Average Error: 16.6 → 3.7
Time: 56.3s
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\log \left(e^{(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*}\right)\right) \cdot R\]

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 16.6

    \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]

  2. Simplified16.6

    \[\leadsto \color{blue}{R \cdot \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)}\]
  3. Using strategy rm

  4. Applied cos-diff3.6

    \[\leadsto R \cdot \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)} + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\]
  5. Using strategy rm

  6. Applied add-log-exp3.7

    \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\log \left(e^{(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*}\right)\right)}\]

  7. Final simplification3.7

    \[\leadsto \cos^{-1} \left(\log \left(e^{(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*}\right)\right) \cdot R\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))

Details

Time bar (total: 51.7s)Debug log

sample825.0ms

Algorithm
intervals

simplify47.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

46.0ms
(* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R)

prune36.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 18.1b

localize75.0ms

Local error

Found 4 expressions with local error:

4.4b
(cos (- lambda1 lambda2))
0.9b
(acos (fma (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)) (* (sin phi2) (sin phi1))))
0.3b
(* R (acos (fma (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)) (* (sin phi2) (sin phi1)))))
0.3b
(fma (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)) (* (sin phi2) (sin phi1)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 45
Calls

4 calls. Slowest were:

3.0ms
(cos (- lambda1 lambda2))
1.0ms
(* R (acos (fma (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)) (* (sin phi2) (sin phi1)))))
1.0ms
(acos (fma (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)) (* (sin phi2) (sin phi1))))

series265.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

190.0ms
(fma (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)) (* (sin phi2) (sin phi1)))
36.0ms
(* R (acos (fma (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)) (* (sin phi2) (sin phi1)))))
21.0ms
(cos (- lambda1 lambda2))
17.0ms
(acos (fma (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)) (* (sin phi2) (sin phi1))))

simplify523.0ms

Counts
18 → 57
Calls

18 calls. Slowest were:

100.0ms
(- (+ (* phi1 phi2) 1) (* 1/2 (pow phi1 2)))
69.0ms
(- (+ 1 (* lambda2 lambda1)) (* 1/2 (pow lambda1 2)))
40.0ms
(* R (acos (fma (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)) (* (sin phi1) (sin phi2)))))

prune1.1s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 3.0b

localize44.0ms

Local error

Found 4 expressions with local error:

0.8b
(acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1))))
0.3b
(* R (acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1)))))
0.3b
(fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1)))
0.2b
(* (sin phi2) (sin phi1))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 53
Calls

4 calls. Slowest were:

7.0ms
(* (sin phi2) (sin phi1))
2.0ms
(* R (acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1)))))
1.0ms
(acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1))))

series380.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

260.0ms
(fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1)))
57.0ms
(* (sin phi2) (sin phi1))
42.0ms
(* R (acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1)))))
20.0ms
(acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1))))

simplify2.8s

Counts
24 → 65
Calls

24 calls. Slowest were:

312.0ms
(acos (fma (* (cos phi1) (cos phi2)) (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))) (* (sin phi1) (sin phi2))))
278.0ms
(+ (* (cos phi1) (* (cos phi2) (* (sin lambda1) (sin lambda2)))) (+ (* (cos phi1) (* (cos phi2) (* (cos lambda1) (cos lambda2)))) (* (sin phi1) (sin phi2))))
254.0ms
(* (acos (fma (* (cos phi1) (cos phi2)) (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))) (* (sin phi1) (sin phi2)))) R)

prune1.7s

Pruning

16 alts after pruning (16 fresh and 0 done)

Merged error: 2.8b

localize22.0ms

Local error

Found 4 expressions with local error:

0.8b
(acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1))))
0.7b
(log (exp (acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1))))))
0.4b
(exp (acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1)))))
0.3b
(* R (log (exp (acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1)))))))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

7.0ms
(* R (log (exp (acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1)))))))
4.0ms
(acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1))))
4.0ms
(log (exp (acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1))))))

series152.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

61.0ms
(exp (acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1)))))
50.0ms
(* R (log (exp (acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1)))))))
21.0ms
(log (exp (acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1))))))
20.0ms
(acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1))))

simplify8.5s

Counts
38 → 77
Calls

38 calls. Slowest were:

644.0ms
(exp (acos (fma (* (cos phi1) (cos phi2)) (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))) (* (sin phi1) (sin phi2)))))
311.0ms
(log (exp (acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1))))))
301.0ms
(exp (sqrt (acos (fma (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin phi2) (sin phi1))))))

prune2.2s

Pruning

19 alts after pruning (19 fresh and 0 done)

Merged error: 2.8b

localize27.0ms

Local error

Found 4 expressions with local error:

0.8b
(acos (+ (* (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2)))) (* (sin phi2) (sin phi1))))
0.3b
(* R (acos (+ (* (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2)))) (* (sin phi2) (sin phi1)))))
0.2b
(* (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))))
0.2b
(* (sin phi2) (sin phi1))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

20.0ms
(* (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))))
6.0ms
(* (sin phi2) (sin phi1))
2.0ms
(acos (+ (* (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2)))) (* (sin phi2) (sin phi1))))

series331.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

199.0ms
(* (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))))
63.0ms
(* R (acos (+ (* (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2)))) (* (sin phi2) (sin phi1)))))
54.0ms
(* (sin phi2) (sin phi1))
16.0ms
(acos (+ (* (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2)))) (* (sin phi2) (sin phi1))))

simplify5.4s

Counts
46 → 84
Calls

46 calls. Slowest were:

442.0ms
(* (+ (cos (+ phi1 phi2)) (cos (- phi1 phi2))) (+ (* (+ (cos (+ lambda1 lambda2)) (cos (- lambda1 lambda2))) 2) (* 2 (- (cos (- lambda1 lambda2)) (cos (+ lambda1 lambda2))))))
386.0ms
(* 2 (+ (* (* (cos lambda1) (cos lambda2)) (* (cos lambda1) (cos lambda2))) (- (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2))) (* (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))))))
286.0ms
(sqrt (acos (+ (* (* (cos phi1) (cos phi2)) (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2)))) (* (sin phi2) (sin phi1)))))

prune2.0s

Pruning

17 alts after pruning (17 fresh and 0 done)

Merged error: 2.8b

regimes1.3s

Accuracy

0% (0.5b remaining)

Error of 3.7b against oracle of 3.2b and baseline of 3.7b

bsearch3.0ms

end0.0ms

sample23.9s

Algorithm
intervals