Average Error: 0 → 0
Time: 8.1m
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 2019016 +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: 8.1m)Debug log

sample15.6s

Algorithm
intervals
Results
2204×(pre true 80)
1948×(body exit 10240)
196×(body real 1280)
46×(body real 640)
11×(body real 320)
3×(body real 160)

simplify9.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
8.0ms
(+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))

prune15.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize54.0ms

Local error

Found 4 expressions with local error:

7.9b
(cos (- lambda1 lambda2))
7.7b
(sin (- lambda1 lambda2))
0.6b
(fma (cos phi2) (cos (- lambda1 lambda2)) (cos phi1))
0.1b
(* (cos phi2) (sin (- lambda1 lambda2)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
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×add-log-exp
4×log1p-expm1-u
4×sub-neg
4×expm1-log1p-u
3×associate-*l*
3×associate-*r*
2×distribute-lft-in
2×cos-sum
2×distribute-rgt-in
1×fma-udef
1×cbrt-unprod
1×*-commutative
1×sin-diff
1×prod-exp
1×pow-prod-down
1×cos-diff
Counts
4 → 57
Calls
4 calls:
Slowest
6.0ms
(* (cos phi2) (sin (- lambda1 lambda2)))
2.0ms
(cos (- lambda1 lambda2))
2.0ms
(sin (- lambda1 lambda2))
0.0ms
(fma (cos phi2) (cos (- lambda1 lambda2)) (cos phi1))

series168.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
91.0ms
(fma (cos phi2) (cos (- lambda1 lambda2)) (cos phi1))
46.0ms
(* (cos phi2) (sin (- lambda1 lambda2)))
17.0ms
(sin (- lambda1 lambda2))
14.0ms
(cos (- lambda1 lambda2))

simplify561.0ms

Counts
30 → 69
Calls
30 calls:
Slowest
190.0ms
(- 2 (+ (* 1/2 (pow lambda1 2)) (* 1/2 (pow phi2 2))))
83.0ms
(- lambda1 (+ lambda2 (* 1/6 (pow lambda1 3))))
54.0ms
(* (* (* (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)))
29.0ms
(* (* (cos lambda1) (sin (- lambda2))) (cos phi2))

prune1.1s

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample7.8m

Algorithm
intervals
Results
68232×(pre true 80)
60231×(body exit 10240)
5402×(body real 1280)
2067×(body real 640)
416×(body real 320)
96×(body real 160)
20×(body real 80)