Average Error: 16.5 → 4.0
Time: 53.2s
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\]

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 16.5

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

  2. Initial simplification16.5

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

  4. Applied cos-diff3.9

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

  6. Applied acos-asin4.0

    \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\]

  7. Final simplification4.0

    \[\leadsto R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\]

Reproduce

herbie shell --seed 2018362 
(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: 20.8s)Debug log

start1.4s

Algorithm
intervals

setup42.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 15.4b

localize75.0ms

Local error

Found 4 expressions with local error:

2.2b
(cos (- lambda1 lambda2))
1.0b
(acos (+ (* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))
0.2b
(* R (acos (+ (* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))
0.2b
(* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2)))

rewrite34.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

20.0ms
(* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2)))
6.0ms
(cos (- lambda1 lambda2))
3.0ms
(acos (+ (* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))

series257.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

158.0ms
(* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2)))
44.0ms
(* R (acos (+ (* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))
35.0ms
(cos (- lambda1 lambda2))
20.0ms
(acos (+ (* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))

simplify1.0s

Counts
27 → 60
Calls

27 calls. Slowest were:

284.0ms
(* (* (* (cos (- lambda1 lambda2)) (cos (- lambda1 lambda2))) (cos (- lambda1 lambda2))) (* (* (* (cos phi1) (cos phi1)) (cos phi1)) (* (* (cos phi2) (cos phi2)) (cos phi2))))
266.0ms
(* (* (* (cos (- lambda1 lambda2)) (cos (- lambda1 lambda2))) (cos (- lambda1 lambda2))) (* (* (* (cos phi1) (cos phi2)) (* (cos phi1) (cos phi2))) (* (cos phi1) (cos phi2))))
82.0ms
(- (+ 1 (* lambda2 lambda1)) (* 1/2 (pow lambda1 2)))

prune1.3s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 4.3b

localize65.0ms

Local error

Found 4 expressions with local error:

1.0b
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))
0.3b
(* R (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))
0.2b
(* (sin lambda1) (sin lambda2))
0.2b
(* (sin phi2) (sin phi1))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

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

series186.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

62.0ms
(* (sin phi2) (sin phi1))
54.0ms
(* (sin lambda1) (sin lambda2))
53.0ms
(* R (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))
17.0ms
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))

simplify1.5s

Counts
33 → 67
Calls

33 calls. Slowest were:

203.0ms
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))
192.0ms
(sqrt (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))
173.0ms
(cbrt (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))

prune1.7s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 4.2b

localize37.0ms

Local error

Found 4 expressions with local error:

26.6b
(cbrt (* (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin lambda1) (sin lambda2))))
1.0b
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (cbrt (* (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin lambda1) (sin lambda2))))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))
0.3b
(* R (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (cbrt (* (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin lambda1) (sin lambda2))))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))
0.2b
(* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2)))

rewrite58.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

30.0ms
(cbrt (* (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin lambda1) (sin lambda2))))
20.0ms
(* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2)))
1.0ms
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (cbrt (* (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin lambda1) (sin lambda2))))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))

series259.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

79.0ms
(* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2)))
74.0ms
(cbrt (* (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin lambda1) (sin lambda2))))
72.0ms
(* R (acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (cbrt (* (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin lambda1) (sin lambda2))))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1)))))
33.0ms
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (cbrt (* (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin lambda1) (sin lambda2))))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))

simplify6.5s

Counts
51 → 80
Calls

51 calls. Slowest were:

707.0ms
(* (* (* (* (sin lambda1) (sin lambda1)) (sin lambda1)) (* (* (sin lambda2) (sin lambda2)) (sin lambda2))) (* (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin lambda1) (sin lambda2))))
684.0ms
(* (* (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin lambda1) (sin lambda2))) (* (* (* (sin lambda1) (sin lambda1)) (sin lambda1)) (* (* (sin lambda2) (sin lambda2)) (sin lambda2))))
549.0ms
(* (* (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin lambda1) (sin lambda2))) (* (* (* (sin lambda1) (sin lambda2)) (* (sin lambda1) (sin lambda2))) (* (sin lambda1) (sin lambda2))))

prune2.2s

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 4.2b

localize49.0ms

Local error

Found 4 expressions with local error:

1.0b
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (* (cbrt (sin lambda1)) (cbrt (sin lambda1))) (* (cbrt (sin lambda1)) (sin lambda2)))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))
0.6b
(cbrt (sin lambda1))
0.6b
(cbrt (sin lambda1))
0.6b
(cbrt (sin lambda1))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
4 → 41
Calls

4 calls. Slowest were:

1.0ms
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (* (cbrt (sin lambda1)) (cbrt (sin lambda1))) (* (cbrt (sin lambda1)) (sin lambda2)))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))
0.0ms
(cbrt (sin lambda1))
0.0ms
(cbrt (sin lambda1))

series550.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

193.0ms
(cbrt (sin lambda1))
189.0ms
(cbrt (sin lambda1))
147.0ms
(cbrt (sin lambda1))
20.0ms
(acos (+ (* (+ (* (cos lambda1) (cos lambda2)) (* (* (cbrt (sin lambda1)) (cbrt (sin lambda1))) (* (cbrt (sin lambda1)) (sin lambda2)))) (* (cos phi1) (cos phi2))) (* (sin phi2) (sin phi1))))

simplify629.0ms

Counts
21 → 53
Calls

21 calls. Slowest were:

133.0ms
(acos (+ (* (cos phi1) (* (cos phi2) (* (sin lambda1) (sin lambda2)))) (+ (* (cos phi1) (* (cos phi2) (* (cos lambda1) (cos lambda2)))) (* (sin phi1) (sin phi2)))))
127.0ms
(acos (+ (* (cos phi1) (* (cos phi2) (* (sin lambda1) (sin lambda2)))) (+ (* (cos phi1) (* (cos phi2) (* (cos lambda1) (cos lambda2)))) (* (sin phi1) (sin phi2)))))
100.0ms
(acos (+ (* (cos phi1) (* (cos phi2) (* (cos lambda1) (cos lambda2)))) (+ (* (cos phi1) (* (cos phi2) (* (sin lambda2) (sin lambda1)))) (* (sin phi1) (sin phi2)))))

prune1.4s

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 4.2b

regimes1.5s

Accuracy

0% (0.4b remaining)

Error of 4.0b against oracle of 3.6b and baseline of 4.0b

bsearch5.0ms