Average Error: 37.0 → 3.8
Time: 22.9s
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 37.0

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

  2. Simplified3.8

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

  3. Taylor expanded around -inf 3.8

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

  4. Simplified3.8

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

  5. 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 2019010 +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: 20.6s)Debug log

sample217.0ms

Algorithm
intervals

simplify136.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
136.0ms
(* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2)))))

prune19.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 3.2b

localize38.0ms

Local error

Found 4 expressions with local error:

2.6b
(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))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×add-exp-log
6×associate-*l*
6×add-cbrt-cube
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-*r*
2×*-commutative
2×associate-*l/
1×flip--
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×flip3--
1×hypot-udef
Counts
4 → 53
Calls
4 calls:
Slowest
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))
0.0ms
(hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2))

series873.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
432.0ms
(* (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2)) R)
333.0ms
(hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2))) (- phi1 phi2))
84.0ms
(* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2)))
24.0ms
(cos (/ (+ phi2 phi1) 2))

simplify1.6s

Counts
24 → 65
Calls
24 calls:
Slowest
303.0ms
(+ (/ (* 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)))))))
239.0ms
(* (* (* (- lambda1 lambda2) (- lambda1 lambda2)) (- lambda1 lambda2)) (* (* (cos (/ (+ phi2 phi1) 2)) (cos (/ (+ phi2 phi1) 2))) (cos (/ (+ phi2 phi1) 2))))
196.0ms
(* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (/ (+ phi2 phi1) 2)))
194.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi2 2))))
136.0ms
(- (+ (* 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))))))

prune927.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 3.0b

localize37.0ms

Local error

Found 4 expressions with local error:

2.6b
(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))

rewrite42.0ms

Algorithm
rewrite-expression-head
Rules
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×add-exp-log
6×associate-*l*
6×add-cbrt-cube
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-*r*
2×*-commutative
2×associate-*l/
1×flip--
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×flip3--
1×hypot-udef
Counts
4 → 53
Calls
4 calls:
Slowest
26.0ms
(* (- lambda1 lambda2) (cos (* (+ phi1 phi2) 1/2)))
10.0ms
(cos (* (+ phi1 phi2) 1/2))
5.0ms
(* (hypot (* (- lambda1 lambda2) (cos (* (+ phi1 phi2) 1/2))) (- phi1 phi2)) R)
0.0ms
(hypot (* (- lambda1 lambda2) (cos (* (+ phi1 phi2) 1/2))) (- phi1 phi2))

series1.0s

Counts
4 → 12
Calls
4 calls:
Slowest
483.0ms
(* (hypot (* (- lambda1 lambda2) (cos (* (+ phi1 phi2) 1/2))) (- phi1 phi2)) R)
387.0ms
(hypot (* (- lambda1 lambda2) (cos (* (+ phi1 phi2) 1/2))) (- phi1 phi2))
106.0ms
(* (- lambda1 lambda2) (cos (* (+ phi1 phi2) 1/2)))
25.0ms
(cos (* (+ phi1 phi2) 1/2))

simplify1.3s

Counts
24 → 65
Calls
24 calls:
Slowest
196.0ms
(* (- (* lambda1 lambda1) (* lambda2 lambda2)) (cos (* (+ phi1 phi2) 1/2)))
184.0ms
(+ (/ (* 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)))))))
166.0ms
(* (* (* (- lambda1 lambda2) (- lambda1 lambda2)) (- lambda1 lambda2)) (* (* (cos (* (+ phi1 phi2) 1/2)) (cos (* (+ phi1 phi2) 1/2))) (cos (* (+ phi1 phi2) 1/2))))
158.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi1 2))))
114.0ms
(- (+ (* 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))))))

prune1.1s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 3.0b

localize16.0ms

Local error

Found 4 expressions with local error:

2.6b
(cos (* (+ phi1 phi2) 1/2))
0.5b
(expm1 (cos (* (+ phi1 phi2) 1/2)))
0.3b
(log1p (expm1 (cos (* (+ phi1 phi2) 1/2))))
0.2b
(* (- lambda1 lambda2) (log1p (expm1 (cos (* (+ phi1 phi2) 1/2)))))

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×add-exp-log
6×add-cbrt-cube
6×*-un-lft-identity
6×pow1
6×add-sqr-sqrt
5×log1p-expm1-u
4×add-log-exp
4×expm1-log1p-u
3×associate-*l*
3×associate-*r*
2×associate-*l/
1×expm1-udef
1×flip--
1×cbrt-unprod
1×*-commutative
1×log1p-expm1
1×prod-exp
1×pow-prod-down
1×flip3--
1×expm1-log1p
1×log1p-udef
Counts
4 → 52
Calls
4 calls:
Slowest
5.0ms
(cos (* (+ phi1 phi2) 1/2))
4.0ms
(* (- lambda1 lambda2) (log1p (expm1 (cos (* (+ phi1 phi2) 1/2)))))
0.0ms
(expm1 (cos (* (+ phi1 phi2) 1/2)))
0.0ms
(log1p (expm1 (cos (* (+ phi1 phi2) 1/2))))

series217.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
103.0ms
(* (- lambda1 lambda2) (log1p (expm1 (cos (* (+ phi1 phi2) 1/2)))))
49.0ms
(log1p (expm1 (cos (* (+ phi1 phi2) 1/2))))
44.0ms
(expm1 (cos (* (+ phi1 phi2) 1/2)))
20.0ms
(cos (* (+ phi1 phi2) 1/2))

simplify1.3s

Counts
24 → 64
Calls
24 calls:
Slowest
283.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi1 2))))
206.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi1 2))))
168.0ms
(* (* (* (- lambda1 lambda2) (- lambda1 lambda2)) (- lambda1 lambda2)) (* (* (log1p (expm1 (cos (* (+ phi1 phi2) 1/2)))) (log1p (expm1 (cos (* (+ phi1 phi2) 1/2))))) (log1p (expm1 (cos (* (+ phi1 phi2) 1/2))))))
164.0ms
(* (- (* lambda1 lambda1) (* lambda2 lambda2)) (log1p (expm1 (cos (* (+ phi1 phi2) 1/2)))))
148.0ms
(- E (+ (* 1/8 (* E (pow phi1 2))) (+ (* 1/4 (* E (* phi1 phi2))) 1)))

prune1.1s

Pruning

13 alts after pruning (11 fresh and 2 done)

Merged error: 3.0b

localize42.0ms

Local error

Found 4 expressions with local error:

2.6b
(cos (* (+ phi1 phi2) 1/2))
2.6b
(cos (* (+ phi1 phi2) 1/2))
2.6b
(cos (* (+ phi1 phi2) 1/2))
0.7b
(* (expm1 (cos (* (+ phi1 phi2) 1/2))) (expm1 (cos (* (+ phi1 phi2) 1/2))))

rewrite37.0ms

Algorithm
rewrite-expression-head
Rules
9×pow1
6×add-cube-cbrt
6×add-exp-log
6×add-cbrt-cube
6×*-un-lft-identity
6×add-sqr-sqrt
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-*l*
3×associate-*r*
1×cbrt-unprod
1×pow-prod-up
1×*-commutative
1×prod-exp
1×pow-prod-down
1×pow-plus
1×pow2
Counts
4 → 49
Calls
4 calls:
Slowest
9.0ms
(cos (* (+ phi1 phi2) 1/2))
9.0ms
(cos (* (+ phi1 phi2) 1/2))
9.0ms
(cos (* (+ phi1 phi2) 1/2))
7.0ms
(* (expm1 (cos (* (+ phi1 phi2) 1/2))) (expm1 (cos (* (+ phi1 phi2) 1/2))))

series143.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
72.0ms
(* (expm1 (cos (* (+ phi1 phi2) 1/2))) (expm1 (cos (* (+ phi1 phi2) 1/2))))
28.0ms
(cos (* (+ phi1 phi2) 1/2))
24.0ms
(cos (* (+ phi1 phi2) 1/2))
19.0ms
(cos (* (+ phi1 phi2) 1/2))

simplify1.2s

Counts
23 → 61
Calls
23 calls:
Slowest
267.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi1 2))))
241.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi1 2))))
239.0ms
(- 1 (+ (* 1/4 (* phi1 phi2)) (* 1/8 (pow phi1 2))))
131.0ms
(- (+ (pow E 2) (+ (* 1/2 (* E (* phi1 phi2))) (+ (* 1/4 (* E (pow phi1 2))) 1))) (+ (* 1/2 (* (pow E 2) (* phi1 phi2))) (+ (* 1/4 (* (pow E 2) (pow phi1 2))) (* 2 E))))
50.0ms
(* (* (* (expm1 (cos (* (+ phi1 phi2) 1/2))) (expm1 (cos (* (+ phi1 phi2) 1/2)))) (expm1 (cos (* (+ phi1 phi2) 1/2)))) (* (* (expm1 (cos (* (+ phi1 phi2) 1/2))) (expm1 (cos (* (+ phi1 phi2) 1/2)))) (expm1 (cos (* (+ phi1 phi2) 1/2)))))

prune1.3s

Pruning

14 alts after pruning (11 fresh and 3 done)

Merged error: 3.0b

regimes814.0ms

Accuracy

0% (0.2b remaining)

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

bsearch2.0ms

end0.0ms

sample7.2s

Algorithm
intervals