Destination given bearing on a great circle

Average Error: 0.2 → 0.1

Time: 29.5s

Precision: binary64

Cost: 65408

Math

\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]

↓

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta - \frac{\sin \left(\phi_1 + \phi_1\right) \cdot \left(\sin delta \cdot \cos theta\right)}{2}} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin theta) (sin delta)) (cos phi1))
   (-
    (cos delta)
    (*
     (sin phi1)
     (sin
      (asin
       (+
        (* (sin phi1) (cos delta))
        (* (* (cos phi1) (sin delta)) (cos theta))))))))))

↓

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi1) (* (sin theta) (sin delta)))
   (-
    (* (* (cos phi1) (cos phi1)) (cos delta))
    (/ (* (sin (+ phi1 phi1)) (* (sin delta) (cos theta))) 2.0)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))));
}

↓

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), (((cos(phi1) * cos(phi1)) * cos(delta)) - ((sin((phi1 + phi1)) * (sin(delta) * cos(theta))) / 2.0)));
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))))
end function

↓

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), (((cos(phi1) * cos(phi1)) * cos(delta)) - ((sin((phi1 + phi1)) * (sin(delta) * cos(theta))) / 2.0d0)))
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + Math.atan2(((Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1)), (Math.cos(delta) - (Math.sin(phi1) * Math.sin(Math.asin(((Math.sin(phi1) * Math.cos(delta)) + ((Math.cos(phi1) * Math.sin(delta)) * Math.cos(theta))))))));
}

↓

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), (((Math.cos(phi1) * Math.cos(phi1)) * Math.cos(delta)) - ((Math.sin((phi1 + phi1)) * (Math.sin(delta) * Math.cos(theta))) / 2.0)));
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2(((math.sin(theta) * math.sin(delta)) * math.cos(phi1)), (math.cos(delta) - (math.sin(phi1) * math.sin(math.asin(((math.sin(phi1) * math.cos(delta)) + ((math.cos(phi1) * math.sin(delta)) * math.cos(theta))))))))

↓

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), (((math.cos(phi1) * math.cos(phi1)) * math.cos(delta)) - ((math.sin((phi1 + phi1)) * (math.sin(delta) * math.cos(theta))) / 2.0)))

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(sin(phi1) * cos(delta)) + Float64(Float64(cos(phi1) * sin(delta)) * cos(theta)))))))))
end

↓

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), Float64(Float64(Float64(cos(phi1) * cos(phi1)) * cos(delta)) - Float64(Float64(sin(Float64(phi1 + phi1)) * Float64(sin(delta) * cos(theta))) / 2.0))))
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))));
end

↓

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), (((cos(phi1) * cos(phi1)) * cos(delta)) - ((sin((phi1 + phi1)) * (sin(delta) * cos(theta))) / 2.0)));
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.2

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]

2. Taylor expanded in phi1 around inf 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \color{blue}{\left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \sin \phi_1}} \]

3. Applied egg-rr 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \color{blue}{\mathsf{fma}\left(\cos delta \cdot \sin \phi_1, \sin \phi_1, \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right) \cdot \sin \phi_1\right)}} \]

4. Taylor expanded in delta around inf 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos delta - \left(\sin \phi_1 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) + \cos delta \cdot {\sin \phi_1}^{2}\right)}} \]

5. Simplified 0.1

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta - \left(\sin \phi_1 \cdot \cos \phi_1\right) \cdot \left(\cos theta \cdot \sin delta\right)}} \]
   Proof
   (-.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi1)) (cos.f64 delta)) (*.f64 (*.f64 (sin.f64 phi1) (cos.f64 phi1)) (*.f64 (cos.f64 theta) (sin.f64 delta)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (Rewrite<= 1-sub-sin_binary64 (-.f64 1 (*.f64 (sin.f64 phi1) (sin.f64 phi1)))) (cos.f64 delta)) (*.f64 (*.f64 (sin.f64 phi1) (cos.f64 phi1)) (*.f64 (cos.f64 theta) (sin.f64 delta)))): 35 points increase in error, 38 points decrease in error
   (-.f64 (*.f64 (-.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 phi1) 2))) (cos.f64 delta)) (*.f64 (*.f64 (sin.f64 phi1) (cos.f64 phi1)) (*.f64 (cos.f64 theta) (sin.f64 delta)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (Rewrite=> sub-neg_binary64 (+.f64 1 (neg.f64 (pow.f64 (sin.f64 phi1) 2)))) (cos.f64 delta)) (*.f64 (*.f64 (sin.f64 phi1) (cos.f64 phi1)) (*.f64 (cos.f64 theta) (sin.f64 delta)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 (pow.f64 (sin.f64 phi1) 2)) 1)) (cos.f64 delta)) (*.f64 (*.f64 (sin.f64 phi1) (cos.f64 phi1)) (*.f64 (cos.f64 theta) (sin.f64 delta)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (cos.f64 delta) (*.f64 (neg.f64 (pow.f64 (sin.f64 phi1) 2)) (cos.f64 delta)))) (*.f64 (*.f64 (sin.f64 phi1) (cos.f64 phi1)) (*.f64 (cos.f64 theta) (sin.f64 delta)))): 12 points increase in error, 11 points decrease in error
   (-.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (cos.f64 delta) (*.f64 (pow.f64 (sin.f64 phi1) 2) (cos.f64 delta)))) (*.f64 (*.f64 (sin.f64 phi1) (cos.f64 phi1)) (*.f64 (cos.f64 theta) (sin.f64 delta)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (cos.f64 delta) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2)))) (*.f64 (*.f64 (sin.f64 phi1) (cos.f64 phi1)) (*.f64 (cos.f64 theta) (sin.f64 delta)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (cos.f64 delta) (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (sin.f64 phi1) (cos.f64 phi1)) (cos.f64 theta)) (sin.f64 delta)))): 3 points increase in error, 1 points decrease in error
   (-.f64 (-.f64 (cos.f64 delta) (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2))) (*.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (sin.f64 phi1) (*.f64 (cos.f64 phi1) (cos.f64 theta)))) (sin.f64 delta))): 2 points increase in error, 2 points decrease in error
   (-.f64 (-.f64 (cos.f64 delta) (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2))) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 delta) (*.f64 (sin.f64 phi1) (*.f64 (cos.f64 phi1) (cos.f64 theta)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--r+_binary64 (-.f64 (cos.f64 delta) (+.f64 (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2)) (*.f64 (sin.f64 delta) (*.f64 (sin.f64 phi1) (*.f64 (cos.f64 phi1) (cos.f64 theta))))))): 16 points increase in error, 16 points decrease in error
   (-.f64 (cos.f64 delta) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 (sin.f64 delta) (*.f64 (sin.f64 phi1) (*.f64 (cos.f64 phi1) (cos.f64 theta)))) (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (cos.f64 delta) (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 (sin.f64 phi1) (*.f64 (cos.f64 phi1) (cos.f64 theta))) (sin.f64 delta))) (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (cos.f64 delta) (+.f64 (Rewrite=> associate-*l*_binary64 (*.f64 (sin.f64 phi1) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 theta)) (sin.f64 delta)))) (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2)))): 5 points increase in error, 2 points decrease in error
   (-.f64 (cos.f64 delta) (+.f64 (*.f64 (sin.f64 phi1) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 delta) (*.f64 (cos.f64 phi1) (cos.f64 theta))))) (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2)))): 0 points increase in error, 0 points decrease in error

6. Applied egg-rr 0.1

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta - \color{blue}{\frac{\left(0 + \sin \left(\phi_1 + \phi_1\right)\right) \cdot \left(\cos theta \cdot \sin delta\right)}{2}}} \]

7. Final simplification 0.1

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta - \frac{\sin \left(\phi_1 + \phi_1\right) \cdot \left(\sin delta \cdot \cos theta\right)}{2}} \]

Alternatives

Alternative 1
Error	3.5
Cost	65152

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta - \sin \phi_1 \cdot \left(\sin delta \cdot \cos \phi_1\right)} \]

Alternative 2
Error	5.0
Cost	51904

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sqrt[3]{{\sin \phi_1}^{6}}} \]

Alternative 3
Error	5.0
Cost	45504

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - {\sin \phi_1}^{2}} \]

Alternative 4
Error	5.5
Cost	45444

\[\begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ \mathbf{if}\;delta \leq -10457042708176.582:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin theta \cdot \cos \phi_1\right)\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 2.973062714492113 \cdot 10^{-19}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \end{array} \]

Alternative 5
Error	5.5
Cost	39240

\[\begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \mathbf{if}\;delta \leq -10457042708176.582:\\ \;\;\;\;t_2\\ \mathbf{elif}\;delta \leq 2.973062714492113 \cdot 10^{-19}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	7.5
Cost	32512

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta} \]

Alternative 7
Error	8.9
Cost	25984

\[\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta} \]

Alternative 8
Error	13.6
Cost	19848

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{if}\;delta \leq -5.171663110881808 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;delta \leq 2.8044676031109692 \cdot 10^{-21}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	16.7
Cost	19584

\[\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta} \]

Alternative 10
Error	19.3
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.69803528859953 \cdot 10^{-200}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\lambda_1 \leq 2.7011053857671446 \cdot 10^{-112}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 11
Error	19.6
Cost	64

\[\lambda_1 \]

Reproduce

herbie shell --seed 2022318 
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))