Average Error: 0 → 0
Time: 7.9m
Precision: 64
Internal Precision: 128
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*}\]

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0

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

  2. Simplified0

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

  3. Final simplification0

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

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))

Details

Time bar (total: 7.9m)Debug log

sample15.0s

Algorithm
intervals

simplify16.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

15.0ms
(+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))

prune55.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize63.0ms

Local error

Found 4 expressions with local error:

6.4b
(sin (- lambda1 lambda2))
6.4b
(cos (- lambda1 lambda2))
0.6b
(fma (cos phi2) (cos (- lambda1 lambda2)) (cos phi1))
0.1b
(* (cos phi2) (sin (- lambda1 lambda2)))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

6.0ms
(* (cos phi2) (sin (- lambda1 lambda2)))
4.0ms
(sin (- lambda1 lambda2))
2.0ms
(cos (- lambda1 lambda2))

series175.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

86.0ms
(fma (cos phi2) (cos (- lambda1 lambda2)) (cos phi1))
49.0ms
(* (cos phi2) (sin (- lambda1 lambda2)))
21.0ms
(cos (- lambda1 lambda2))
19.0ms
(sin (- lambda1 lambda2))

simplify521.0ms

Counts
30 → 69
Calls

30 calls. Slowest were:

157.0ms
(- 2 (+ (* 1/2 (pow lambda1 2)) (* 1/2 (pow phi2 2))))
81.0ms
(- lambda1 (+ lambda2 (* 1/6 (pow lambda1 3))))
53.0ms
(* (* (* (cos phi2) (cos phi2)) (cos phi2)) (* (* (sin (- lambda1 lambda2)) (sin (- lambda1 lambda2))) (sin (- lambda1 lambda2))))

prune763.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample7.6m

Algorithm
intervals