Average Error: 0 → 0
Time: 7.6m
Precision: 64
Internal Precision: 128
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1} + \lambda_1\]

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. Final simplification0

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

Reproduce

herbie shell --seed 2019010 
(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: 7.6m)Debug log

sample18.2s

Algorithm
intervals

simplify12.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
11.0ms
(+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize39.0ms

Local error

Found 4 expressions with local error:

7.1b
(sin (- lambda1 lambda2))
6.9b
(cos (- lambda1 lambda2))
0.5b
(+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))
0.2b
(* (cos phi2) (sin (- lambda1 lambda2)))

rewrite31.0ms

Algorithm
rewrite-expression-head
Rules
6×add-log-exp
6×add-cube-cbrt
6×add-exp-log
6×sin-sum
6×add-cbrt-cube
6×*-un-lft-identity
6×pow1
6×add-sqr-sqrt
4×sub-neg
3×associate-*l*
3×associate-*r*
3×distribute-lft-in
3×cos-diff
3×distribute-rgt-in
2×associate-+r+
2×cos-sum
1×flip-+
1×cbrt-unprod
1×*-commutative
1×sin-diff
1×prod-exp
1×pow-prod-down
1×flip3-+
1×sum-log
1×+-commutative
Counts
4 → 54
Calls
4 calls:
Slowest
21.0ms
(* (cos phi2) (sin (- lambda1 lambda2)))
5.0ms
(+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))
2.0ms
(sin (- lambda1 lambda2))
2.0ms
(cos (- lambda1 lambda2))

series259.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
140.0ms
(+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))
65.0ms
(* (cos phi2) (sin (- lambda1 lambda2)))
30.0ms
(sin (- lambda1 lambda2))
24.0ms
(cos (- lambda1 lambda2))

simplify595.0ms

Counts
33 → 66
Calls
33 calls:
Slowest
137.0ms
(- 2 (+ (* 1/2 (pow phi2 2)) (* 1/2 (pow phi1 2))))
78.0ms
(- lambda1 (+ lambda2 (* 1/6 (pow lambda1 3))))
71.0ms
(* (* (* (cos phi2) (cos phi2)) (cos phi2)) (* (* (sin (- lambda1 lambda2)) (sin (- lambda1 lambda2))) (sin (- lambda1 lambda2))))
50.0ms
(- (+ 1 (* lambda2 lambda1)) (* 1/2 (pow lambda1 2)))
42.0ms
(* (* (cos lambda1) (sin (- lambda2))) (cos phi2))

prune1.0s

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample7.3m

Algorithm
intervals