Destination given bearing on a great circle

Percentage Accurate: 99.7% → 99.8%

Time: 23.1s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

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))))))))))

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))))))));
}

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

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))))))));
}

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))))))))

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

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

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]

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

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))))))))))

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))))))));
}

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

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))))))));
}

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))))))))

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

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

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]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right), -\sin \phi_1, \cos delta\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Sin[delta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision]) + N[Cos[delta], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. +-commutative 99.7%

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

   3. sin-asin 99.7%

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

   4. *-commutative 99.7%

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

   5. distribute-rgt-neg-in 99.7%

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

   6. fma-def 99.7%

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

3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \mathsf{fma}\left(\sin delta \cdot \cos theta, \cos \phi_1, \cos delta \cdot \sin \phi_1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sin-asin 99.7%

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

   2. +-commutative 99.7%

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

   3. associate-*l* 99.7%

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

   4. *-commutative 99.7%

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

   5. fma-def 99.7%

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

   6. *-commutative 99.7%

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

3. Applied egg-rr 99.7%

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

4. Taylor expanded in theta around inf 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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(delta) * (sin(theta) * cos(phi1))), (cos(delta) - (sin(phi1) * ((sin(delta) * (cos(phi1) * cos(theta))) + (cos(delta) * sin(phi1))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sin-asin 99.7%

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

   2. +-commutative 99.7%

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

   3. associate-*l* 99.7%

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

   4. *-commutative 99.7%

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

   5. fma-def 99.7%

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

   6. *-commutative 99.7%

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

3. Applied egg-rr 99.7%

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

4. Taylor expanded in theta around inf 99.7%

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

5. Taylor expanded in delta around inf 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 94.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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(delta) * (sin(theta) * cos(phi1))), (cos(delta) - (sin(phi1) * ((cos(delta) * sin(phi1)) + (sin(delta) * cos(phi1))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sin-asin 99.7%

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

   2. +-commutative 99.7%

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

   3. associate-*l* 99.7%

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

   4. *-commutative 99.7%

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

   5. fma-def 99.7%

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

   6. *-commutative 99.7%

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

3. Applied egg-rr 99.7%

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

4. Taylor expanded in theta around inf 99.7%

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

5. Taylor expanded in theta around 0 94.6%

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

6. Final simplification 94.6%

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

Alternative 5: 91.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ \mathbf{if}\;delta \leq -3.4:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.000108:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos delta\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin theta) (sin delta)))))
   (if (<= delta -3.4)
     (+ lambda1 (atan2 t_1 (cos delta)))
     (if (<= delta 0.000108)
       (+ lambda1 (atan2 t_1 (pow (cos phi1) 2.0)))
       (+ lambda1 (atan2 t_1 (expm1 (log1p (cos delta)))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(theta) * sin(delta));
	double tmp;
	if (delta <= -3.4) {
		tmp = lambda1 + atan2(t_1, cos(delta));
	} else if (delta <= 0.000108) {
		tmp = lambda1 + atan2(t_1, pow(cos(phi1), 2.0));
	} else {
		tmp = lambda1 + atan2(t_1, expm1(log1p(cos(delta))));
	}
	return tmp;
}

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta));
	double tmp;
	if (delta <= -3.4) {
		tmp = lambda1 + Math.atan2(t_1, Math.cos(delta));
	} else if (delta <= 0.000108) {
		tmp = lambda1 + Math.atan2(t_1, Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = lambda1 + Math.atan2(t_1, Math.expm1(Math.log1p(Math.cos(delta))));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.cos(phi1) * (math.sin(theta) * math.sin(delta))
	tmp = 0
	if delta <= -3.4:
		tmp = lambda1 + math.atan2(t_1, math.cos(delta))
	elif delta <= 0.000108:
		tmp = lambda1 + math.atan2(t_1, math.pow(math.cos(phi1), 2.0))
	else:
		tmp = lambda1 + math.atan2(t_1, math.expm1(math.log1p(math.cos(delta))))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(theta) * sin(delta)))
	tmp = 0.0
	if (delta <= -3.4)
		tmp = Float64(lambda1 + atan(t_1, cos(delta)));
	elseif (delta <= 0.000108)
		tmp = Float64(lambda1 + atan(t_1, (cos(phi1) ^ 2.0)));
	else
		tmp = Float64(lambda1 + atan(t_1, expm1(log1p(cos(delta)))));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -3.4], N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 0.000108], N[(lambda1 + N[ArcTan[t$95$1 / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(Exp[N[Log[1 + N[Cos[delta], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if delta < -3.39999999999999991

   1. Initial program 99.8%

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

   2. Taylor expanded in phi1 around 0 88.6%

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

   if -3.39999999999999991 < delta < 1.08e-4

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. sin-asin 99.6%

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

      4. *-commutative 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. fma-def 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in delta around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. sub-neg 99.2%

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

      3. unpow2 99.2%

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

      4. 1-sub-sin 99.4%

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

      5. unpow2 99.4%

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

   6. Simplified 99.4%

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

   if 1.08e-4 < delta

   1. Initial program 99.6%

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

      1. expm1-log1p-u 99.6%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}} \]

      2. sub-neg 99.6%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos delta + \left(-\sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)\right)} \]

      3. +-commutative 99.6%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(-\sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \cos delta}\right)\right)} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\sin \phi_1, -\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right), \cos delta\right)\right)\right)}} \]

   4. Taylor expanded in phi1 around 0 78.1%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\color{blue}{\log \left(\cos delta + 1\right)}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 78.1%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\log \color{blue}{\left(1 + \cos delta\right)}\right)} \]

      2. log1p-def 78.2%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\cos delta\right)}\right)} \]

   6. Simplified 78.2%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\cos delta\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -3.4:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.000108:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos delta\right)\right)}\\ \end{array} \]

Alternative 6: 92.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(delta) - pow(sin(phi1), 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((cos(phi1) * (sin(theta) * sin(delta))), (cos(delta) - (sin(phi1) ** 2.0d0)))
end function

Java

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(delta) - Math.pow(Math.sin(phi1), 2.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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[Cos[delta], $MachinePrecision] - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in delta around 0 92.2%

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

3. Final simplification 92.2%

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

Alternative 7: 91.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ \mathbf{if}\;delta \leq -3.4 \lor \neg \left(delta \leq 9.5 \cdot 10^{-5}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{{\cos \phi_1}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin theta) (sin delta)))))
   (if (or (<= delta -3.4) (not (<= delta 9.5e-5)))
     (+ lambda1 (atan2 t_1 (cos delta)))
     (+ lambda1 (atan2 t_1 (pow (cos phi1) 2.0))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(theta) * sin(delta));
	double tmp;
	if ((delta <= -3.4) || !(delta <= 9.5e-5)) {
		tmp = lambda1 + atan2(t_1, cos(delta));
	} else {
		tmp = lambda1 + atan2(t_1, pow(cos(phi1), 2.0));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(phi1) * (sin(theta) * sin(delta))
    if ((delta <= (-3.4d0)) .or. (.not. (delta <= 9.5d-5))) then
        tmp = lambda1 + atan2(t_1, cos(delta))
    else
        tmp = lambda1 + atan2(t_1, (cos(phi1) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta));
	double tmp;
	if ((delta <= -3.4) || !(delta <= 9.5e-5)) {
		tmp = lambda1 + Math.atan2(t_1, Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2(t_1, Math.pow(Math.cos(phi1), 2.0));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.cos(phi1) * (math.sin(theta) * math.sin(delta))
	tmp = 0
	if (delta <= -3.4) or not (delta <= 9.5e-5):
		tmp = lambda1 + math.atan2(t_1, math.cos(delta))
	else:
		tmp = lambda1 + math.atan2(t_1, math.pow(math.cos(phi1), 2.0))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(theta) * sin(delta)))
	tmp = 0.0
	if ((delta <= -3.4) || !(delta <= 9.5e-5))
		tmp = Float64(lambda1 + atan(t_1, cos(delta)));
	else
		tmp = Float64(lambda1 + atan(t_1, (cos(phi1) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = cos(phi1) * (sin(theta) * sin(delta));
	tmp = 0.0;
	if ((delta <= -3.4) || ~((delta <= 9.5e-5)))
		tmp = lambda1 + atan2(t_1, cos(delta));
	else
		tmp = lambda1 + atan2(t_1, (cos(phi1) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[delta, -3.4], N[Not[LessEqual[delta, 9.5e-5]], $MachinePrecision]], N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if delta < -3.39999999999999991 or 9.5000000000000005e-5 < delta

   1. Initial program 99.7%

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

   2. Taylor expanded in phi1 around 0 83.4%

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

   if -3.39999999999999991 < delta < 9.5000000000000005e-5

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. sin-asin 99.6%

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

      4. *-commutative 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. fma-def 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in delta around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. sub-neg 99.2%

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

      3. unpow2 99.2%

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

      4. 1-sub-sin 99.4%

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

      5. unpow2 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -3.4 \lor \neg \left(delta \leq 9.5 \cdot 10^{-5}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{{\cos \phi_1}^{2}}\\ \end{array} \]

Alternative 8: 91.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ \mathbf{if}\;delta \leq -3.4 \lor \neg \left(delta \leq 0.00024\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin theta) (sin delta)))))
   (if (or (<= delta -3.4) (not (<= delta 0.00024)))
     (+ lambda1 (atan2 t_1 (cos delta)))
     (+ lambda1 (atan2 t_1 (+ 0.5 (* 0.5 (cos (+ phi1 phi1)))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(theta) * sin(delta));
	double tmp;
	if ((delta <= -3.4) || !(delta <= 0.00024)) {
		tmp = lambda1 + atan2(t_1, cos(delta));
	} else {
		tmp = lambda1 + atan2(t_1, (0.5 + (0.5 * cos((phi1 + phi1)))));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(phi1) * (sin(theta) * sin(delta))
    if ((delta <= (-3.4d0)) .or. (.not. (delta <= 0.00024d0))) then
        tmp = lambda1 + atan2(t_1, cos(delta))
    else
        tmp = lambda1 + atan2(t_1, (0.5d0 + (0.5d0 * cos((phi1 + phi1)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta));
	double tmp;
	if ((delta <= -3.4) || !(delta <= 0.00024)) {
		tmp = lambda1 + Math.atan2(t_1, Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2(t_1, (0.5 + (0.5 * Math.cos((phi1 + phi1)))));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.cos(phi1) * (math.sin(theta) * math.sin(delta))
	tmp = 0
	if (delta <= -3.4) or not (delta <= 0.00024):
		tmp = lambda1 + math.atan2(t_1, math.cos(delta))
	else:
		tmp = lambda1 + math.atan2(t_1, (0.5 + (0.5 * math.cos((phi1 + phi1)))))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(theta) * sin(delta)))
	tmp = 0.0
	if ((delta <= -3.4) || !(delta <= 0.00024))
		tmp = Float64(lambda1 + atan(t_1, cos(delta)));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(0.5 + Float64(0.5 * cos(Float64(phi1 + phi1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = cos(phi1) * (sin(theta) * sin(delta));
	tmp = 0.0;
	if ((delta <= -3.4) || ~((delta <= 0.00024)))
		tmp = lambda1 + atan2(t_1, cos(delta));
	else
		tmp = lambda1 + atan2(t_1, (0.5 + (0.5 * cos((phi1 + phi1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[delta, -3.4], N[Not[LessEqual[delta, 0.00024]], $MachinePrecision]], N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(0.5 + N[(0.5 * N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if delta < -3.39999999999999991 or 2.40000000000000006e-4 < delta

   1. Initial program 99.7%

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

   2. Taylor expanded in phi1 around 0 83.4%

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

   if -3.39999999999999991 < delta < 2.40000000000000006e-4

   1. Initial program 99.6%

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

      1. expm1-log1p-u 99.6%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}} \]

      2. sub-neg 99.6%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos delta + \left(-\sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)\right)} \]

      3. +-commutative 99.6%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(-\sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \cos delta}\right)\right)} \]

   3. Applied egg-rr 99.6%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\sin \phi_1, -\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right), \cos delta\right)\right)\right)}} \]

   4. Taylor expanded in delta around 0 99.1%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\color{blue}{\log \left(2 + -1 \cdot {\sin \phi_1}^{2}\right)}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 99.1%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\log \left(2 + \color{blue}{\left(-{\sin \phi_1}^{2}\right)}\right)\right)} \]

      2. unsub-neg 99.1%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\log \color{blue}{\left(2 - {\sin \phi_1}^{2}\right)}\right)} \]

   6. Simplified 99.1%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\color{blue}{\log \left(2 - {\sin \phi_1}^{2}\right)}\right)} \]
   7. Step-by-step derivation

      1. unpow2 99.1%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\log \left(2 - \color{blue}{\sin \phi_1 \cdot \sin \phi_1}\right)\right)} \]

      2. sin-mult 99.2%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\log \left(2 - \color{blue}{\frac{\cos \left(\phi_1 - \phi_1\right) - \cos \left(\phi_1 + \phi_1\right)}{2}}\right)\right)} \]

   8. Applied egg-rr 99.2%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\log \left(2 - \color{blue}{\frac{\cos \left(\phi_1 - \phi_1\right) - \cos \left(\phi_1 + \phi_1\right)}{2}}\right)\right)} \]
   9. Step-by-step derivation

      1. div-sub 99.2%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\log \left(2 - \color{blue}{\left(\frac{\cos \left(\phi_1 - \phi_1\right)}{2} - \frac{\cos \left(\phi_1 + \phi_1\right)}{2}\right)}\right)\right)} \]

      2. +-inverses 99.2%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\log \left(2 - \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(\phi_1 + \phi_1\right)}{2}\right)\right)\right)} \]

      3. cos-0 99.2%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\log \left(2 - \left(\frac{\color{blue}{1}}{2} - \frac{\cos \left(\phi_1 + \phi_1\right)}{2}\right)\right)\right)} \]

      4. metadata-eval 99.2%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\log \left(2 - \left(\color{blue}{0.5} - \frac{\cos \left(\phi_1 + \phi_1\right)}{2}\right)\right)\right)} \]

   10. Simplified 99.2%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{expm1}\left(\log \left(2 - \color{blue}{\left(0.5 - \frac{\cos \left(\phi_1 + \phi_1\right)}{2}\right)}\right)\right)} \]

   11. Taylor expanded in phi1 around inf 99.3%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{0.5 + 0.5 \cdot \cos \left(2 \cdot \phi_1\right)}} \]
   12. Step-by-step derivation

       1. count-2 99.3%

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{0.5 + 0.5 \cdot \cos \color{blue}{\left(\phi_1 + \phi_1\right)}} \]

   13. Simplified 99.3%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -3.4 \lor \neg \left(delta \leq 0.00024\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}\\ \end{array} \]

Alternative 9: 88.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

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((cos(phi1) * (sin(theta) * sin(delta))), cos(delta))
end function

Java

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(delta));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in phi1 around 0 87.5%

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

3. Final simplification 87.5%

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

Alternative 10: 85.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

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(delta))
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(delta));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in phi1 around 0 87.5%

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

3. Taylor expanded in phi1 around 0 85.7%

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

4. Final simplification 85.7%

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

Alternative 11: 80.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;theta \leq -2.3 \cdot 10^{+51} \lor \neg \left(theta \leq 0.037\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= theta -2.3e+51) (not (<= theta 0.037)))
   (+ lambda1 (atan2 (* (sin theta) (sin delta)) 1.0))
   (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((theta <= -2.3e+51) || !(theta <= 0.037)) {
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), 1.0);
	} else {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((theta <= (-2.3d+51)) .or. (.not. (theta <= 0.037d0))) then
        tmp = lambda1 + atan2((sin(theta) * sin(delta)), 1.0d0)
    else
        tmp = lambda1 + atan2((theta * sin(delta)), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((theta <= -2.3e+51) || !(theta <= 0.037)) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * Math.sin(delta)), 1.0);
	} else {
		tmp = lambda1 + Math.atan2((theta * Math.sin(delta)), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (theta <= -2.3e+51) or not (theta <= 0.037):
		tmp = lambda1 + math.atan2((math.sin(theta) * math.sin(delta)), 1.0)
	else:
		tmp = lambda1 + math.atan2((theta * math.sin(delta)), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((theta <= -2.3e+51) || !(theta <= 0.037))
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * sin(delta)), 1.0));
	else
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((theta <= -2.3e+51) || ~((theta <= 0.037)))
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), 1.0);
	else
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[theta, -2.3e+51], N[Not[LessEqual[theta, 0.037]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / 1.0], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if theta < -2.30000000000000005e51 or 0.0369999999999999982 < theta

   1. Initial program 99.5%

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

   2. Taylor expanded in phi1 around 0 83.9%

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

   3. Taylor expanded in phi1 around 0 81.8%

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

   4. Taylor expanded in delta around 0 74.0%

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

   if -2.30000000000000005e51 < theta < 0.0369999999999999982

   1. Initial program 99.8%

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

   2. Taylor expanded in phi1 around 0 90.3%

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

   3. Taylor expanded in phi1 around 0 88.8%

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

   4. Taylor expanded in theta around 0 87.6%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot theta}}{\cos delta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -2.3 \cdot 10^{+51} \lor \neg \left(theta \leq 0.037\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]

Alternative 12: 79.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;theta \leq -1.8 \cdot 10^{+51}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{elif}\;theta \leq 0.0069:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= theta -1.8e+51)
   (+ lambda1 (atan2 (* (sin theta) delta) (cos delta)))
   (if (<= theta 0.0069)
     (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))
     (+ lambda1 (atan2 (* (sin theta) (sin delta)) 1.0)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (theta <= -1.8e+51) {
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	} else if (theta <= 0.0069) {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	} else {
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), 1.0);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (theta <= (-1.8d+51)) then
        tmp = lambda1 + atan2((sin(theta) * delta), cos(delta))
    else if (theta <= 0.0069d0) then
        tmp = lambda1 + atan2((theta * sin(delta)), cos(delta))
    else
        tmp = lambda1 + atan2((sin(theta) * sin(delta)), 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (theta <= -1.8e+51) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * delta), Math.cos(delta));
	} else if (theta <= 0.0069) {
		tmp = lambda1 + Math.atan2((theta * Math.sin(delta)), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * Math.sin(delta)), 1.0);
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if theta <= -1.8e+51:
		tmp = lambda1 + math.atan2((math.sin(theta) * delta), math.cos(delta))
	elif theta <= 0.0069:
		tmp = lambda1 + math.atan2((theta * math.sin(delta)), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((math.sin(theta) * math.sin(delta)), 1.0)
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (theta <= -1.8e+51)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), cos(delta)));
	elseif (theta <= 0.0069)
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * sin(delta)), 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (theta <= -1.8e+51)
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	elseif (theta <= 0.0069)
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	else
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[theta, -1.8e+51], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[theta, 0.0069], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / 1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if theta < -1.80000000000000005e51

   1. Initial program 99.8%

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

   2. Taylor expanded in phi1 around 0 83.4%

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

   3. Taylor expanded in phi1 around 0 81.1%

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

   4. Taylor expanded in delta around 0 71.2%

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

   if -1.80000000000000005e51 < theta < 0.0068999999999999999

   1. Initial program 99.8%

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

   2. Taylor expanded in phi1 around 0 90.3%

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

   3. Taylor expanded in phi1 around 0 88.8%

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

   4. Taylor expanded in theta around 0 87.6%

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

   if 0.0068999999999999999 < theta

   1. Initial program 99.3%

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

   2. Taylor expanded in phi1 around 0 84.3%

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

   3. Taylor expanded in phi1 around 0 82.2%

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

   4. Taylor expanded in delta around 0 76.1%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -1.8 \cdot 10^{+51}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{elif}\;theta \leq 0.0069:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \end{array} \]

Alternative 13: 72.5% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

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((theta * sin(delta)), cos(delta))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in phi1 around 0 87.5%

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

3. Taylor expanded in phi1 around 0 85.7%

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

4. Taylor expanded in theta around 0 76.1%

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

5. Final simplification 76.1%

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

Alternative 14: 71.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{0}{\cos delta} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+ lambda1 (atan2 0.0 (cos delta))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2(0.0, cos(delta));
}

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(0.0d0, cos(delta))
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + Math.atan2(0.0, Math.cos(delta));
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2(0.0, math.cos(delta))

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(0.0, cos(delta)))
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2(0.0, cos(delta));
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[0.0 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in phi1 around 0 87.5%

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

3. Taylor expanded in phi1 around 0 85.7%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \sin theta}}{\cos delta} \]
4. Step-by-step derivation

   1. add-log-exp 78.3%

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

5. Applied egg-rr 78.3%

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

6. Taylor expanded in delta around 0 73.1%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\log \color{blue}{1}}{\cos delta} \]

7. Final simplification 73.1%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{0}{\cos delta} \]

Alternative 15: 69.5% accurate, 1320.0× speedup

Math

\[\begin{array}{l} \\ \lambda_1 \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta) :precision binary64 lambda1)

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1;
}

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
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return lambda1
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := lambda1

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in phi1 around 0 87.5%

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

3. Taylor expanded in lambda1 around inf 72.2%

   \[\leadsto \color{blue}{\lambda_1} \]

4. Final simplification 72.2%

   \[\leadsto \lambda_1 \]

Reproduce

herbie shell --seed 2023187 
(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))))))))))