
11×add-cube-cbrt
11×*-un-lft-identity
11×pow1
9×associate-*r*
7×cbrt-unprod
7×prod-exp
6×sqrt-prod
4×add-log-exp
4×log1p-expm1-u
4×pow-prod-down
4×associate-*l*
4×expm1-log1p-u
3×div-exp
3×times-frac
3×associate-/r*
3×cbrt-undiv
2×*-commutative
1×associate-/l*
1×div-inv
1×hypot-def
1×fma-udef
1×pow1/2
1×frac-2neg
1×clear-num
1×rem-sqrt-square
Counts
4 → 82
Calls
4 calls:
Slowest
28.0ms
(/ (* (* (cos th) (sqrt (fma a1 a1 (* a2 a2)))) (hypot a1 a2)) (sqrt 2))
20.0ms
(* (* (cos th) (sqrt (fma a1 a1 (* a2 a2)))) (hypot a1 a2))
8.0ms
(* (cos th) (sqrt (fma a1 a1 (* a2 a2))))
2.0ms
(sqrt (fma a1 a1 (* a2 a2)))

series612.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
261.0ms
(/ (* (* (cos th) (sqrt (fma a1 a1 (* a2 a2)))) (hypot a1 a2)) (sqrt 2))
200.0ms
(* (* (cos th) (sqrt (fma a1 a1 (* a2 a2)))) (hypot a1 a2))
113.0ms
(* (cos th) (sqrt (fma a1 a1 (* a2 a2))))
36.0ms
(sqrt (fma a1 a1 (* a2 a2)))

simplify2.4s

Counts
53 → 94
Calls
53 calls:
Slowest
633.0ms
(* (* (* (* (cos th) (cos th)) (cos th)) (* (* (sqrt (fma a1 a1 (* a2 a2))) (sqrt (fma a1 a1 (* a2 a2)))) (sqrt (fma a1 a1 (* a2 a2))))) (* (* (hypot a1 a2) (hypot a1 a2)) (hypot a1 a2)))
236.0ms
(* (* (* (* (cos th) (sqrt (fma a1 a1 (* a2 a2)))) (* (cos th) (sqrt (fma a1 a1 (* a2 a2))))) (* (cos th) (sqrt (fma a1 a1 (* a2 a2))))) (* (* (hypot a1 a2) (hypot a1 a2)) (hypot a1 a2)))
220.0ms
(* (* (* (cos th) (cos th)) (cos th)) (* (* (sqrt (fma a1 a1 (* a2 a2))) (sqrt (fma a1 a1 (* a2 a2)))) (sqrt (fma a1 a1 (* a2 a2)))))
180.0ms
(- (log (* (* (cos th) (sqrt (fma a1 a1 (* a2 a2)))) (hypot a1 a2))) (log (sqrt 2)))
147.0ms
(- (+ (+ (log (cos th)) (log (sqrt (fma a1 a1 (* a2 a2))))) (log (hypot a1 a2))) (log (sqrt 2)))

prune1.2s

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0b

regimes364.0ms

Accuracy

0% (0.4b remaining)

Error of 0.5b against oracle of 0.1b and baseline of 0.5b

bsearch2.0ms

end0.0ms

sample8.7s

Algorithm
intervals
Results
2.9s2541×body1280valid
2.2s2152×body640valid
1.6s8102×body80valid
1.1s1033×body320valid
619.0ms485×body160valid