Average Error: 16.6 → 4.0
Time: 55.8s
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \log \left(e^{\cos^{-1} \left(\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right) + \sin \phi_2 \cdot \sin \phi_1\right)}\right)\]

Error

Bits error versus R

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 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. Initial simplification16.6

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

  4. Applied cos-diff4.0

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

  6. Applied add-log-exp4.0

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

  7. Taylor expanded around inf 4.0

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

  8. Simplified4.0

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

  9. Final simplification4.0

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

Reproduce

herbie shell --seed 2018360 
(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: 25.5s)Debug log

start855.0ms

Algorithm
intervals

setup106.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 18.6b

localize61.0ms

Local error

Found 4 expressions with local error:

3.3b
(cos (- lambda1 lambda2))
2.4b
(acos (+ (* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))
0.2b
(* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2)))
0.2b
(* R (acos (+ (* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

10.0ms
(* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2)))
3.0ms
(cos (- lambda1 lambda2))
1.0ms
(acos (+ (* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))

series198.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

107.0ms
(* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2)))
46.0ms
(* R (acos (+ (* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))
33.0ms
(cos (- lambda1 lambda2))
11.0ms
(acos (+ (* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))

simplify949.0ms

Counts
27 → 60
Calls

27 calls. Slowest were:

259.0ms
(* (* (* (cos (- lambda1 lambda2)) (cos (- lambda1 lambda2))) (cos (- lambda1 lambda2))) (* (* (* (cos phi1) (cos phi1)) (cos phi1)) (* (* (cos phi2) (cos phi2)) (cos phi2))))
240.0ms
(* (* (* (cos (- lambda1 lambda2)) (cos (- lambda1 lambda2))) (cos (- lambda1 lambda2))) (* (* (* (cos phi1) (cos phi2)) (* (cos phi1) (cos phi2))) (* (cos phi1) (cos phi2))))
55.0ms
(- (+ 1 (* lambda2 lambda1)) (* 1/2 (pow lambda1 2)))

prune1.3s

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 4.6b

localize71.0ms

Local error

Found 4 expressions with local error:

2.4b
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))
0.2b
(* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2)))
0.2b
(* R (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))
0.2b
(* (sin lambda1) (sin lambda2))

rewrite53.0ms

Algorithm
rewrite-expression-head
Counts
4 → 62
Calls

4 calls. Slowest were:

35.0ms
(* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2)))
11.0ms
(* (sin lambda1) (sin lambda2))
3.0ms
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))

series317.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

190.0ms
(* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2)))
61.0ms
(* (sin lambda1) (sin lambda2))
37.0ms
(* R (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))
29.0ms
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))

simplify3.9s

Counts
42 → 74
Calls

42 calls. Slowest were:

411.0ms
(* (- (* (* (cos lambda1) (cos lambda2)) (* (cos lambda1) (cos lambda2))) (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2)))) (+ (cos (+ phi1 phi2)) (cos (- phi1 phi2))))
288.0ms
(* (+ (pow (* (cos lambda1) (cos lambda2)) 3) (pow (* (sin lambda1) (sin lambda2)) 3)) (+ (cos (+ phi1 phi2)) (cos (- phi1 phi2))))
282.0ms
(* (+ (* (+ (cos (+ lambda1 lambda2)) (cos (- lambda1 lambda2))) 2) (* 2 (- (cos (- lambda1 lambda2)) (cos (+ lambda1 lambda2))))) (+ (cos (+ phi1 phi2)) (cos (- phi1 phi2))))

prune1.9s

Pruning

15 alts after pruning (15 fresh and 0 done)

Merged error: 4.5b

localize12.0ms

Local error

Found 4 expressions with local error:

2.4b
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))
0.6b
(log (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))
0.3b
(exp (log (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))))
0.2b
(* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2)))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

22.0ms
(* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2)))
2.0ms
(exp (log (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))))
2.0ms
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))

series347.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

227.0ms
(* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2)))
74.0ms
(log (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))
30.0ms
(exp (log (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))))
16.0ms
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))

simplify5.2s

Counts
42 → 72
Calls

42 calls. Slowest were:

722.0ms
(* (+ (pow (* (cos lambda1) (cos lambda2)) 3) (pow (* (sin lambda1) (sin lambda2)) 3)) (+ (cos (+ phi1 phi2)) (cos (- phi1 phi2))))
372.0ms
(* (- (* (* (cos lambda1) (cos lambda2)) (* (cos lambda1) (cos lambda2))) (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2)))) (+ (cos (+ phi1 phi2)) (cos (- phi1 phi2))))
246.0ms
(* (+ (* (+ (cos (+ lambda1 lambda2)) (cos (- lambda1 lambda2))) 2) (* 2 (- (cos (- lambda1 lambda2)) (cos (+ lambda1 lambda2))))) (+ (cos (+ phi1 phi2)) (cos (- phi1 phi2))))

prune2.2s

Pruning

17 alts after pruning (17 fresh and 0 done)

Merged error: 4.5b

localize21.0ms

Local error

Found 4 expressions with local error:

2.4b
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))
1.4b
(log (exp (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))))
0.4b
(exp (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))
0.2b
(* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2)))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

19.0ms
(* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2)))
5.0ms
(exp (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))
4.0ms
(log (exp (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))))

series313.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

226.0ms
(* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2)))
53.0ms
(exp (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))
19.0ms
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))
15.0ms
(log (exp (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))))

simplify4.5s

Counts
43 → 73
Calls

43 calls. Slowest were:

363.0ms
(* (- (* (* (cos lambda1) (cos lambda2)) (* (cos lambda1) (cos lambda2))) (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2)))) (+ (cos (+ phi1 phi2)) (cos (- phi1 phi2))))
296.0ms
(* (+ (pow (* (cos lambda1) (cos lambda2)) 3) (pow (* (sin lambda1) (sin lambda2)) 3)) (+ (cos (+ phi1 phi2)) (cos (- phi1 phi2))))
254.0ms
(* (+ (* (+ (cos (+ lambda1 lambda2)) (cos (- lambda1 lambda2))) 2) (* 2 (- (cos (- lambda1 lambda2)) (cos (+ lambda1 lambda2))))) (+ (cos (+ phi1 phi2)) (cos (- phi1 phi2))))

prune1.8s

Pruning

19 alts after pruning (19 fresh and 0 done)

Merged error: 4.5b

regimes1.3s

Accuracy

0% (0.4b remaining)

Error of 4.0b against oracle of 3.7b and baseline of 4.0b

bsearch3.0ms