Average Error: 0 → 0
Time: 8.6m
Precision: 64
Internal Precision: 128
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} + \lambda_1\]

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Initial simplification0

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

  3. Final simplification0

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

Reproduce

herbie shell --seed 2018362 
(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: 16.9s)Debug log

start14.8s

Algorithm
intervals

setup44.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize95.0ms

Local error

Found 4 expressions with local error:

7.9b
(sin (- lambda1 lambda2))
7.6b
(cos (- lambda1 lambda2))
0.6b
(+ (* (cos phi2) (cos (- lambda1 lambda2))) (cos phi1))
0.2b
(* (cos phi2) (sin (- lambda1 lambda2)))

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

11.0ms
(+ (* (cos phi2) (cos (- lambda1 lambda2))) (cos phi1))
9.0ms
(* (cos phi2) (sin (- lambda1 lambda2)))
5.0ms
(sin (- lambda1 lambda2))

series243.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

144.0ms
(+ (* (cos phi2) (cos (- lambda1 lambda2))) (cos phi1))
67.0ms
(* (cos phi2) (sin (- lambda1 lambda2)))
19.0ms
(sin (- lambda1 lambda2))
13.0ms
(cos (- lambda1 lambda2))

simplify650.0ms

Counts
33 → 66
Calls

33 calls. Slowest were:

154.0ms
(- 2 (+ (* 1/2 (pow lambda1 2)) (* 1/2 (pow phi2 2))))
74.0ms
(* (* (* (cos phi2) (cos phi2)) (cos phi2)) (* (* (sin (- lambda1 lambda2)) (sin (- lambda1 lambda2))) (sin (- lambda1 lambda2))))
68.0ms
(- lambda1 (+ lambda2 (* 1/6 (pow lambda1 3))))

prune1.1s

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b