Average Error: 37.0 → 0.1
Time: 1.2m
Precision: 64
Internal Precision: 128
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) - \log \left(e^{\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

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 37.0

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

  2. Simplified3.6

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

  3. Taylor expanded around -inf 3.6

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

  5. Applied distribute-rgt-in3.6

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

  6. Applied cos-sum0.1

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

  8. Applied add-log-exp0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019004 +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: 1.2m)Debug log

sample191.0ms

Algorithm
intervals

simplify153.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

153.0ms
(* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2)))))

prune20.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 4.2b

localize49.0ms

Local error

Found 3 expressions with local error:

2.2b
(cos (/ (+ phi2 phi1) 2))
0.2b
(* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))
0.1b
(* (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2)) R)

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
3 → 43
Calls

3 calls. Slowest were:

10.0ms
(* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))
5.0ms
(* (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2)) R)
4.0ms
(cos (/ (+ phi2 phi1) 2))

series570.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

455.0ms
(* (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2)) R)
89.0ms
(* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))
26.0ms
(cos (/ (+ phi2 phi1) 2))

simplify1.3s

Counts
21 → 52
Calls

21 calls. Slowest were:

230.0ms
(* (* (* (- lambda1 lambda2) (- lambda1 lambda2)) (- lambda1 lambda2)) (* (* (cos (/ (+ phi2 phi1) 2)) (cos (/ (+ phi2 phi1) 2))) (cos (/ (+ phi2 phi1) 2))))
213.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi2 2))))
187.0ms
(- (+ (* lambda2 (* (cos (* 1/2 (+ phi1 phi2))) R)) (+ (/ (* phi1 (* R phi2)) (* (cos (* 1/2 (+ phi1 phi2))) lambda1)) (/ (* phi1 (* lambda2 (* R phi2))) (* (cos (* 1/2 (+ phi1 phi2))) (pow lambda1 2))))))

prune720.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 4.1b

localize20.0ms

Local error

Found 3 expressions with local error:

2.2b
(cos (* 1/2 (+ phi1 phi2)))
0.2b
(* (- lambda1 lambda2) (cos (* 1/2 (+ phi1 phi2))))
0.1b
(* (hypot (* (- lambda1 lambda2) (cos (* 1/2 (+ phi1 phi2)))) (- phi1 phi2)) R)

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
3 → 45
Calls

3 calls. Slowest were:

16.0ms
(* (- lambda1 lambda2) (cos (* 1/2 (+ phi1 phi2))))
6.0ms
(cos (* 1/2 (+ phi1 phi2)))
5.0ms
(* (hypot (* (- lambda1 lambda2) (cos (* 1/2 (+ phi1 phi2)))) (- phi1 phi2)) R)

series649.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

527.0ms
(* (hypot (* (- lambda1 lambda2) (cos (* 1/2 (+ phi1 phi2)))) (- phi1 phi2)) R)
92.0ms
(* (- lambda1 lambda2) (cos (* 1/2 (+ phi1 phi2))))
29.0ms
(cos (* 1/2 (+ phi1 phi2)))

simplify1.4s

Counts
23 → 54
Calls

23 calls. Slowest were:

269.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi1 2))))
206.0ms
(* (* (* (- lambda1 lambda2) (- lambda1 lambda2)) (- lambda1 lambda2)) (* (* (cos (* 1/2 (+ phi1 phi2))) (cos (* 1/2 (+ phi1 phi2)))) (cos (* 1/2 (+ phi1 phi2)))))
196.0ms
(* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (* 1/2 (+ phi1 phi2))))

prune797.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.1b

localize36.0ms

Local error

Found 4 expressions with local error:

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

rewrite86.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

45.0ms
(* (- lambda1 lambda2) (- (* (cos (* phi1 1/2)) (cos (* phi2 1/2))) (* (sin (* phi1 1/2)) (sin (* phi2 1/2)))))
27.0ms
(- (* (cos (* phi1 1/2)) (cos (* phi2 1/2))) (* (sin (* phi1 1/2)) (sin (* phi2 1/2))))
7.0ms
(* (sin (* phi1 1/2)) (sin (* phi2 1/2)))

series18.3s

Counts
4 → 12
Calls

4 calls. Slowest were:

17.8s
(* (hypot (* (- lambda1 lambda2) (- (* (cos (* phi1 1/2)) (cos (* phi2 1/2))) (* (sin (* phi1 1/2)) (sin (* phi2 1/2))))) (- phi1 phi2)) R)
307.0ms
(* (- lambda1 lambda2) (- (* (cos (* phi1 1/2)) (cos (* phi2 1/2))) (* (sin (* phi1 1/2)) (sin (* phi2 1/2)))))
104.0ms
(- (* (cos (* phi1 1/2)) (cos (* phi2 1/2))) (* (sin (* phi1 1/2)) (sin (* phi2 1/2))))
80.0ms
(* (sin (* phi1 1/2)) (sin (* phi2 1/2)))

simplify11.3s

Counts
65 → 102
Calls

65 calls. Slowest were:

697.0ms
(* (- lambda1 lambda2) (- (* (sin (* phi1 1/2)) (sin (* phi2 1/2)))))
649.0ms
(* (- (* lambda1 lambda1) (* lambda2 lambda2)) (- (+ (cos (+ (* phi1 1/2) (* phi2 1/2))) (cos (- (* phi1 1/2) (* phi2 1/2)))) (- (cos (- (* phi1 1/2) (* phi2 1/2))) (cos (+ (* phi1 1/2) (* phi2 1/2))))))
592.0ms
(* (fma (- (sin (* phi2 1/2))) (sin (* phi1 1/2)) (* (sin (* phi2 1/2)) (sin (* phi1 1/2)))) (- lambda1 lambda2))

prune2.3s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize19.0ms

Local error

Found 4 expressions with local error:

2.4b
(* (* (sin (* phi2 1/2)) (sin (* phi1 1/2))) (- lambda2 lambda1))
0.2b
(* (sin (* phi2 1/2)) (sin (* phi1 1/2)))
0.2b
(* (- lambda1 lambda2) (* (cos (* phi1 1/2)) (cos (* phi2 1/2))))
0.1b
(* (hypot (+ (* (- lambda1 lambda2) (* (cos (* phi1 1/2)) (cos (* phi2 1/2)))) (* (* (sin (* phi2 1/2)) (sin (* phi1 1/2))) (- lambda2 lambda1))) (- phi1 phi2)) R)

rewrite84.0ms

Algorithm
rewrite-expression-head
Counts
4 → 87
Calls

4 calls. Slowest were:

35.0ms
(* (* (sin (* phi2 1/2)) (sin (* phi1 1/2))) (- lambda2 lambda1))
29.0ms
(* (- lambda1 lambda2) (* (cos (* phi1 1/2)) (cos (* phi2 1/2))))
13.0ms
(* (sin (* phi2 1/2)) (sin (* phi1 1/2)))

series18.7s

Counts
4 → 12
Calls

4 calls. Slowest were:

18.3s
(* (hypot (+ (* (- lambda1 lambda2) (* (cos (* phi1 1/2)) (cos (* phi2 1/2)))) (* (* (sin (* phi2 1/2)) (sin (* phi1 1/2))) (- lambda2 lambda1))) (- phi1 phi2)) R)
205.0ms
(* (* (sin (* phi2 1/2)) (sin (* phi1 1/2))) (- lambda2 lambda1))
140.0ms
(* (- lambda1 lambda2) (* (cos (* phi1 1/2)) (cos (* phi2 1/2))))
50.0ms
(* (sin (* phi2 1/2)) (sin (* phi1 1/2)))

simplify5.6s

Counts
59 → 99
Calls

59 calls. Slowest were:

565.0ms
(* (* (* (- lambda1 lambda2) (- lambda1 lambda2)) (- lambda1 lambda2)) (* (* (* (cos (* phi1 1/2)) (cos (* phi1 1/2))) (cos (* phi1 1/2))) (* (* (cos (* phi2 1/2)) (cos (* phi2 1/2))) (cos (* phi2 1/2)))))
546.0ms
(* (* (* (- lambda1 lambda2) (- lambda1 lambda2)) (- lambda1 lambda2)) (* (* (* (cos (* phi1 1/2)) (cos (* phi2 1/2))) (* (cos (* phi1 1/2)) (cos (* phi2 1/2)))) (* (cos (* phi1 1/2)) (cos (* phi2 1/2)))))
469.0ms
(* (- (cos (- (* phi2 1/2) (* phi1 1/2))) (cos (+ (* phi2 1/2) (* phi1 1/2)))) (- lambda2 lambda1))

prune2.0s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

regimes809.0ms

Accuracy

0% (0.1b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.1b

bsearch2.0ms

end0.0ms

sample6.6s

Algorithm
intervals