Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 20.1s

Alternatives: 22

Speedup: 1.2×

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: 99.8% 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, 1.2× speedup

Math

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

FPCore

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

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), Float64(cos(delta) - Float64(sin(phi1) * fma(sin(phi1), cos(delta), Float64(cos(phi1) * Float64(sin(delta) * cos(theta))))))))
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[(N[Sin[phi1], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. sin-asin N/A

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

   12. *-commutative N/A

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 84.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin theta \cdot \sin delta\\ t_2 := \tan^{-1}_* \frac{t\_1}{\cos delta}\\ t_3 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot t\_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)}\\ \mathbf{if}\;t\_3 \leq -3:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{elif}\;t\_3 \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{-22}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1}\\ \mathbf{elif}\;t\_3 \leq 3.14:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin theta) (sin delta)))
        (t_2 (atan2 t_1 (cos delta)))
        (t_3
         (+
          lambda1
          (atan2
           (* (cos phi1) t_1)
           (-
            (cos delta)
            (*
             (sin phi1)
             (sin
              (asin
               (+
                (* (cos delta) (sin phi1))
                (* (cos theta) (* (sin delta) (cos phi1))))))))))))
   (if (<= t_3 -3.0)
     (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))
     (if (<= t_3 -0.05)
       t_2
       (if (<= t_3 1e-22)
         (+ lambda1 (atan2 t_1 1.0))
         (if (<= t_3 3.14)
           t_2
           (+ lambda1 (atan2 t_1 (fma delta (* delta -0.5) 1.0)))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(theta) * sin(delta);
	double t_2 = atan2(t_1, cos(delta));
	double t_3 = lambda1 + atan2((cos(phi1) * t_1), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (sin(delta) * cos(phi1)))))))));
	double tmp;
	if (t_3 <= -3.0) {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	} else if (t_3 <= -0.05) {
		tmp = t_2;
	} else if (t_3 <= 1e-22) {
		tmp = lambda1 + atan2(t_1, 1.0);
	} else if (t_3 <= 3.14) {
		tmp = t_2;
	} else {
		tmp = lambda1 + atan2(t_1, fma(delta, (delta * -0.5), 1.0));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(theta) * sin(delta))
	t_2 = atan(t_1, cos(delta))
	t_3 = Float64(lambda1 + atan(Float64(cos(phi1) * t_1), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(cos(delta) * sin(phi1)) + Float64(cos(theta) * Float64(sin(delta) * cos(phi1))))))))))
	tmp = 0.0
	if (t_3 <= -3.0)
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	elseif (t_3 <= -0.05)
		tmp = t_2;
	elseif (t_3 <= 1e-22)
		tmp = Float64(lambda1 + atan(t_1, 1.0));
	elseif (t_3 <= 3.14)
		tmp = t_2;
	else
		tmp = Float64(lambda1 + atan(t_1, fma(delta, Float64(delta * -0.5), 1.0)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -3.0], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.05], t$95$2, If[LessEqual[t$95$3, 1e-22], N[(lambda1 + N[ArcTan[t$95$1 / 1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 3.14], t$95$2, N[(lambda1 + N[ArcTan[t$95$1 / N[(delta * N[(delta * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < -3

   1. Initial program 100.0%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 98.3

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in theta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 98.3

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

   11. Applied rewrites 98.3%

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

   if -3 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < -0.050000000000000003 or 1e-22 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 3.14000000000000012

   1. Initial program 99.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 64.9

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

   8. Applied rewrites 64.9%

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

   9. Taylor expanded in lambda1 around 0

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

       1. lower-atan2.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-cos.f64 64.7

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

   11. Applied rewrites 64.7%

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

   if -0.050000000000000003 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 1e-22

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 78.3

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 75.3%

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

   if 3.14000000000000012 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 98.3

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in delta around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{1 + \frac{-1}{2} \cdot {delta}^{2}}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{\frac{-1}{2} \cdot {delta}^{2} + 1}} \]

       2. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{{delta}^{2} \cdot \frac{-1}{2}} + 1} \]

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 98.6

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

   11. Applied rewrites 98.6%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)} \leq -3:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)} \leq -0.05:\\ \;\;\;\;\tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)} \leq 10^{-22}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)} \leq 3.14:\\ \;\;\;\;\tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. sin-asin N/A

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

   12. *-commutative N/A

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

4. Applied rewrites 99.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.6

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 94.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \cos \phi_1\\ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(\sin \phi_1, \cos delta, t\_1\right), -\sin \phi_1, \cos delta\right)} \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (cos phi1))))
   (+
    lambda1
    (atan2
     (* (sin theta) t_1)
     (fma (fma (sin phi1) (cos delta) t_1) (- (sin phi1)) (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * cos(phi1);
	return lambda1 + atan2((sin(theta) * t_1), fma(fma(sin(phi1), cos(delta), t_1), -sin(phi1), cos(delta)));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * cos(phi1))
	return Float64(lambda1 + atan(Float64(sin(theta) * t_1), fma(fma(sin(phi1), cos(delta), t_1), Float64(-sin(phi1)), cos(delta))))
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision] + t$95$1), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. sin-asin N/A

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

   12. *-commutative N/A

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

4. Applied rewrites 99.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.6

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in theta around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lower-cos.f64 95.2

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

9. Applied rewrites 95.2%

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

10. Final simplification 95.2%

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

Alternative 5: 91.7% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi1) (* (sin theta) (sin delta)))
   (fma (fma (sin phi1) (cos delta) (sin delta)) (- (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(sin(phi1), cos(delta), sin(delta)), -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(sin(phi1), cos(delta), sin(delta)), 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[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[Sin[delta], $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. sin-asin N/A

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

   12. *-commutative N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in phi1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-cos.f64 92.7

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

7. Applied rewrites 92.7%

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

8. Taylor expanded in theta around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 92.5

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

10. Applied rewrites 92.5%

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

11. Final simplification 92.5%

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

Alternative 6: 92.0% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi1) (* (sin theta) (sin delta)))
   (fma (fma (sin phi1) 1.0 (sin delta)) (- (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(sin(phi1), 1.0, sin(delta)), -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(sin(phi1), 1.0, sin(delta)), 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[Sin[phi1], $MachinePrecision] * 1.0 + N[Sin[delta], $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. sin-asin N/A

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

   12. *-commutative N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in phi1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-cos.f64 92.7

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

7. Applied rewrites 92.7%

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

8. Taylor expanded in theta around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 92.5

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

10. Applied rewrites 92.5%

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

11. Taylor expanded in delta around 0

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

13. Applied rewrites 91.7%

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

14. Final simplification 91.7%

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

Alternative 7: 92.3% 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. Add Preprocessing

3. Taylor expanded in delta around 0

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

   1. lower-pow.f64 N/A

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

   2. lower-sin.f64 91.1

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

5. Applied rewrites 91.1%

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

6. Final simplification 91.1%

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

Alternative 8: 92.3% accurate, 1.9× speedup

Math

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

FPCore

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. sin-asin N/A

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

   12. *-commutative N/A

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

4. Applied rewrites 99.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.6

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in delta around 0

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

   1. lower-pow.f64 N/A

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

   2. lower-sin.f64 91.1

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

9. Applied rewrites 91.1%

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

10. Final simplification 91.1%

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

Alternative 9: 91.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -0.033:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.24:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -0.033)
   (+ lambda1 (atan2 (* (sin theta) (* (sin delta) (cos phi1))) (cos delta)))
   (if (<= delta 0.24)
     (+
      lambda1
      (atan2 (* (cos phi1) (* (sin theta) (sin delta))) (pow (cos phi1) 2.0)))
     (+
      lambda1
      (atan2 (* (sin delta) (* (sin theta) (cos phi1))) (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -0.033) {
		tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), cos(delta));
	} else if (delta <= 0.24) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), pow(cos(phi1), 2.0));
	} else {
		tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), 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 (delta <= (-0.033d0)) then
        tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), cos(delta))
    else if (delta <= 0.24d0) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(phi1) ** 2.0d0))
    else
        tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), 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 (delta <= -0.033) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * (Math.sin(delta) * Math.cos(phi1))), Math.cos(delta));
	} else if (delta <= 0.24) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * (Math.sin(theta) * Math.cos(phi1))), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= -0.033:
		tmp = lambda1 + math.atan2((math.sin(theta) * (math.sin(delta) * math.cos(phi1))), math.cos(delta))
	elif delta <= 0.24:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), math.pow(math.cos(phi1), 2.0))
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * (math.sin(theta) * math.cos(phi1))), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -0.033)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), cos(delta)));
	elseif (delta <= 0.24)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), (cos(phi1) ^ 2.0)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * Float64(sin(theta) * cos(phi1))), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= -0.033)
		tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), cos(delta));
	elseif (delta <= 0.24)
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(phi1) ^ 2.0));
	else
		tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -0.033], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 0.24], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -0.033000000000000002

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.1

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

   5. Applied rewrites 90.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 90.1

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

   7. Applied rewrites 90.1%

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

   if -0.033000000000000002 < delta < 0.23999999999999999

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. sin-asin N/A

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

      12. *-commutative N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in delta around 0

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

      1. unpow2 N/A

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

      2. 1-sub-sin N/A

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

      3. unpow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 98.9

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

   7. Applied rewrites 98.9%

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

   if 0.23999999999999999 < 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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 82.6

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

   5. Applied rewrites 82.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 82.6

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

   7. Applied rewrites 82.6%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.033:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.24:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -0.033:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.24:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -0.033)
   (+ lambda1 (atan2 (* (sin theta) (* (sin delta) (cos phi1))) (cos delta)))
   (if (<= delta 0.24)
     (+
      lambda1
      (atan2 (* (cos phi1) (* (sin theta) delta)) (pow (cos phi1) 2.0)))
     (+
      lambda1
      (atan2 (* (sin delta) (* (sin theta) (cos phi1))) (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -0.033) {
		tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), cos(delta));
	} else if (delta <= 0.24) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), pow(cos(phi1), 2.0));
	} else {
		tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), 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 (delta <= (-0.033d0)) then
        tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), cos(delta))
    else if (delta <= 0.24d0) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (cos(phi1) ** 2.0d0))
    else
        tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), 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 (delta <= -0.033) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * (Math.sin(delta) * Math.cos(phi1))), Math.cos(delta));
	} else if (delta <= 0.24) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * delta)), Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * (Math.sin(theta) * Math.cos(phi1))), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= -0.033:
		tmp = lambda1 + math.atan2((math.sin(theta) * (math.sin(delta) * math.cos(phi1))), math.cos(delta))
	elif delta <= 0.24:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * delta)), math.pow(math.cos(phi1), 2.0))
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * (math.sin(theta) * math.cos(phi1))), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -0.033)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), cos(delta)));
	elseif (delta <= 0.24)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), (cos(phi1) ^ 2.0)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * Float64(sin(theta) * cos(phi1))), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= -0.033)
		tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), cos(delta));
	elseif (delta <= 0.24)
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (cos(phi1) ^ 2.0));
	else
		tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -0.033], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 0.24], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -0.033000000000000002

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.1

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

   5. Applied rewrites 90.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 90.1

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

   7. Applied rewrites 90.1%

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

   if -0.033000000000000002 < delta < 0.23999999999999999

   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. Add Preprocessing

   3. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in delta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 98.5

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

   8. Applied rewrites 98.5%

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

      1. lift-sin.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. 1-sub-sin N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 98.9

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

   10. Applied rewrites 98.9%

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

   if 0.23999999999999999 < 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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 82.6

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

   5. Applied rewrites 82.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 82.6

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

   7. Applied rewrites 82.6%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.033:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.24:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -0.033:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.18 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -0.033)
   (+ lambda1 (atan2 (* (sin theta) (* (sin delta) (cos phi1))) (cos delta)))
   (if (<= delta 1.18e-5)
     (+
      lambda1
      (atan2
       (* (cos phi1) (* (sin theta) delta))
       (+ 0.5 (* 0.5 (cos (+ phi1 phi1))))))
     (+
      lambda1
      (atan2 (* (sin delta) (* (sin theta) (cos phi1))) (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -0.033) {
		tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), cos(delta));
	} else if (delta <= 1.18e-5) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5 + (0.5 * cos((phi1 + phi1)))));
	} else {
		tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), 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 (delta <= (-0.033d0)) then
        tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), cos(delta))
    else if (delta <= 1.18d-5) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5d0 + (0.5d0 * cos((phi1 + phi1)))))
    else
        tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), 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 (delta <= -0.033) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * (Math.sin(delta) * Math.cos(phi1))), Math.cos(delta));
	} else if (delta <= 1.18e-5) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * delta)), (0.5 + (0.5 * Math.cos((phi1 + phi1)))));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * (Math.sin(theta) * Math.cos(phi1))), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= -0.033:
		tmp = lambda1 + math.atan2((math.sin(theta) * (math.sin(delta) * math.cos(phi1))), math.cos(delta))
	elif delta <= 1.18e-5:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * delta)), (0.5 + (0.5 * math.cos((phi1 + phi1)))))
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * (math.sin(theta) * math.cos(phi1))), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -0.033)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), cos(delta)));
	elseif (delta <= 1.18e-5)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), Float64(0.5 + Float64(0.5 * cos(Float64(phi1 + phi1))))));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * Float64(sin(theta) * cos(phi1))), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= -0.033)
		tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), cos(delta));
	elseif (delta <= 1.18e-5)
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5 + (0.5 * cos((phi1 + phi1)))));
	else
		tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -0.033], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 1.18e-5], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -0.033000000000000002

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.1

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

   5. Applied rewrites 90.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 90.1

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

   7. Applied rewrites 90.1%

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

   if -0.033000000000000002 < delta < 1.18000000000000005e-5

   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. Add Preprocessing

   3. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in delta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 99.2

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

   8. Applied rewrites 99.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-atan2.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.2

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

   10. Applied rewrites 99.3%

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

   if 1.18000000000000005e-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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 81.7

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

   5. Applied rewrites 81.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 81.7

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

   7. Applied rewrites 81.7%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.033:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.18 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -0.033:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 1.18 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2 (* (sin delta) (* (sin theta) (cos phi1))) (cos delta)))))
   (if (<= delta -0.033)
     t_1
     (if (<= delta 1.18e-5)
       (+
        lambda1
        (atan2
         (* (cos phi1) (* (sin theta) delta))
         (+ 0.5 (* 0.5 (cos (+ phi1 phi1))))))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta));
	double tmp;
	if (delta <= -0.033) {
		tmp = t_1;
	} else if (delta <= 1.18e-5) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5 + (0.5 * cos((phi1 + phi1)))));
	} else {
		tmp = t_1;
	}
	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 = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta))
    if (delta <= (-0.033d0)) then
        tmp = t_1
    else if (delta <= 1.18d-5) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5d0 + (0.5d0 * cos((phi1 + phi1)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2((Math.sin(delta) * (Math.sin(theta) * Math.cos(phi1))), Math.cos(delta));
	double tmp;
	if (delta <= -0.033) {
		tmp = t_1;
	} else if (delta <= 1.18e-5) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * delta)), (0.5 + (0.5 * Math.cos((phi1 + phi1)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((math.sin(delta) * (math.sin(theta) * math.cos(phi1))), math.cos(delta))
	tmp = 0
	if delta <= -0.033:
		tmp = t_1
	elif delta <= 1.18e-5:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * delta)), (0.5 + (0.5 * math.cos((phi1 + phi1)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(sin(delta) * Float64(sin(theta) * cos(phi1))), cos(delta)))
	tmp = 0.0
	if (delta <= -0.033)
		tmp = t_1;
	elseif (delta <= 1.18e-5)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), Float64(0.5 + Float64(0.5 * cos(Float64(phi1 + phi1))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta));
	tmp = 0.0;
	if (delta <= -0.033)
		tmp = t_1;
	elseif (delta <= 1.18e-5)
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5 + (0.5 * cos((phi1 + phi1)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -0.033], t$95$1, If[LessEqual[delta, 1.18e-5], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -0.033000000000000002 or 1.18000000000000005e-5 < delta

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 85.4

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

   5. Applied rewrites 85.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 85.4

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

   7. Applied rewrites 85.4%

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

   if -0.033000000000000002 < delta < 1.18000000000000005e-5

   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. Add Preprocessing

   3. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in delta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 99.2

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

   8. Applied rewrites 99.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-atan2.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.2

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

   10. Applied rewrites 99.3%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.033:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.18 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{if}\;delta \leq -0.037:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* (sin theta) (sin delta)) (cos delta)))))
   (if (<= delta -0.037)
     t_1
     (if (<= delta 1.4e-5)
       (+
        lambda1
        (atan2
         (* (cos phi1) (* (sin theta) delta))
         (+ 0.5 (* 0.5 (cos (+ phi1 phi1))))))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((sin(theta) * sin(delta)), cos(delta));
	double tmp;
	if (delta <= -0.037) {
		tmp = t_1;
	} else if (delta <= 1.4e-5) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5 + (0.5 * cos((phi1 + phi1)))));
	} else {
		tmp = t_1;
	}
	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 = lambda1 + atan2((sin(theta) * sin(delta)), cos(delta))
    if (delta <= (-0.037d0)) then
        tmp = t_1
    else if (delta <= 1.4d-5) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5d0 + (0.5d0 * cos((phi1 + phi1)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2((Math.sin(theta) * Math.sin(delta)), Math.cos(delta));
	double tmp;
	if (delta <= -0.037) {
		tmp = t_1;
	} else if (delta <= 1.4e-5) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * delta)), (0.5 + (0.5 * Math.cos((phi1 + phi1)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((math.sin(theta) * math.sin(delta)), math.cos(delta))
	tmp = 0
	if delta <= -0.037:
		tmp = t_1
	elif delta <= 1.4e-5:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * delta)), (0.5 + (0.5 * math.cos((phi1 + phi1)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(sin(theta) * sin(delta)), cos(delta)))
	tmp = 0.0
	if (delta <= -0.037)
		tmp = t_1;
	elseif (delta <= 1.4e-5)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), Float64(0.5 + Float64(0.5 * cos(Float64(phi1 + phi1))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((sin(theta) * sin(delta)), cos(delta));
	tmp = 0.0;
	if (delta <= -0.037)
		tmp = t_1;
	elseif (delta <= 1.4e-5)
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5 + (0.5 * cos((phi1 + phi1)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -0.037], t$95$1, If[LessEqual[delta, 1.4e-5], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -0.0369999999999999982 or 1.39999999999999998e-5 < delta

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 83.9

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

   8. Applied rewrites 83.9%

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

   if -0.0369999999999999982 < delta < 1.39999999999999998e-5

   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. Add Preprocessing

   3. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in delta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 99.2

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

   8. Applied rewrites 99.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-atan2.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.2

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

   10. Applied rewrites 99.3%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.037:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 86.7% 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. Add Preprocessing

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 87.6

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

5. Applied rewrites 87.6%

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

6. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sin.f64 86.5

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

8. Applied rewrites 86.5%

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

9. Final simplification 86.5%

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

Alternative 15: 81.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin theta \cdot \sin delta\\ \mathbf{if}\;theta \leq -1.8:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1}\\ \mathbf{elif}\;theta \leq 0.78:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \left(\mathsf{fma}\left(-0.16666666666666666, theta \cdot theta, 1\right) + \left(theta \cdot theta\right) \cdot \left(\left(theta \cdot theta\right) \cdot 0.008333333333333333\right)\right)\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin theta) (sin delta))))
   (if (<= theta -1.8)
     (+ lambda1 (atan2 t_1 1.0))
     (if (<= theta 0.78)
       (+
        lambda1
        (atan2
         (*
          theta
          (*
           (sin delta)
           (+
            (fma -0.16666666666666666 (* theta theta) 1.0)
            (* (* theta theta) (* (* theta theta) 0.008333333333333333)))))
         (cos delta)))
       (+ lambda1 (atan2 t_1 (fma delta (* delta -0.5) 1.0)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(theta) * sin(delta);
	double tmp;
	if (theta <= -1.8) {
		tmp = lambda1 + atan2(t_1, 1.0);
	} else if (theta <= 0.78) {
		tmp = lambda1 + atan2((theta * (sin(delta) * (fma(-0.16666666666666666, (theta * theta), 1.0) + ((theta * theta) * ((theta * theta) * 0.008333333333333333))))), cos(delta));
	} else {
		tmp = lambda1 + atan2(t_1, fma(delta, (delta * -0.5), 1.0));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(theta) * sin(delta))
	tmp = 0.0
	if (theta <= -1.8)
		tmp = Float64(lambda1 + atan(t_1, 1.0));
	elseif (theta <= 0.78)
		tmp = Float64(lambda1 + atan(Float64(theta * Float64(sin(delta) * Float64(fma(-0.16666666666666666, Float64(theta * theta), 1.0) + Float64(Float64(theta * theta) * Float64(Float64(theta * theta) * 0.008333333333333333))))), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(t_1, fma(delta, Float64(delta * -0.5), 1.0)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[theta, -1.8], N[(lambda1 + N[ArcTan[t$95$1 / 1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[theta, 0.78], N[(lambda1 + N[ArcTan[N[(theta * N[(N[Sin[delta], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(theta * theta), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(theta * theta), $MachinePrecision] * N[(N[(theta * theta), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(delta * N[(delta * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if theta < -1.80000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 88.8

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

   8. Applied rewrites 88.8%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 76.7%

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

   if -1.80000000000000004 < theta < 0.78000000000000003

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 89.5

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in theta around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{theta \cdot \left(\sin delta + {theta}^{2} \cdot \left(\frac{-1}{6} \cdot \sin delta + \frac{1}{120} \cdot \left({theta}^{2} \cdot \sin delta\right)\right)\right)}}{\cos delta} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{theta \cdot \left(\sin delta + {theta}^{2} \cdot \left(\frac{-1}{6} \cdot \sin delta + \frac{1}{120} \cdot \left({theta}^{2} \cdot \sin delta\right)\right)\right)}}{\cos delta} \]

       2. distribute-lft-in N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta + \color{blue}{\left({theta}^{2} \cdot \left(\frac{-1}{6} \cdot \sin delta\right) + {theta}^{2} \cdot \left(\frac{1}{120} \cdot \left({theta}^{2} \cdot \sin delta\right)\right)\right)}\right)}{\cos delta} \]

       3. associate-+r+ N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \color{blue}{\left(\left(\sin delta + {theta}^{2} \cdot \left(\frac{-1}{6} \cdot \sin delta\right)\right) + {theta}^{2} \cdot \left(\frac{1}{120} \cdot \left({theta}^{2} \cdot \sin delta\right)\right)\right)}}{\cos delta} \]

       4. associate-*r* N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\left(\sin delta + \color{blue}{\left({theta}^{2} \cdot \frac{-1}{6}\right) \cdot \sin delta}\right) + {theta}^{2} \cdot \left(\frac{1}{120} \cdot \left({theta}^{2} \cdot \sin delta\right)\right)\right)}{\cos delta} \]

       5. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\left(\sin delta + \color{blue}{\left(\frac{-1}{6} \cdot {theta}^{2}\right)} \cdot \sin delta\right) + {theta}^{2} \cdot \left(\frac{1}{120} \cdot \left({theta}^{2} \cdot \sin delta\right)\right)\right)}{\cos delta} \]

       6. distribute-rgt1-in N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {theta}^{2} + 1\right) \cdot \sin delta} + {theta}^{2} \cdot \left(\frac{1}{120} \cdot \left({theta}^{2} \cdot \sin delta\right)\right)\right)}{\cos delta} \]

       7. associate-*r* N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\left(\frac{-1}{6} \cdot {theta}^{2} + 1\right) \cdot \sin delta + {theta}^{2} \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {theta}^{2}\right) \cdot \sin delta\right)}\right)}{\cos delta} \]

       8. associate-*r* N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\left(\frac{-1}{6} \cdot {theta}^{2} + 1\right) \cdot \sin delta + \color{blue}{\left({theta}^{2} \cdot \left(\frac{1}{120} \cdot {theta}^{2}\right)\right) \cdot \sin delta}\right)}{\cos delta} \]

       9. distribute-rgt-out N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \color{blue}{\left(\sin delta \cdot \left(\left(\frac{-1}{6} \cdot {theta}^{2} + 1\right) + {theta}^{2} \cdot \left(\frac{1}{120} \cdot {theta}^{2}\right)\right)\right)}}{\cos delta} \]

       10. lower-*.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \color{blue}{\left(\sin delta \cdot \left(\left(\frac{-1}{6} \cdot {theta}^{2} + 1\right) + {theta}^{2} \cdot \left(\frac{1}{120} \cdot {theta}^{2}\right)\right)\right)}}{\cos delta} \]

       11. lower-sin.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\color{blue}{\sin delta} \cdot \left(\left(\frac{-1}{6} \cdot {theta}^{2} + 1\right) + {theta}^{2} \cdot \left(\frac{1}{120} \cdot {theta}^{2}\right)\right)\right)}{\cos delta} \]

       12. lower-+.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {theta}^{2} + 1\right) + {theta}^{2} \cdot \left(\frac{1}{120} \cdot {theta}^{2}\right)\right)}\right)}{\cos delta} \]

   11. Applied rewrites 89.5%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{theta \cdot \left(\sin delta \cdot \left(\mathsf{fma}\left(-0.16666666666666666, theta \cdot theta, 1\right) + \left(theta \cdot theta\right) \cdot \left(\left(theta \cdot theta\right) \cdot 0.008333333333333333\right)\right)\right)}}{\cos delta} \]

   if 0.78000000000000003 < theta

   1. Initial program 99.0%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 77.0

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

   8. Applied rewrites 77.0%

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

   9. Taylor expanded in delta around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{1 + \frac{-1}{2} \cdot {delta}^{2}}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{\frac{-1}{2} \cdot {delta}^{2} + 1}} \]

       2. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{{delta}^{2} \cdot \frac{-1}{2}} + 1} \]

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 64.3

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

   11. Applied rewrites 64.3%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -1.8:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \mathbf{elif}\;theta \leq 0.78:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \left(\mathsf{fma}\left(-0.16666666666666666, theta \cdot theta, 1\right) + \left(theta \cdot theta\right) \cdot \left(\left(theta \cdot theta\right) \cdot 0.008333333333333333\right)\right)\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 81.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin theta \cdot \sin delta\\ \mathbf{if}\;theta \leq -1.8:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1}\\ \mathbf{elif}\;theta \leq 0.315:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, -0.16666666666666666 \cdot \left(theta \cdot theta\right), theta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin theta) (sin delta))))
   (if (<= theta -1.8)
     (+ lambda1 (atan2 t_1 1.0))
     (if (<= theta 0.315)
       (+
        lambda1
        (atan2
         (*
          (sin delta)
          (fma theta (* -0.16666666666666666 (* theta theta)) theta))
         (cos delta)))
       (+ lambda1 (atan2 t_1 (fma delta (* delta -0.5) 1.0)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(theta) * sin(delta);
	double tmp;
	if (theta <= -1.8) {
		tmp = lambda1 + atan2(t_1, 1.0);
	} else if (theta <= 0.315) {
		tmp = lambda1 + atan2((sin(delta) * fma(theta, (-0.16666666666666666 * (theta * theta)), theta)), cos(delta));
	} else {
		tmp = lambda1 + atan2(t_1, fma(delta, (delta * -0.5), 1.0));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(theta) * sin(delta))
	tmp = 0.0
	if (theta <= -1.8)
		tmp = Float64(lambda1 + atan(t_1, 1.0));
	elseif (theta <= 0.315)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * fma(theta, Float64(-0.16666666666666666 * Float64(theta * theta)), theta)), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(t_1, fma(delta, Float64(delta * -0.5), 1.0)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[theta, -1.8], N[(lambda1 + N[ArcTan[t$95$1 / 1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[theta, 0.315], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(theta * N[(-0.16666666666666666 * N[(theta * theta), $MachinePrecision]), $MachinePrecision] + theta), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(delta * N[(delta * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if theta < -1.80000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 88.8

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

   8. Applied rewrites 88.8%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 76.7%

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

   if -1.80000000000000004 < theta < 0.315000000000000002

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 89.5

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in theta around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{theta \cdot \left(\sin delta + \frac{-1}{6} \cdot \left({theta}^{2} \cdot \sin delta\right)\right)}}{\cos delta} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({theta}^{2} \cdot \sin delta\right) + \sin delta\right)}}{\cos delta} \]

       2. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\color{blue}{\left({theta}^{2} \cdot \sin delta\right) \cdot \frac{-1}{6}} + \sin delta\right)}{\cos delta} \]

       3. associate-*r* N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\color{blue}{{theta}^{2} \cdot \left(\sin delta \cdot \frac{-1}{6}\right)} + \sin delta\right)}{\cos delta} \]

       4. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \left({theta}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \sin delta\right)} + \sin delta\right)}{\cos delta} \]

       5. distribute-lft-in N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{theta \cdot \left({theta}^{2} \cdot \left(\frac{-1}{6} \cdot \sin delta\right)\right) + theta \cdot \sin delta}}{\cos delta} \]

       6. associate-*r* N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \color{blue}{\left(\left({theta}^{2} \cdot \frac{-1}{6}\right) \cdot \sin delta\right)} + theta \cdot \sin delta}{\cos delta} \]

       7. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {theta}^{2}\right)} \cdot \sin delta\right) + theta \cdot \sin delta}{\cos delta} \]

       8. associate-*r* N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(theta \cdot \left(\frac{-1}{6} \cdot {theta}^{2}\right)\right) \cdot \sin delta} + theta \cdot \sin delta}{\cos delta} \]

       9. distribute-rgt-out N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \left(theta \cdot \left(\frac{-1}{6} \cdot {theta}^{2}\right) + theta\right)}}{\cos delta} \]

       10. lower-*.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \left(theta \cdot \left(\frac{-1}{6} \cdot {theta}^{2}\right) + theta\right)}}{\cos delta} \]

       11. lower-sin.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta} \cdot \left(theta \cdot \left(\frac{-1}{6} \cdot {theta}^{2}\right) + theta\right)}{\cos delta} \]

       12. lower-fma.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\mathsf{fma}\left(theta, \frac{-1}{6} \cdot {theta}^{2}, theta\right)}}{\cos delta} \]

       13. lower-*.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, \color{blue}{\frac{-1}{6} \cdot {theta}^{2}}, theta\right)}{\cos delta} \]

       14. unpow2 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, \frac{-1}{6} \cdot \color{blue}{\left(theta \cdot theta\right)}, theta\right)}{\cos delta} \]

       15. lower-*.f64 89.3

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, -0.16666666666666666 \cdot \color{blue}{\left(theta \cdot theta\right)}, theta\right)}{\cos delta} \]

   11. Applied rewrites 89.3%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \mathsf{fma}\left(theta, -0.16666666666666666 \cdot \left(theta \cdot theta\right), theta\right)}}{\cos delta} \]

   if 0.315000000000000002 < theta

   1. Initial program 99.0%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 77.0

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

   8. Applied rewrites 77.0%

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

   9. Taylor expanded in delta around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{1 + \frac{-1}{2} \cdot {delta}^{2}}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{\frac{-1}{2} \cdot {delta}^{2} + 1}} \]

       2. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{{delta}^{2} \cdot \frac{-1}{2}} + 1} \]

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 64.3

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

   11. Applied rewrites 64.3%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -1.8:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \mathbf{elif}\;theta \leq 0.315:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, -0.16666666666666666 \cdot \left(theta \cdot theta\right), theta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 81.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin theta \cdot \sin delta\\ \mathbf{if}\;theta \leq -1.8:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1}\\ \mathbf{elif}\;theta \leq 0.064:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin theta) (sin delta))))
   (if (<= theta -1.8)
     (+ lambda1 (atan2 t_1 1.0))
     (if (<= theta 0.064)
       (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))
       (+ lambda1 (atan2 t_1 (fma delta (* delta -0.5) 1.0)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(theta) * sin(delta);
	double tmp;
	if (theta <= -1.8) {
		tmp = lambda1 + atan2(t_1, 1.0);
	} else if (theta <= 0.064) {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	} else {
		tmp = lambda1 + atan2(t_1, fma(delta, (delta * -0.5), 1.0));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(theta) * sin(delta))
	tmp = 0.0
	if (theta <= -1.8)
		tmp = Float64(lambda1 + atan(t_1, 1.0));
	elseif (theta <= 0.064)
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(t_1, fma(delta, Float64(delta * -0.5), 1.0)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[theta, -1.8], N[(lambda1 + N[ArcTan[t$95$1 / 1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[theta, 0.064], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(delta * N[(delta * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if theta < -1.80000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 88.8

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

   8. Applied rewrites 88.8%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 76.7%

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

   if -1.80000000000000004 < theta < 0.064000000000000001

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 89.5

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in theta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 89.1

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

   11. Applied rewrites 89.1%

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

   if 0.064000000000000001 < theta

   1. Initial program 99.0%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 77.0

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

   8. Applied rewrites 77.0%

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

   9. Taylor expanded in delta around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{1 + \frac{-1}{2} \cdot {delta}^{2}}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{\frac{-1}{2} \cdot {delta}^{2} + 1}} \]

       2. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{{delta}^{2} \cdot \frac{-1}{2}} + 1} \]

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 64.3

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

   11. Applied rewrites 64.3%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -1.8:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \mathbf{elif}\;theta \leq 0.064:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 80.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \mathbf{if}\;theta \leq -1.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 1.4 \cdot 10^{-28}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* (sin theta) (sin delta)) 1.0))))
   (if (<= theta -1.8)
     t_1
     (if (<= theta 1.4e-28)
       (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((sin(theta) * sin(delta)), 1.0);
	double tmp;
	if (theta <= -1.8) {
		tmp = t_1;
	} else if (theta <= 1.4e-28) {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	} else {
		tmp = t_1;
	}
	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 = lambda1 + atan2((sin(theta) * sin(delta)), 1.0d0)
    if (theta <= (-1.8d0)) then
        tmp = t_1
    else if (theta <= 1.4d-28) then
        tmp = lambda1 + atan2((theta * sin(delta)), cos(delta))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2((Math.sin(theta) * Math.sin(delta)), 1.0);
	double tmp;
	if (theta <= -1.8) {
		tmp = t_1;
	} else if (theta <= 1.4e-28) {
		tmp = lambda1 + Math.atan2((theta * Math.sin(delta)), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((math.sin(theta) * math.sin(delta)), 1.0)
	tmp = 0
	if theta <= -1.8:
		tmp = t_1
	elif theta <= 1.4e-28:
		tmp = lambda1 + math.atan2((theta * math.sin(delta)), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(sin(theta) * sin(delta)), 1.0))
	tmp = 0.0
	if (theta <= -1.8)
		tmp = t_1;
	elseif (theta <= 1.4e-28)
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((sin(theta) * sin(delta)), 1.0);
	tmp = 0.0;
	if (theta <= -1.8)
		tmp = t_1;
	elseif (theta <= 1.4e-28)
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / 1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[theta, -1.8], t$95$1, If[LessEqual[theta, 1.4e-28], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -1.80000000000000004 or 1.3999999999999999e-28 < 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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 83.4

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

   8. Applied rewrites 83.4%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 71.4%

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

   if -1.80000000000000004 < theta < 1.3999999999999999e-28

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 89.7

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

   8. Applied rewrites 89.7%

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

   9. Taylor expanded in theta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 89.3

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

   11. Applied rewrites 89.3%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -1.8:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \mathbf{elif}\;theta \leq 1.4 \cdot 10^{-28}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 80.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{if}\;delta \leq -1.55 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 0.0146:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))))
   (if (<= delta -1.55e+43)
     t_1
     (if (<= delta 0.0146)
       (+ lambda1 (atan2 (* (sin theta) delta) (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((theta * sin(delta)), cos(delta));
	double tmp;
	if (delta <= -1.55e+43) {
		tmp = t_1;
	} else if (delta <= 0.0146) {
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	} else {
		tmp = t_1;
	}
	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 = lambda1 + atan2((theta * sin(delta)), cos(delta))
    if (delta <= (-1.55d+43)) then
        tmp = t_1
    else if (delta <= 0.0146d0) then
        tmp = lambda1 + atan2((sin(theta) * delta), cos(delta))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2((theta * Math.sin(delta)), Math.cos(delta));
	double tmp;
	if (delta <= -1.55e+43) {
		tmp = t_1;
	} else if (delta <= 0.0146) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * delta), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((theta * math.sin(delta)), math.cos(delta))
	tmp = 0
	if delta <= -1.55e+43:
		tmp = t_1
	elif delta <= 0.0146:
		tmp = lambda1 + math.atan2((math.sin(theta) * delta), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)))
	tmp = 0.0
	if (delta <= -1.55e+43)
		tmp = t_1;
	elseif (delta <= 0.0146)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((theta * sin(delta)), cos(delta));
	tmp = 0.0;
	if (delta <= -1.55e+43)
		tmp = t_1;
	elseif (delta <= 0.0146)
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -1.55e+43], t$95$1, If[LessEqual[delta, 0.0146], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -1.5500000000000001e43 or 0.0146000000000000001 < delta

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 83.4

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

   8. Applied rewrites 83.4%

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

   9. Taylor expanded in theta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 70.4

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

   11. Applied rewrites 70.4%

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

   if -1.5500000000000001e43 < delta < 0.0146000000000000001

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.1

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 89.4

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

   8. Applied rewrites 89.4%

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

   9. Taylor expanded in delta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 88.1

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

   11. Applied rewrites 88.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -1.55 \cdot 10^{+43}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.0146:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 74.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(0.041666666666666664, delta \cdot delta, -0.5\right), 1\right)} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (sin theta) delta)
   (fma (* delta delta) (fma 0.041666666666666664 (* delta delta) -0.5) 1.0))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((sin(theta) * delta), fma((delta * delta), fma(0.041666666666666664, (delta * delta), -0.5), 1.0));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(sin(theta) * delta), fma(Float64(delta * delta), fma(0.041666666666666664, Float64(delta * delta), -0.5), 1.0)))
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[(N[(delta * delta), $MachinePrecision] * N[(0.041666666666666664 * N[(delta * delta), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $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. Add Preprocessing

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 87.6

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

5. Applied rewrites 87.6%

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

6. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sin.f64 86.5

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

8. Applied rewrites 86.5%

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

9. Taylor expanded in delta around 0

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

    1. lower-*.f64 N/A

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

    2. lower-sin.f64 70.7

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

11. Applied rewrites 70.7%

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

12. Taylor expanded in delta around 0

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{1 + {delta}^{2} \cdot \left(\frac{1}{24} \cdot {delta}^{2} - \frac{1}{2}\right)}} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{{delta}^{2} \cdot \left(\frac{1}{24} \cdot {delta}^{2} - \frac{1}{2}\right) + 1}} \]

    2. lower-fma.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{\mathsf{fma}\left({delta}^{2}, \frac{1}{24} \cdot {delta}^{2} - \frac{1}{2}, 1\right)}} \]

    3. unpow2 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(\color{blue}{delta \cdot delta}, \frac{1}{24} \cdot {delta}^{2} - \frac{1}{2}, 1\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(\color{blue}{delta \cdot delta}, \frac{1}{24} \cdot {delta}^{2} - \frac{1}{2}, 1\right)} \]

    5. sub-neg N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \color{blue}{\frac{1}{24} \cdot {delta}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)} \]

    6. metadata-eval N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \frac{1}{24} \cdot {delta}^{2} + \color{blue}{\frac{-1}{2}}, 1\right)} \]

    7. lower-fma.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \color{blue}{\mathsf{fma}\left(\frac{1}{24}, {delta}^{2}, \frac{-1}{2}\right)}, 1\right)} \]

    8. unpow2 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{1}{24}, \color{blue}{delta \cdot delta}, \frac{-1}{2}\right), 1\right)} \]

    9. lower-*.f64 72.5

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(0.041666666666666664, \color{blue}{delta \cdot delta}, -0.5\right), 1\right)} \]

14. Applied rewrites 72.5%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(0.041666666666666664, delta \cdot delta, -0.5\right), 1\right)}} \]

15. Final simplification 72.5%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(0.041666666666666664, delta \cdot delta, -0.5\right), 1\right)} \]
16. Add Preprocessing

Alternative 21: 73.4% accurate, 6.4× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+ lambda1 (atan2 (* (sin theta) delta) 1.0)))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((sin(theta) * delta), 1.0);
}

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) * delta), 1.0d0)
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + Math.atan2((Math.sin(theta) * delta), 1.0);
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2((math.sin(theta) * delta), 1.0)

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(sin(theta) * delta), 1.0))
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2((sin(theta) * delta), 1.0);
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / 1.0], $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. Add Preprocessing

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 87.6

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

5. Applied rewrites 87.6%

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

6. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sin.f64 86.5

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

8. Applied rewrites 86.5%

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

9. Taylor expanded in delta around 0

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

    1. lower-*.f64 N/A

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

    2. lower-sin.f64 70.7

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

11. Applied rewrites 70.7%

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

12. Taylor expanded in delta around 0

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

14. Applied rewrites 69.9%

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

15. Final simplification 69.9%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{1} \]
16. Add Preprocessing

Alternative 22: 67.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(theta * delta), $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. Add Preprocessing

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 87.6

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

5. Applied rewrites 87.6%

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

6. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sin.f64 86.5

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

8. Applied rewrites 86.5%

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

9. Taylor expanded in delta around 0

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

    1. lower-*.f64 N/A

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

    2. lower-sin.f64 70.7

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

11. Applied rewrites 70.7%

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

12. Taylor expanded in theta around 0

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

    1. lower-*.f64 62.1

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

14. Applied rewrites 62.1%

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

15. Final simplification 62.1%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot delta}{\cos delta} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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))))))))))