Average Error: 37.1 → 3.8
Time: 23.7s
Precision: 64
Internal Precision: 128
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

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 37.1

    \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

  2. Simplified3.8

    \[\leadsto \color{blue}{\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R}\]
  3. Using strategy rm

  4. Applied add-log-exp3.9

    \[\leadsto \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)}\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

  5. Taylor expanded around -inf 3.8

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

  6. Simplified3.8

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

  7. Final simplification3.8

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

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))

Details

Time bar (total: 21.3s)Debug log

sample187.0ms

Algorithm
intervals

simplify179.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

179.0ms
(* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2)))))

prune27.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 4.0b

localize34.0ms

Local error

Found 4 expressions with local error:

3.3b
(cos (/ (+ phi2 phi1) 2))
0.2b
(* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))
0.1b
(* (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2)) R)
0.0b
(hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 53
Calls

4 calls. Slowest were:

5.0ms
(* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))
2.0ms
(* (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2)) R)
2.0ms
(cos (/ (+ phi2 phi1) 2))

series987.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

472.0ms
(* (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2)) R)
402.0ms
(hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2))
86.0ms
(* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))
27.0ms
(cos (/ (+ phi2 phi1) 2))

simplify1.6s

Counts
24 → 65
Calls

24 calls. Slowest were:

282.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi2 2))))
274.0ms
(+ (/ (* lambda2 (* phi1 phi2)) (* (pow lambda1 2) (cos (* 1/2 (+ phi1 phi2))))) (+ (* lambda2 (cos (* 1/2 (+ phi1 phi2)))) (/ (* phi1 phi2) (* lambda1 (cos (* 1/2 (+ phi1 phi2)))))))
213.0ms
(* (* (* (- lambda1 lambda2) (- lambda1 lambda2)) (- lambda1 lambda2)) (* (* (cos (/ (+ phi2 phi1) 2)) (cos (/ (+ phi2 phi1) 2))) (cos (/ (+ phi2 phi1) 2))))

prune973.0ms

Pruning

15 alts after pruning (15 fresh and 0 done)

Merged error: 3.8b

localize36.0ms

Local error

Found 4 expressions with local error:

3.3b
(cos (/ (+ phi2 phi1) 2))
3.3b
(cos (/ (+ phi2 phi1) 2))
3.3b
(cos (/ (+ phi2 phi1) 2))
0.5b
(cbrt (* (* (cos (/ (+ phi2 phi1) 2)) (cos (/ (+ phi2 phi1) 2))) (cos (/ (+ phi2 phi1) 2))))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 40
Calls

4 calls. Slowest were:

15.0ms
(cbrt (* (* (cos (/ (+ phi2 phi1) 2)) (cos (/ (+ phi2 phi1) 2))) (cos (/ (+ phi2 phi1) 2))))
4.0ms
(cos (/ (+ phi2 phi1) 2))
4.0ms
(cos (/ (+ phi2 phi1) 2))

series70.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

22.0ms
(cos (/ (+ phi2 phi1) 2))
17.0ms
(cbrt (* (* (cos (/ (+ phi2 phi1) 2)) (cos (/ (+ phi2 phi1) 2))) (cos (/ (+ phi2 phi1) 2))))
16.0ms
(cos (/ (+ phi2 phi1) 2))
15.0ms
(cos (/ (+ phi2 phi1) 2))

simplify1.1s

Counts
15 → 52
Calls

15 calls. Slowest were:

278.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi2 2))))
225.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi2 2))))
224.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi2 2))))

prune929.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 3.8b

localize16.0ms

Local error

Found 4 expressions with local error:

3.3b
(cos (/ (+ phi2 phi1) 2))
0.5b
(log (exp (cos (/ (+ phi2 phi1) 2))))
0.2b
(* (- lambda1 lambda2) (log (exp (cos (/ (+ phi2 phi1) 2)))))
0.1b
(exp (cos (/ (+ phi2 phi1) 2)))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

12.0ms
(* (- lambda1 lambda2) (log (exp (cos (/ (+ phi2 phi1) 2)))))
4.0ms
(exp (cos (/ (+ phi2 phi1) 2)))
4.0ms
(log (exp (cos (/ (+ phi2 phi1) 2))))

series142.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

76.0ms
(* (- lambda1 lambda2) (log (exp (cos (/ (+ phi2 phi1) 2)))))
36.0ms
(exp (cos (/ (+ phi2 phi1) 2)))
15.0ms
(cos (/ (+ phi2 phi1) 2))
15.0ms
(log (exp (cos (/ (+ phi2 phi1) 2))))

simplify1.4s

Counts
43 → 82
Calls

43 calls. Slowest were:

201.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi2 2))))
182.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi2 2))))
174.0ms
(* (* (* (- lambda1 lambda2) (- lambda1 lambda2)) (- lambda1 lambda2)) (* (* (log (exp (cos (/ (+ phi2 phi1) 2)))) (log (exp (cos (/ (+ phi2 phi1) 2))))) (log (exp (cos (/ (+ phi2 phi1) 2))))))

prune1.6s

Pruning

16 alts after pruning (15 fresh and 1 done)

Merged error: 3.8b

localize17.0ms

Local error

Found 4 expressions with local error:

3.3b
(cos (* (+ phi1 phi2) 1/2))
0.2b
(* (- lambda1 lambda2) (cos (* (+ phi1 phi2) 1/2)))
0.1b
(* (hypot (* (- lambda1 lambda2) (cos (* (+ phi1 phi2) 1/2))) (- phi1 phi2)) R)
0.0b
(hypot (* (- lambda1 lambda2) (cos (* (+ phi1 phi2) 1/2))) (- phi1 phi2))

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 53
Calls

4 calls. Slowest were:

22.0ms
(* (- lambda1 lambda2) (cos (* (+ phi1 phi2) 1/2)))
9.0ms
(cos (* (+ phi1 phi2) 1/2))
5.0ms
(* (hypot (* (- lambda1 lambda2) (cos (* (+ phi1 phi2) 1/2))) (- phi1 phi2)) R)

series949.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

438.0ms
(* (hypot (* (- lambda1 lambda2) (cos (* (+ phi1 phi2) 1/2))) (- phi1 phi2)) R)
407.0ms
(hypot (* (- lambda1 lambda2) (cos (* (+ phi1 phi2) 1/2))) (- phi1 phi2))
83.0ms
(* (- lambda1 lambda2) (cos (* (+ phi1 phi2) 1/2)))
21.0ms
(cos (* (+ phi1 phi2) 1/2))

simplify1.7s

Counts
24 → 65
Calls

24 calls. Slowest were:

318.0ms
(+ (/ (* lambda2 (* phi1 phi2)) (* (pow lambda1 2) (cos (* 1/2 (+ phi1 phi2))))) (+ (* lambda2 (cos (* 1/2 (+ phi1 phi2)))) (/ (* phi1 phi2) (* lambda1 (cos (* 1/2 (+ phi1 phi2)))))))
212.0ms
(* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (* (+ phi1 phi2) 1/2)))
212.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi1 2))))

prune1.0s

Pruning

16 alts after pruning (14 fresh and 2 done)

Merged error: 3.8b

regimes985.0ms

Accuracy

0% (0.1b remaining)

Error of 3.8b against oracle of 3.6b and baseline of 3.8b

bsearch2.0ms

end0.0ms

sample7.3s

Algorithm
intervals