Spherical law of cosines

Percentage Accurate: 74.1% → 94.1%

Time: 20.5s

Alternatives: 19

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
  R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2))))) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Initial Program: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
  R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2))))) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Alternative 1: 94.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right), \cos \lambda_2, \mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (fma
    (* (cos lambda1) (* (cos phi2) (cos phi1)))
    (cos lambda2)
    (fma
     (* (* (sin lambda1) (sin lambda2)) (cos phi1))
     (cos phi2)
     (* (sin phi2) (sin phi1)))))
  R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(fma((cos(lambda1) * (cos(phi2) * cos(phi1))), cos(lambda2), fma(((sin(lambda1) * sin(lambda2)) * cos(phi1)), cos(phi2), (sin(phi2) * sin(phi1))))) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(fma(Float64(cos(lambda1) * Float64(cos(phi2) * cos(phi1))), cos(lambda2), fma(Float64(Float64(sin(lambda1) * sin(lambda2)) * cos(phi1)), cos(phi2), Float64(sin(phi2) * sin(phi1))))) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 73.0%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

   6. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

   7. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

   8. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

   9. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

   10. lower-sin.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)\right) \cdot R \]

   11. lower-sin.f64 94.4

       \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

4. Applied rewrites 94.4%

   \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \cdot R \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)} \cdot R \]

   2. +-commutative N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]

   3. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-in N/A

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

   7. associate-+l+ N/A

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

6. Applied rewrites 94.4%

   \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right), \cos \lambda_2, \mathsf{fma}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)} \cdot R \]
7. Add Preprocessing

Alternative 2: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (fma
    (*
     (fma (sin lambda2) (sin lambda1) (* (cos lambda2) (cos lambda1)))
     (cos phi2))
    (cos phi1)
    (* (sin phi2) (sin phi1))))
  R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(fma((fma(sin(lambda2), sin(lambda1), (cos(lambda2) * cos(lambda1))) * cos(phi2)), cos(phi1), (sin(phi2) * sin(phi1)))) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(fma(Float64(fma(sin(lambda2), sin(lambda1), Float64(cos(lambda2) * cos(lambda1))) * cos(phi2)), cos(phi1), Float64(sin(phi2) * sin(phi1)))) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 73.0%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

   6. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

   7. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

   8. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

   9. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

   10. lower-sin.f64 N/A

       \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)\right) \cdot R \]

   11. lower-sin.f64 94.4

       \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

4. Applied rewrites 94.4%

   \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \cdot R \]

5. Taylor expanded in lambda1 around inf

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

7. Applied rewrites 94.4%

   \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]
8. Add Preprocessing

Alternative 3: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R\\ t_1 := \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)\\ t_2 := \cos^{-1} \left(t\_1 \cdot \cos \phi_2\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -7000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 5.6 \cdot 10^{-33}:\\ \;\;\;\;\cos^{-1} \left(t\_1 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{+17}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 4.2 \cdot 10^{+247}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 2.1 \cdot 10^{+255}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 1.1 \cdot 10^{+301}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (acos
           (fma
            (sin phi2)
            (sin phi1)
            (* (* (cos (- lambda2 lambda1)) (cos phi1)) (cos phi2))))
          R))
        (t_1 (fma (sin lambda2) (sin lambda1) (* (cos lambda2) (cos lambda1))))
        (t_2 (* (acos (* t_1 (cos phi2))) R)))
   (if (<= phi2 -7000000000000.0)
     t_0
     (if (<= phi2 5.6e-33)
       (* (acos (* t_1 (cos phi1))) R)
       (if (<= phi2 6.5e+17)
         t_2
         (if (<= phi2 4.2e+247)
           t_0
           (if (<= phi2 2.1e+255)
             t_2
             (if (<= phi2 1.1e+301) t_0 (* (acos (cos phi1)) R)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos(fma(sin(phi2), sin(phi1), ((cos((lambda2 - lambda1)) * cos(phi1)) * cos(phi2)))) * R;
	double t_1 = fma(sin(lambda2), sin(lambda1), (cos(lambda2) * cos(lambda1)));
	double t_2 = acos((t_1 * cos(phi2))) * R;
	double tmp;
	if (phi2 <= -7000000000000.0) {
		tmp = t_0;
	} else if (phi2 <= 5.6e-33) {
		tmp = acos((t_1 * cos(phi1))) * R;
	} else if (phi2 <= 6.5e+17) {
		tmp = t_2;
	} else if (phi2 <= 4.2e+247) {
		tmp = t_0;
	} else if (phi2 <= 2.1e+255) {
		tmp = t_2;
	} else if (phi2 <= 1.1e+301) {
		tmp = t_0;
	} else {
		tmp = acos(cos(phi1)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi1)) * cos(phi2)))) * R)
	t_1 = fma(sin(lambda2), sin(lambda1), Float64(cos(lambda2) * cos(lambda1)))
	t_2 = Float64(acos(Float64(t_1 * cos(phi2))) * R)
	tmp = 0.0
	if (phi2 <= -7000000000000.0)
		tmp = t_0;
	elseif (phi2 <= 5.6e-33)
		tmp = Float64(acos(Float64(t_1 * cos(phi1))) * R);
	elseif (phi2 <= 6.5e+17)
		tmp = t_2;
	elseif (phi2 <= 4.2e+247)
		tmp = t_0;
	elseif (phi2 <= 2.1e+255)
		tmp = t_2;
	elseif (phi2 <= 1.1e+301)
		tmp = t_0;
	else
		tmp = Float64(acos(cos(phi1)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcCos[N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -7000000000000.0], t$95$0, If[LessEqual[phi2, 5.6e-33], N[(N[ArcCos[N[(t$95$1 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 6.5e+17], t$95$2, If[LessEqual[phi2, 4.2e+247], t$95$0, If[LessEqual[phi2, 2.1e+255], t$95$2, If[LessEqual[phi2, 1.1e+301], t$95$0, N[(N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -7e12 or 6.5e17 < phi2 < 4.2e247 or 2.1e255 < phi2 < 1.1e301

   1. Initial program 77.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 77.1

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

      5. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right) \cdot R \]

      6. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right)\right) \cdot R \]

      7. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)}\right)\right) \cdot R \]

      8. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2}\right)\right) \cdot R \]

      10. lower-*.f64 77.1

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot \cos \phi_2\right)\right) \cdot R \]

      11. lift-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      12. lift--.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      13. cos-diff N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      14. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      15. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      16. cos-diff-rev N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      17. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      18. lower--.f64 77.1

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right) \cdot R \]

   4. Applied rewrites 77.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2\right)\right)} \cdot R \]

   if -7e12 < phi2 < 5.6e-33

   1. Initial program 69.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)\right) \cdot R \]

      11. lower-sin.f64 90.2

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

   4. Applied rewrites 90.2%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \cdot R \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\color{blue}{\sin \lambda_2 \cdot \sin \lambda_1} + \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R \]

      5. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      6. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\sin \lambda_2}, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R \]

      7. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \color{blue}{\sin \lambda_1}, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      10. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2} \cdot \cos \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      11. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \color{blue}{\cos \lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 88.0

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   7. Applied rewrites 88.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   if 5.6e-33 < phi2 < 6.5e17 or 4.2e247 < phi2 < 2.1e255

   1. Initial program 67.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)\right) \cdot R \]

      11. lower-sin.f64 92.1

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

   4. Applied rewrites 92.1%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right)} \cdot R \]

      3. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)} \cdot \cos \phi_2\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\color{blue}{\sin \lambda_2 \cdot \sin \lambda_1} + \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right) \cdot R \]

      5. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)} \cdot \cos \phi_2\right) \cdot R \]

      6. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\sin \lambda_2}, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right) \cdot R \]

      7. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \color{blue}{\sin \lambda_1}, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_2\right) \cdot R \]

      10. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2} \cdot \cos \lambda_1\right) \cdot \cos \phi_2\right) \cdot R \]

      11. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \color{blue}{\cos \lambda_1}\right) \cdot \cos \phi_2\right) \cdot R \]

      12. lower-cos.f64 73.5

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]

   7. Applied rewrites 73.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2\right)} \cdot R \]

   if 1.1e301 < phi2

   1. Initial program 66.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 67.7

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 67.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 26.8%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 19.6%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 18.9%

       \[\leadsto \cos^{-1} \cos \phi_1 \cdot R \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_1\right) \cdot R\\ t_1 := \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right) \cdot R\\ \mathbf{if}\;\lambda_1 \leq -1.25 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\lambda_1 \leq 7.8 \cdot 10^{-26}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_1 \leq 2.6 \cdot 10^{+40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq 2.2 \cdot 10^{+237}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\lambda_1 \leq 1.9 \cdot 10^{+252}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq 2.2 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (acos
           (*
            (fma (sin lambda2) (sin lambda1) (* (cos lambda2) (cos lambda1)))
            (cos phi1)))
          R))
        (t_1
         (*
          (acos
           (fma
            (sin phi2)
            (sin phi1)
            (* (cos lambda1) (* (cos phi2) (cos phi1)))))
          R)))
   (if (<= lambda1 -1.25e-7)
     t_1
     (if (<= lambda1 7.8e-26)
       (*
        (acos
         (fma
          (* (cos lambda2) (cos phi2))
          (cos phi1)
          (* (sin phi2) (sin phi1))))
        R)
       (if (<= lambda1 2.6e+40)
         t_0
         (if (<= lambda1 2.2e+237)
           t_1
           (if (<= lambda1 1.9e+252)
             t_0
             (if (<= lambda1 2.2e+306) t_1 (* (acos (cos phi1)) R)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos((fma(sin(lambda2), sin(lambda1), (cos(lambda2) * cos(lambda1))) * cos(phi1))) * R;
	double t_1 = acos(fma(sin(phi2), sin(phi1), (cos(lambda1) * (cos(phi2) * cos(phi1))))) * R;
	double tmp;
	if (lambda1 <= -1.25e-7) {
		tmp = t_1;
	} else if (lambda1 <= 7.8e-26) {
		tmp = acos(fma((cos(lambda2) * cos(phi2)), cos(phi1), (sin(phi2) * sin(phi1)))) * R;
	} else if (lambda1 <= 2.6e+40) {
		tmp = t_0;
	} else if (lambda1 <= 2.2e+237) {
		tmp = t_1;
	} else if (lambda1 <= 1.9e+252) {
		tmp = t_0;
	} else if (lambda1 <= 2.2e+306) {
		tmp = t_1;
	} else {
		tmp = acos(cos(phi1)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(Float64(fma(sin(lambda2), sin(lambda1), Float64(cos(lambda2) * cos(lambda1))) * cos(phi1))) * R)
	t_1 = Float64(acos(fma(sin(phi2), sin(phi1), Float64(cos(lambda1) * Float64(cos(phi2) * cos(phi1))))) * R)
	tmp = 0.0
	if (lambda1 <= -1.25e-7)
		tmp = t_1;
	elseif (lambda1 <= 7.8e-26)
		tmp = Float64(acos(fma(Float64(cos(lambda2) * cos(phi2)), cos(phi1), Float64(sin(phi2) * sin(phi1)))) * R);
	elseif (lambda1 <= 2.6e+40)
		tmp = t_0;
	elseif (lambda1 <= 2.2e+237)
		tmp = t_1;
	elseif (lambda1 <= 1.9e+252)
		tmp = t_0;
	elseif (lambda1 <= 2.2e+306)
		tmp = t_1;
	else
		tmp = Float64(acos(cos(phi1)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[ArcCos[N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[lambda1, -1.25e-7], t$95$1, If[LessEqual[lambda1, 7.8e-26], N[(N[ArcCos[N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda1, 2.6e+40], t$95$0, If[LessEqual[lambda1, 2.2e+237], t$95$1, If[LessEqual[lambda1, 1.9e+252], t$95$0, If[LessEqual[lambda1, 2.2e+306], t$95$1, N[(N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if lambda1 < -1.24999999999999994e-7 or 2.6000000000000001e40 < lambda1 < 2.2e237 or 1.89999999999999986e252 < lambda1 < 2.2e306

   1. Initial program 62.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 62.8

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 62.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]
   6. Step-by-step derivation

   7. Applied rewrites 62.8%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \color{blue}{\sin \phi_1}, \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right) \cdot R \]

   if -1.24999999999999994e-7 < lambda1 < 7.79999999999999973e-26

   1. Initial program 87.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \phi_2}, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \phi_2}, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2} \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2} \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_2}, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      9. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      12. lower-sin.f64 87.6

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 87.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   if 7.79999999999999973e-26 < lambda1 < 2.6000000000000001e40 or 2.2e237 < lambda1 < 1.89999999999999986e252

   1. Initial program 67.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)\right) \cdot R \]

      11. lower-sin.f64 99.2

          \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)\right) \cdot R \]

   4. Applied rewrites 99.2%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \cdot R \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot R \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\color{blue}{\sin \lambda_2 \cdot \sin \lambda_1} + \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R \]

      5. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      6. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\sin \lambda_2}, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R \]

      7. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \color{blue}{\sin \lambda_1}, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      10. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \color{blue}{\cos \lambda_2} \cdot \cos \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      11. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \color{blue}{\cos \lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 60.1

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   7. Applied rewrites 60.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   if 2.2e306 < lambda1

   1. Initial program 60.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 61.0

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 61.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 25.8%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 19.3%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 19.3%

       \[\leadsto \cos^{-1} \cos \phi_1 \cdot R \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right) \cdot R\\ \mathbf{if}\;\lambda_1 \leq -1.25 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq 7.8 \cdot 10^{-26}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_1 \leq 2.2 \cdot 10^{+306}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (acos
           (fma
            (sin phi2)
            (sin phi1)
            (* (cos lambda1) (* (cos phi2) (cos phi1)))))
          R)))
   (if (<= lambda1 -1.25e-7)
     t_0
     (if (<= lambda1 7.8e-26)
       (*
        (acos
         (fma
          (* (cos lambda2) (cos phi2))
          (cos phi1)
          (* (sin phi2) (sin phi1))))
        R)
       (if (<= lambda1 2.2e+306) t_0 (* (acos (cos phi1)) R))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos(fma(sin(phi2), sin(phi1), (cos(lambda1) * (cos(phi2) * cos(phi1))))) * R;
	double tmp;
	if (lambda1 <= -1.25e-7) {
		tmp = t_0;
	} else if (lambda1 <= 7.8e-26) {
		tmp = acos(fma((cos(lambda2) * cos(phi2)), cos(phi1), (sin(phi2) * sin(phi1)))) * R;
	} else if (lambda1 <= 2.2e+306) {
		tmp = t_0;
	} else {
		tmp = acos(cos(phi1)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(fma(sin(phi2), sin(phi1), Float64(cos(lambda1) * Float64(cos(phi2) * cos(phi1))))) * R)
	tmp = 0.0
	if (lambda1 <= -1.25e-7)
		tmp = t_0;
	elseif (lambda1 <= 7.8e-26)
		tmp = Float64(acos(fma(Float64(cos(lambda2) * cos(phi2)), cos(phi1), Float64(sin(phi2) * sin(phi1)))) * R);
	elseif (lambda1 <= 2.2e+306)
		tmp = t_0;
	else
		tmp = Float64(acos(cos(phi1)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[lambda1, -1.25e-7], t$95$0, If[LessEqual[lambda1, 7.8e-26], N[(N[ArcCos[N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda1, 2.2e+306], t$95$0, N[(N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.24999999999999994e-7 or 7.79999999999999973e-26 < lambda1 < 2.2e306

   1. Initial program 63.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 62.8

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 62.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]
   6. Step-by-step derivation

   7. Applied rewrites 62.8%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \color{blue}{\sin \phi_1}, \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right) \cdot R \]

   if -1.24999999999999994e-7 < lambda1 < 7.79999999999999973e-26

   1. Initial program 87.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \phi_2}, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \cos \phi_2}, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2} \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2} \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_2}, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      9. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      12. lower-sin.f64 87.6

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 87.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   if 2.2e306 < lambda1

   1. Initial program 60.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 61.0

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 61.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 25.8%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 19.3%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 19.3%

       \[\leadsto \cos^{-1} \cos \phi_1 \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 63.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.00088:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 1.95 \cdot 10^{+220}:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_2 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 7.5 \cdot 10^{+291}:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_2 \cdot \cos \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -0.00088)
   (* (acos (* (cos (- lambda2 lambda1)) (cos phi1))) R)
   (if (<= lambda2 2e-7)
     (*
      (acos
       (fma (sin phi2) (sin phi1) (* (cos lambda1) (* (cos phi2) (cos phi1)))))
      R)
     (if (<= lambda2 1.95e+220)
       (* (acos (* (cos lambda2) (cos phi1))) R)
       (if (<= lambda2 7.5e+291)
         (* (acos (* (cos lambda2) (cos phi2))) R)
         (* (acos (cos phi1)) R))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -0.00088) {
		tmp = acos((cos((lambda2 - lambda1)) * cos(phi1))) * R;
	} else if (lambda2 <= 2e-7) {
		tmp = acos(fma(sin(phi2), sin(phi1), (cos(lambda1) * (cos(phi2) * cos(phi1))))) * R;
	} else if (lambda2 <= 1.95e+220) {
		tmp = acos((cos(lambda2) * cos(phi1))) * R;
	} else if (lambda2 <= 7.5e+291) {
		tmp = acos((cos(lambda2) * cos(phi2))) * R;
	} else {
		tmp = acos(cos(phi1)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -0.00088)
		tmp = Float64(acos(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi1))) * R);
	elseif (lambda2 <= 2e-7)
		tmp = Float64(acos(fma(sin(phi2), sin(phi1), Float64(cos(lambda1) * Float64(cos(phi2) * cos(phi1))))) * R);
	elseif (lambda2 <= 1.95e+220)
		tmp = Float64(acos(Float64(cos(lambda2) * cos(phi1))) * R);
	elseif (lambda2 <= 7.5e+291)
		tmp = Float64(acos(Float64(cos(lambda2) * cos(phi2))) * R);
	else
		tmp = Float64(acos(cos(phi1)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -0.00088], N[(N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda2, 2e-7], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda2, 1.95e+220], N[(N[ArcCos[N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[lambda2, 7.5e+291], N[(N[ArcCos[N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if lambda2 < -8.80000000000000031e-4

   1. Initial program 50.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 35.3

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 35.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   if -8.80000000000000031e-4 < lambda2 < 1.9999999999999999e-7

   1. Initial program 89.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 89.1

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 89.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]
   6. Step-by-step derivation

   7. Applied rewrites 89.1%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \color{blue}{\sin \phi_1}, \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right) \cdot R \]

   if 1.9999999999999999e-7 < lambda2 < 1.95000000000000008e220

   1. Initial program 63.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 34.8

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 34.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 34.6%

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   if 1.95000000000000008e220 < lambda2 < 7.5000000000000001e291

   1. Initial program 57.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{\cos \phi_2 \cdot \left({\sin \lambda_1}^{2} \cdot {\sin \lambda_2}^{2} - {\cos \lambda_1}^{2} \cdot {\cos \lambda_2}^{2}\right)}{\sin \lambda_1 \cdot \sin \lambda_2 - \cos \lambda_1 \cdot \cos \lambda_2}\right)} \cdot R \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(-{\cos \lambda_1}^{2}, {\cos \lambda_2}^{2}, {\sin \lambda_2}^{2} \cdot {\sin \lambda_1}^{2}\right) \cdot \cos \phi_2}{\mathsf{fma}\left(-\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right)}\right)} \cdot R \]

   8. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 34.8%

       \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]

   if 7.5000000000000001e291 < lambda2

   1. Initial program 44.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 22.8

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 22.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 9.4%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 9.4%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 3.2%

       \[\leadsto \cos^{-1} \cos \phi_1 \cdot R \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 53.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := t\_0 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -15.5:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -2 \cdot 10^{-70}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, t\_1\right)\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 0.0155:\\ \;\;\;\;\cos^{-1} t\_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))) (t_1 (* t_0 (cos phi2))))
   (if (<= phi1 -15.5)
     (* (acos (* t_0 (cos phi1))) R)
     (if (<= phi1 -2e-70)
       (* (acos (fma (sin phi2) phi1 t_1)) R)
       (if (<= phi1 0.0155) (* (acos t_1) R) (* (acos (cos phi1)) R))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = t_0 * cos(phi2);
	double tmp;
	if (phi1 <= -15.5) {
		tmp = acos((t_0 * cos(phi1))) * R;
	} else if (phi1 <= -2e-70) {
		tmp = acos(fma(sin(phi2), phi1, t_1)) * R;
	} else if (phi1 <= 0.0155) {
		tmp = acos(t_1) * R;
	} else {
		tmp = acos(cos(phi1)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(t_0 * cos(phi2))
	tmp = 0.0
	if (phi1 <= -15.5)
		tmp = Float64(acos(Float64(t_0 * cos(phi1))) * R);
	elseif (phi1 <= -2e-70)
		tmp = Float64(acos(fma(sin(phi2), phi1, t_1)) * R);
	elseif (phi1 <= 0.0155)
		tmp = Float64(acos(t_1) * R);
	else
		tmp = Float64(acos(cos(phi1)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -15.5], N[(N[ArcCos[N[(t$95$0 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, -2e-70], N[(N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * phi1 + t$95$1), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 0.0155], N[(N[ArcCos[t$95$1], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -15.5

   1. Initial program 84.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 47.0

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 47.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   if -15.5 < phi1 < -1.99999999999999999e-70

   1. Initial program 58.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]

      3. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\sin \phi_2}, \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}\right)\right) \cdot R \]

      5. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}\right)\right) \cdot R \]

      6. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_2\right)\right) \cdot R \]

      7. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      8. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      9. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      10. remove-double-neg N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      12. distribute-neg-in N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      13. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      14. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      15. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_2\right)\right) \cdot R \]

      16. remove-double-neg N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_2\right)\right) \cdot R \]

      17. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_2\right)\right) \cdot R \]

      18. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      19. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_2\right)\right) \cdot R \]

      20. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_2\right)\right) \cdot R \]

      21. lower-cos.f64 59.1

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_2}\right)\right) \cdot R \]

   5. Applied rewrites 59.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_2, \phi_1, \cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right)\right)} \cdot R \]

   if -1.99999999999999999e-70 < phi1 < 0.0155

   1. Initial program 67.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_2\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_2\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_2\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_2\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      18. lower-cos.f64 67.7

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]

   5. Applied rewrites 67.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right)} \cdot R \]

   if 0.0155 < phi1

   1. Initial program 74.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 56.1

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 56.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 27.9%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 22.3%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 22.2%

       \[\leadsto \cos^{-1} \cos \phi_1 \cdot R \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 48.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\lambda_1 - \lambda_2\right) \leq 0.99998:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (cos (- lambda1 lambda2)) 0.99998)
   (* (acos (* (cos (- lambda2 lambda1)) (cos phi1))) R)
   (* (acos (cos (- phi2 phi1))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (cos((lambda1 - lambda2)) <= 0.99998) {
		tmp = acos((cos((lambda2 - lambda1)) * cos(phi1))) * R;
	} else {
		tmp = acos(cos((phi2 - phi1))) * R;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (cos((lambda1 - lambda2)) <= 0.99998d0) then
        tmp = acos((cos((lambda2 - lambda1)) * cos(phi1))) * r
    else
        tmp = acos(cos((phi2 - phi1))) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (Math.cos((lambda1 - lambda2)) <= 0.99998) {
		tmp = Math.acos((Math.cos((lambda2 - lambda1)) * Math.cos(phi1))) * R;
	} else {
		tmp = Math.acos(Math.cos((phi2 - phi1))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if math.cos((lambda1 - lambda2)) <= 0.99998:
		tmp = math.acos((math.cos((lambda2 - lambda1)) * math.cos(phi1))) * R
	else:
		tmp = math.acos(math.cos((phi2 - phi1))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (cos(Float64(lambda1 - lambda2)) <= 0.99998)
		tmp = Float64(acos(Float64(cos(Float64(lambda2 - lambda1)) * cos(phi1))) * R);
	else
		tmp = Float64(acos(cos(Float64(phi2 - phi1))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (cos((lambda1 - lambda2)) <= 0.99998)
		tmp = acos((cos((lambda2 - lambda1)) * cos(phi1))) * R;
	else
		tmp = acos(cos((phi2 - phi1))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision], 0.99998], N[(N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 lambda1 lambda2)) < 0.99997999999999998

   1. Initial program 72.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 46.2

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 46.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   if 0.99997999999999998 < (cos.f64 (-.f64 lambda1 lambda2))

   1. Initial program 76.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 76.6

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 76.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 73.4%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 51.3%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 39.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right)\\ t_1 := \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot R\\ t_2 := \cos^{-1} \left(t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -32500:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_2 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -8 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq -3.3 \cdot 10^{-63}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \cos \lambda_2\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.9 \cdot 10^{-283}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 7.2 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 2.35 \cdot 10^{-263}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 3.6 \cdot 10^{-239}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma -0.5 (* phi1 phi1) 1.0))
        (t_1 (* (acos (* (cos lambda1) (cos phi2))) R))
        (t_2 (* (acos (* t_0 (cos (- lambda1 lambda2)))) R)))
   (if (<= phi1 -32500.0)
     (* (acos (* (cos lambda2) (cos phi1))) R)
     (if (<= phi1 -8e-47)
       t_1
       (if (<= phi1 -3.3e-63)
         (* (acos (* t_0 (cos lambda2))) R)
         (if (<= phi1 -1.9e-283)
           t_1
           (if (<= phi1 7.2e-308)
             t_2
             (if (<= phi1 2.35e-263)
               t_1
               (if (<= phi1 3.6e-239)
                 t_2
                 (if (<= phi1 2.1e-16) t_1 (* (acos (cos phi1)) R)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(-0.5, (phi1 * phi1), 1.0);
	double t_1 = acos((cos(lambda1) * cos(phi2))) * R;
	double t_2 = acos((t_0 * cos((lambda1 - lambda2)))) * R;
	double tmp;
	if (phi1 <= -32500.0) {
		tmp = acos((cos(lambda2) * cos(phi1))) * R;
	} else if (phi1 <= -8e-47) {
		tmp = t_1;
	} else if (phi1 <= -3.3e-63) {
		tmp = acos((t_0 * cos(lambda2))) * R;
	} else if (phi1 <= -1.9e-283) {
		tmp = t_1;
	} else if (phi1 <= 7.2e-308) {
		tmp = t_2;
	} else if (phi1 <= 2.35e-263) {
		tmp = t_1;
	} else if (phi1 <= 3.6e-239) {
		tmp = t_2;
	} else if (phi1 <= 2.1e-16) {
		tmp = t_1;
	} else {
		tmp = acos(cos(phi1)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(-0.5, Float64(phi1 * phi1), 1.0)
	t_1 = Float64(acos(Float64(cos(lambda1) * cos(phi2))) * R)
	t_2 = Float64(acos(Float64(t_0 * cos(Float64(lambda1 - lambda2)))) * R)
	tmp = 0.0
	if (phi1 <= -32500.0)
		tmp = Float64(acos(Float64(cos(lambda2) * cos(phi1))) * R);
	elseif (phi1 <= -8e-47)
		tmp = t_1;
	elseif (phi1 <= -3.3e-63)
		tmp = Float64(acos(Float64(t_0 * cos(lambda2))) * R);
	elseif (phi1 <= -1.9e-283)
		tmp = t_1;
	elseif (phi1 <= 7.2e-308)
		tmp = t_2;
	elseif (phi1 <= 2.35e-263)
		tmp = t_1;
	elseif (phi1 <= 3.6e-239)
		tmp = t_2;
	elseif (phi1 <= 2.1e-16)
		tmp = t_1;
	else
		tmp = Float64(acos(cos(phi1)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(-0.5 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcCos[N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -32500.0], N[(N[ArcCos[N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, -8e-47], t$95$1, If[LessEqual[phi1, -3.3e-63], N[(N[ArcCos[N[(t$95$0 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, -1.9e-283], t$95$1, If[LessEqual[phi1, 7.2e-308], t$95$2, If[LessEqual[phi1, 2.35e-263], t$95$1, If[LessEqual[phi1, 3.6e-239], t$95$2, If[LessEqual[phi1, 2.1e-16], t$95$1, N[(N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if phi1 < -32500

   1. Initial program 84.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 47.0

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 47.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 33.2%

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   if -32500 < phi1 < -7.9999999999999998e-47 or -3.29999999999999994e-63 < phi1 < -1.9000000000000001e-283 or 7.1999999999999997e-308 < phi1 < 2.34999999999999985e-263 or 3.6000000000000001e-239 < phi1 < 2.1000000000000001e-16

   1. Initial program 65.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 50.1

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 50.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 49.8%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]

   if -7.9999999999999998e-47 < phi1 < -3.29999999999999994e-63

   1. Initial program 51.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 36.4

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 36.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \color{blue}{\frac{-1}{2} \cdot \left({\phi_1}^{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 36.4%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\frac{-1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot R \]
   10. Step-by-step derivation

   11. Applied rewrites 36.2%

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \lambda_2\right) \cdot R \]

   if -1.9000000000000001e-283 < phi1 < 7.1999999999999997e-308 or 2.34999999999999985e-263 < phi1 < 3.6000000000000001e-239

   1. Initial program 72.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 54.1

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 54.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \color{blue}{\frac{-1}{2} \cdot \left({\phi_1}^{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 54.1%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

   if 2.1000000000000001e-16 < phi1

   1. Initial program 75.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 57.4

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 57.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 27.4%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 22.0%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 21.7%

       \[\leadsto \cos^{-1} \cos \phi_1 \cdot R \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 10: 40.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -15.5:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.9 \cdot 10^{-283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 7.2 \cdot 10^{-308}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 2.35 \cdot 10^{-263}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 9.5 \cdot 10^{-216}:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_2 \cdot \cos \phi_2\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 0.0155:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (acos (* (cos lambda1) (cos phi2))) R)))
   (if (<= phi1 -15.5)
     (* (acos (* (cos lambda1) (cos phi1))) R)
     (if (<= phi1 -1.9e-283)
       t_0
       (if (<= phi1 7.2e-308)
         (*
          (acos (* (fma -0.5 (* phi1 phi1) 1.0) (cos (- lambda1 lambda2))))
          R)
         (if (<= phi1 2.35e-263)
           t_0
           (if (<= phi1 9.5e-216)
             (* (acos (* (cos lambda2) (cos phi2))) R)
             (if (<= phi1 0.0155) t_0 (* (acos (cos phi1)) R)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos((cos(lambda1) * cos(phi2))) * R;
	double tmp;
	if (phi1 <= -15.5) {
		tmp = acos((cos(lambda1) * cos(phi1))) * R;
	} else if (phi1 <= -1.9e-283) {
		tmp = t_0;
	} else if (phi1 <= 7.2e-308) {
		tmp = acos((fma(-0.5, (phi1 * phi1), 1.0) * cos((lambda1 - lambda2)))) * R;
	} else if (phi1 <= 2.35e-263) {
		tmp = t_0;
	} else if (phi1 <= 9.5e-216) {
		tmp = acos((cos(lambda2) * cos(phi2))) * R;
	} else if (phi1 <= 0.0155) {
		tmp = t_0;
	} else {
		tmp = acos(cos(phi1)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(Float64(cos(lambda1) * cos(phi2))) * R)
	tmp = 0.0
	if (phi1 <= -15.5)
		tmp = Float64(acos(Float64(cos(lambda1) * cos(phi1))) * R);
	elseif (phi1 <= -1.9e-283)
		tmp = t_0;
	elseif (phi1 <= 7.2e-308)
		tmp = Float64(acos(Float64(fma(-0.5, Float64(phi1 * phi1), 1.0) * cos(Float64(lambda1 - lambda2)))) * R);
	elseif (phi1 <= 2.35e-263)
		tmp = t_0;
	elseif (phi1 <= 9.5e-216)
		tmp = Float64(acos(Float64(cos(lambda2) * cos(phi2))) * R);
	elseif (phi1 <= 0.0155)
		tmp = t_0;
	else
		tmp = Float64(acos(cos(phi1)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -15.5], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, -1.9e-283], t$95$0, If[LessEqual[phi1, 7.2e-308], N[(N[ArcCos[N[(N[(-0.5 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 2.35e-263], t$95$0, If[LessEqual[phi1, 9.5e-216], N[(N[ArcCos[N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 0.0155], t$95$0, N[(N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if phi1 < -15.5

   1. Initial program 84.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 47.0

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 47.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 41.7%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

   if -15.5 < phi1 < -1.9000000000000001e-283 or 7.1999999999999997e-308 < phi1 < 2.34999999999999985e-263 or 9.49999999999999943e-216 < phi1 < 0.0155

   1. Initial program 64.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 49.7

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 49.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 49.3%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]

   if -1.9000000000000001e-283 < phi1 < 7.1999999999999997e-308

   1. Initial program 83.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 68.6

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 68.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \color{blue}{\frac{-1}{2} \cdot \left({\phi_1}^{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 68.6%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

   if 2.34999999999999985e-263 < phi1 < 9.49999999999999943e-216

   1. Initial program 74.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{\cos \phi_2 \cdot \left({\sin \lambda_1}^{2} \cdot {\sin \lambda_2}^{2} - {\cos \lambda_1}^{2} \cdot {\cos \lambda_2}^{2}\right)}{\sin \lambda_1 \cdot \sin \lambda_2 - \cos \lambda_1 \cdot \cos \lambda_2}\right)} \cdot R \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

   7. Applied rewrites 99.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(-{\cos \lambda_1}^{2}, {\cos \lambda_2}^{2}, {\sin \lambda_2}^{2} \cdot {\sin \lambda_1}^{2}\right) \cdot \cos \phi_2}{\mathsf{fma}\left(-\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right)}\right)} \cdot R \]

   8. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 53.0%

       \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]

   if 0.0155 < phi1

   1. Initial program 74.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 56.1

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 56.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 27.9%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 22.3%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 22.2%

       \[\leadsto \cos^{-1} \cos \phi_1 \cdot R \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 11: 39.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot R\\ t_1 := \cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -15.5:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.9 \cdot 10^{-283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 7.2 \cdot 10^{-308}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 2.35 \cdot 10^{-263}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 3.6 \cdot 10^{-239}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (acos (* (cos lambda1) (cos phi2))) R))
        (t_1
         (*
          (acos (* (fma -0.5 (* phi1 phi1) 1.0) (cos (- lambda1 lambda2))))
          R)))
   (if (<= phi1 -15.5)
     (* (acos (* (cos lambda1) (cos phi1))) R)
     (if (<= phi1 -1.9e-283)
       t_0
       (if (<= phi1 7.2e-308)
         t_1
         (if (<= phi1 2.35e-263)
           t_0
           (if (<= phi1 3.6e-239)
             t_1
             (if (<= phi1 2.1e-16) t_0 (* (acos (cos phi1)) R)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos((cos(lambda1) * cos(phi2))) * R;
	double t_1 = acos((fma(-0.5, (phi1 * phi1), 1.0) * cos((lambda1 - lambda2)))) * R;
	double tmp;
	if (phi1 <= -15.5) {
		tmp = acos((cos(lambda1) * cos(phi1))) * R;
	} else if (phi1 <= -1.9e-283) {
		tmp = t_0;
	} else if (phi1 <= 7.2e-308) {
		tmp = t_1;
	} else if (phi1 <= 2.35e-263) {
		tmp = t_0;
	} else if (phi1 <= 3.6e-239) {
		tmp = t_1;
	} else if (phi1 <= 2.1e-16) {
		tmp = t_0;
	} else {
		tmp = acos(cos(phi1)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(Float64(cos(lambda1) * cos(phi2))) * R)
	t_1 = Float64(acos(Float64(fma(-0.5, Float64(phi1 * phi1), 1.0) * cos(Float64(lambda1 - lambda2)))) * R)
	tmp = 0.0
	if (phi1 <= -15.5)
		tmp = Float64(acos(Float64(cos(lambda1) * cos(phi1))) * R);
	elseif (phi1 <= -1.9e-283)
		tmp = t_0;
	elseif (phi1 <= 7.2e-308)
		tmp = t_1;
	elseif (phi1 <= 2.35e-263)
		tmp = t_0;
	elseif (phi1 <= 3.6e-239)
		tmp = t_1;
	elseif (phi1 <= 2.1e-16)
		tmp = t_0;
	else
		tmp = Float64(acos(cos(phi1)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcCos[N[(N[(-0.5 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -15.5], N[(N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, -1.9e-283], t$95$0, If[LessEqual[phi1, 7.2e-308], t$95$1, If[LessEqual[phi1, 2.35e-263], t$95$0, If[LessEqual[phi1, 3.6e-239], t$95$1, If[LessEqual[phi1, 2.1e-16], t$95$0, N[(N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -15.5

   1. Initial program 84.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 47.0

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 47.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 41.7%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \cos \color{blue}{\phi_1}\right) \cdot R \]

   if -15.5 < phi1 < -1.9000000000000001e-283 or 7.1999999999999997e-308 < phi1 < 2.34999999999999985e-263 or 3.6000000000000001e-239 < phi1 < 2.1000000000000001e-16

   1. Initial program 65.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 48.6

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 48.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]

   if -1.9000000000000001e-283 < phi1 < 7.1999999999999997e-308 or 2.34999999999999985e-263 < phi1 < 3.6000000000000001e-239

   1. Initial program 72.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 54.1

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 54.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \color{blue}{\frac{-1}{2} \cdot \left({\phi_1}^{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 54.1%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

   if 2.1000000000000001e-16 < phi1

   1. Initial program 75.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 57.4

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 57.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 27.4%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 22.0%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 21.7%

       \[\leadsto \cos^{-1} \cos \phi_1 \cdot R \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 52.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -15.5:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 1.55 \cdot 10^{-27}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \cos \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi1 -15.5)
     (* (acos (* t_0 (cos phi1))) R)
     (if (<= phi1 1.55e-27)
       (* (acos (* t_0 (cos phi2))) R)
       (* (acos (cos phi1)) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -15.5) {
		tmp = acos((t_0 * cos(phi1))) * R;
	} else if (phi1 <= 1.55e-27) {
		tmp = acos((t_0 * cos(phi2))) * R;
	} else {
		tmp = acos(cos(phi1)) * R;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (phi1 <= (-15.5d0)) then
        tmp = acos((t_0 * cos(phi1))) * r
    else if (phi1 <= 1.55d-27) then
        tmp = acos((t_0 * cos(phi2))) * r
    else
        tmp = acos(cos(phi1)) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -15.5) {
		tmp = Math.acos((t_0 * Math.cos(phi1))) * R;
	} else if (phi1 <= 1.55e-27) {
		tmp = Math.acos((t_0 * Math.cos(phi2))) * R;
	} else {
		tmp = Math.acos(Math.cos(phi1)) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi1 <= -15.5:
		tmp = math.acos((t_0 * math.cos(phi1))) * R
	elif phi1 <= 1.55e-27:
		tmp = math.acos((t_0 * math.cos(phi2))) * R
	else:
		tmp = math.acos(math.cos(phi1)) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -15.5)
		tmp = Float64(acos(Float64(t_0 * cos(phi1))) * R);
	elseif (phi1 <= 1.55e-27)
		tmp = Float64(acos(Float64(t_0 * cos(phi2))) * R);
	else
		tmp = Float64(acos(cos(phi1)) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi1 <= -15.5)
		tmp = acos((t_0 * cos(phi1))) * R;
	elseif (phi1 <= 1.55e-27)
		tmp = acos((t_0 * cos(phi2))) * R;
	else
		tmp = acos(cos(phi1)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -15.5], N[(N[ArcCos[N[(t$95$0 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 1.55e-27], N[(N[ArcCos[N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -15.5

   1. Initial program 84.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 47.0

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 47.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   if -15.5 < phi1 < 1.5499999999999999e-27

   1. Initial program 65.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_2\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_2\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_2\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_2\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      18. lower-cos.f64 65.2

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]

   5. Applied rewrites 65.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right)} \cdot R \]

   if 1.5499999999999999e-27 < phi1

   1. Initial program 75.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 57.2

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 57.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 26.7%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 21.5%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 21.1%

       \[\leadsto \cos^{-1} \cos \phi_1 \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 33.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -32500:\\ \;\;\;\;\cos^{-1} \left(\cos \lambda_2 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 2.6 \cdot 10^{-127}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -32500.0)
   (* (acos (* (cos lambda2) (cos phi1))) R)
   (if (<= phi1 2.6e-127)
     (* (acos (* (fma -0.5 (* phi1 phi1) 1.0) (cos (- lambda1 lambda2)))) R)
     (* (acos (cos phi1)) R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -32500.0) {
		tmp = acos((cos(lambda2) * cos(phi1))) * R;
	} else if (phi1 <= 2.6e-127) {
		tmp = acos((fma(-0.5, (phi1 * phi1), 1.0) * cos((lambda1 - lambda2)))) * R;
	} else {
		tmp = acos(cos(phi1)) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -32500.0)
		tmp = Float64(acos(Float64(cos(lambda2) * cos(phi1))) * R);
	elseif (phi1 <= 2.6e-127)
		tmp = Float64(acos(Float64(fma(-0.5, Float64(phi1 * phi1), 1.0) * cos(Float64(lambda1 - lambda2)))) * R);
	else
		tmp = Float64(acos(cos(phi1)) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -32500.0], N[(N[ArcCos[N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 2.6e-127], N[(N[ArcCos[N[(N[(-0.5 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -32500

   1. Initial program 84.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 47.0

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 47.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 33.2%

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   if -32500 < phi1 < 2.59999999999999991e-127

   1. Initial program 66.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 38.7

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 38.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \color{blue}{\frac{-1}{2} \cdot \left({\phi_1}^{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 38.5%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

   if 2.59999999999999991e-127 < phi1

   1. Initial program 72.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 55.0

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 55.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 25.6%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 21.6%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 17.6%

       \[\leadsto \cos^{-1} \cos \phi_1 \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 24.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right)\\ t_1 := \cos^{-1} \cos \phi_1 \cdot R\\ \mathbf{if}\;\phi_1 \leq -0.54:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 6.2 \cdot 10^{-284}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 4.6 \cdot 10^{-235}:\\ \;\;\;\;\cos^{-1} \left(t\_0 \cdot \cos \lambda_2\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 1.05 \cdot 10^{-114}:\\ \;\;\;\;\cos^{-1} \cos \phi_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma -0.5 (* phi1 phi1) 1.0)) (t_1 (* (acos (cos phi1)) R)))
   (if (<= phi1 -0.54)
     t_1
     (if (<= phi1 6.2e-284)
       (* (acos (* t_0 (cos lambda1))) R)
       (if (<= phi1 4.6e-235)
         (* (acos (* t_0 (cos lambda2))) R)
         (if (<= phi1 1.05e-114) (* (acos (cos phi2)) R) t_1))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(-0.5, (phi1 * phi1), 1.0);
	double t_1 = acos(cos(phi1)) * R;
	double tmp;
	if (phi1 <= -0.54) {
		tmp = t_1;
	} else if (phi1 <= 6.2e-284) {
		tmp = acos((t_0 * cos(lambda1))) * R;
	} else if (phi1 <= 4.6e-235) {
		tmp = acos((t_0 * cos(lambda2))) * R;
	} else if (phi1 <= 1.05e-114) {
		tmp = acos(cos(phi2)) * R;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(-0.5, Float64(phi1 * phi1), 1.0)
	t_1 = Float64(acos(cos(phi1)) * R)
	tmp = 0.0
	if (phi1 <= -0.54)
		tmp = t_1;
	elseif (phi1 <= 6.2e-284)
		tmp = Float64(acos(Float64(t_0 * cos(lambda1))) * R);
	elseif (phi1 <= 4.6e-235)
		tmp = Float64(acos(Float64(t_0 * cos(lambda2))) * R);
	elseif (phi1 <= 1.05e-114)
		tmp = Float64(acos(cos(phi2)) * R);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(-0.5 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -0.54], t$95$1, If[LessEqual[phi1, 6.2e-284], N[(N[ArcCos[N[(t$95$0 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 4.6e-235], N[(N[ArcCos[N[(t$95$0 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 1.05e-114], N[(N[ArcCos[N[Cos[phi2], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -0.54000000000000004 or 1.04999999999999996e-114 < phi1

   1. Initial program 76.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 60.3

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 60.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 32.5%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 24.0%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 22.2%

       \[\leadsto \cos^{-1} \cos \phi_1 \cdot R \]

   if -0.54000000000000004 < phi1 < 6.1999999999999996e-284

   1. Initial program 69.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 39.9

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 39.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \color{blue}{\frac{-1}{2} \cdot \left({\phi_1}^{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 39.9%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\frac{-1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \lambda_1\right) \cdot R \]
   10. Step-by-step derivation

   11. Applied rewrites 31.7%

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \lambda_1\right) \cdot R \]

   if 6.1999999999999996e-284 < phi1 < 4.59999999999999995e-235

   1. Initial program 61.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 43.9

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 43.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \color{blue}{\frac{-1}{2} \cdot \left({\phi_1}^{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 43.9%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\frac{-1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot R \]
   10. Step-by-step derivation

   11. Applied rewrites 23.9%

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \lambda_2\right) \cdot R \]

   if 4.59999999999999995e-235 < phi1 < 1.04999999999999996e-114

   1. Initial program 65.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 44.9

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 44.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 26.9%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 26.9%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto \cos^{-1} \cos \phi_2 \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 26.9%

       \[\leadsto \cos^{-1} \cos \phi_2 \cdot R \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 30.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -32500 \lor \neg \left(\phi_1 \leq 1.32 \cdot 10^{-129}\right):\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -32500.0) (not (<= phi1 1.32e-129)))
   (* (acos (cos phi1)) R)
   (* (acos (* (fma -0.5 (* phi1 phi1) 1.0) (cos (- lambda1 lambda2)))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -32500.0) || !(phi1 <= 1.32e-129)) {
		tmp = acos(cos(phi1)) * R;
	} else {
		tmp = acos((fma(-0.5, (phi1 * phi1), 1.0) * cos((lambda1 - lambda2)))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -32500.0) || !(phi1 <= 1.32e-129))
		tmp = Float64(acos(cos(phi1)) * R);
	else
		tmp = Float64(acos(Float64(fma(-0.5, Float64(phi1 * phi1), 1.0) * cos(Float64(lambda1 - lambda2)))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -32500.0], N[Not[LessEqual[phi1, 1.32e-129]], $MachinePrecision]], N[(N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[(N[(-0.5 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -32500 or 1.31999999999999992e-129 < phi1

   1. Initial program 77.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 61.0

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 61.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 32.8%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 24.3%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 22.0%

       \[\leadsto \cos^{-1} \cos \phi_1 \cdot R \]

   if -32500 < phi1 < 1.31999999999999992e-129

   1. Initial program 66.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 38.7

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 38.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \color{blue}{\frac{-1}{2} \cdot \left({\phi_1}^{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 38.5%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -32500 \lor \neg \left(\phi_1 \leq 1.32 \cdot 10^{-129}\right):\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 25.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \cos \phi_1 \cdot R\\ \mathbf{if}\;\phi_1 \leq -0.54:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 3.15 \cdot 10^{-217}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 3.4 \cdot 10^{-36}:\\ \;\;\;\;\cos^{-1} \cos \phi_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (acos (cos phi1)) R)))
   (if (<= phi1 -0.54)
     t_0
     (if (<= phi1 3.15e-217)
       (* (acos (* (fma -0.5 (* phi1 phi1) 1.0) (cos lambda1))) R)
       (if (<= phi1 3.4e-36) (* (acos (cos phi2)) R) t_0)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos(cos(phi1)) * R;
	double tmp;
	if (phi1 <= -0.54) {
		tmp = t_0;
	} else if (phi1 <= 3.15e-217) {
		tmp = acos((fma(-0.5, (phi1 * phi1), 1.0) * cos(lambda1))) * R;
	} else if (phi1 <= 3.4e-36) {
		tmp = acos(cos(phi2)) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(cos(phi1)) * R)
	tmp = 0.0
	if (phi1 <= -0.54)
		tmp = t_0;
	elseif (phi1 <= 3.15e-217)
		tmp = Float64(acos(Float64(fma(-0.5, Float64(phi1 * phi1), 1.0) * cos(lambda1))) * R);
	elseif (phi1 <= 3.4e-36)
		tmp = Float64(acos(cos(phi2)) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -0.54], t$95$0, If[LessEqual[phi1, 3.15e-217], N[(N[ArcCos[N[(N[(-0.5 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 3.4e-36], N[(N[ArcCos[N[Cos[phi2], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.54000000000000004 or 3.4000000000000003e-36 < phi1

   1. Initial program 79.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 62.7

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 62.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 34.2%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 24.6%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 24.2%

       \[\leadsto \cos^{-1} \cos \phi_1 \cdot R \]

   if -0.54000000000000004 < phi1 < 3.14999999999999999e-217

   1. Initial program 69.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. cos-neg-rev N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. *-lft-identity N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{1 \cdot \lambda_2}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      7. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + -1 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)\right)\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right)\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      12. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. remove-double-neg N/A

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

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      15. metadata-eval N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{1} \cdot \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

      16. *-lft-identity N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \color{blue}{\lambda_1}\right) \cdot \cos \phi_1\right) \cdot R \]

      17. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      18. lower-cos.f64 39.9

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 39.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \color{blue}{\frac{-1}{2} \cdot \left({\phi_1}^{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 39.9%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\frac{-1}{2}, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \lambda_1\right) \cdot R \]
   10. Step-by-step derivation

   11. Applied rewrites 30.5%

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(-0.5, \phi_1 \cdot \phi_1, 1\right) \cdot \cos \lambda_1\right) \cdot R \]

   if 3.14999999999999999e-217 < phi1 < 3.4000000000000003e-36

   1. Initial program 58.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 43.2

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 43.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 21.6%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 21.6%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto \cos^{-1} \cos \phi_2 \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 21.6%

       \[\leadsto \cos^{-1} \cos \phi_2 \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 24.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -15.5 \lor \neg \left(\phi_1 \leq 3 \cdot 10^{-158}\right):\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -15.5) (not (<= phi1 3e-158)))
   (* (acos (cos phi1)) R)
   (* (acos (cos phi2)) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -15.5) || !(phi1 <= 3e-158)) {
		tmp = acos(cos(phi1)) * R;
	} else {
		tmp = acos(cos(phi2)) * R;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((phi1 <= (-15.5d0)) .or. (.not. (phi1 <= 3d-158))) then
        tmp = acos(cos(phi1)) * r
    else
        tmp = acos(cos(phi2)) * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -15.5) || !(phi1 <= 3e-158)) {
		tmp = Math.acos(Math.cos(phi1)) * R;
	} else {
		tmp = Math.acos(Math.cos(phi2)) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi1 <= -15.5) or not (phi1 <= 3e-158):
		tmp = math.acos(math.cos(phi1)) * R
	else:
		tmp = math.acos(math.cos(phi2)) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -15.5) || !(phi1 <= 3e-158))
		tmp = Float64(acos(cos(phi1)) * R);
	else
		tmp = Float64(acos(cos(phi2)) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi1 <= -15.5) || ~((phi1 <= 3e-158)))
		tmp = acos(cos(phi1)) * R;
	else
		tmp = acos(cos(phi2)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -15.5], N[Not[LessEqual[phi1, 3e-158]], $MachinePrecision]], N[(N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(N[ArcCos[N[Cos[phi2], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -15.5 or 3e-158 < phi1

   1. Initial program 75.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 59.1

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 59.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 31.3%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 23.3%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 21.0%

       \[\leadsto \cos^{-1} \cos \phi_1 \cdot R \]

   if -15.5 < phi1 < 3e-158

   1. Initial program 68.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      11. lower-sin.f64 50.8

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 50.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 23.4%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
   9. Step-by-step derivation

   10. Applied rewrites 23.0%

       \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto \cos^{-1} \cos \phi_2 \cdot R \]
   12. Step-by-step derivation

   13. Applied rewrites 22.8%

       \[\leadsto \cos^{-1} \cos \phi_2 \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -15.5 \lor \neg \left(\phi_1 \leq 3 \cdot 10^{-158}\right):\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 26.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (acos (cos (- phi2 phi1))) R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(cos((phi2 - phi1))) * R;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = acos(cos((phi2 - phi1))) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos(Math.cos((phi2 - phi1))) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(math.cos((phi2 - phi1))) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(cos(Float64(phi2 - phi1))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(cos((phi2 - phi1))) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 73.0%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Add Preprocessing

3. Taylor expanded in lambda2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

   2. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   3. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   4. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   5. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   6. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   7. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   8. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

   9. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

   10. lower-sin.f64 N/A

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

   11. lower-sin.f64 56.1

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

5. Applied rewrites 56.1%

   \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

6. Taylor expanded in lambda1 around 0

   \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
7. Step-by-step derivation

8. Applied rewrites 28.5%

   \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
9. Step-by-step derivation

10. Applied rewrites 23.2%

    \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]
11. Add Preprocessing

Alternative 19: 17.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \cos \phi_1 \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (acos (cos phi1)) R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(cos(phi1)) * R;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = acos(cos(phi1)) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos(Math.cos(phi1)) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(math.cos(phi1)) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(cos(phi1)) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(cos(phi1)) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 73.0%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Add Preprocessing

3. Taylor expanded in lambda2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

   2. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   3. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   4. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   5. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   6. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1}, \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   7. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   8. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

   9. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

   10. lower-sin.f64 N/A

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

   11. lower-sin.f64 56.1

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

5. Applied rewrites 56.1%

   \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

6. Taylor expanded in lambda1 around 0

   \[\leadsto \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right) \cdot R \]
7. Step-by-step derivation

8. Applied rewrites 28.5%

   \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1}, \sin \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]
9. Step-by-step derivation

10. Applied rewrites 23.2%

    \[\leadsto \color{blue}{\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R} \]

11. Taylor expanded in phi2 around 0

    \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\phi_1\right)\right) \cdot R \]
12. Step-by-step derivation

13. Applied rewrites 15.8%

    \[\leadsto \cos^{-1} \cos \phi_1 \cdot R \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024338 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  :precision binary64
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))