Spherical law of cosines

Percentage Accurate: 73.6% → 96.5%

Time: 24.1s

Alternatives: 27

Speedup: 1.0×

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: 73.6% 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: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\cos^{-1} \left(t\_0 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + t\_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (* (cos phi1) (cos phi2))))
   (if (<= (acos (+ t_0 (* t_1 (cos (- lambda1 lambda2))))) 0.0)
     (* (- lambda2 lambda1) R)
     (*
      R
      (acos
       (+
        t_0
        (*
         t_1
         (fma
          (sin lambda2)
          (sin lambda1)
          (* (cos lambda1) (cos lambda2))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if (acos((t_0 + (t_1 * cos((lambda1 - lambda2))))) <= 0.0) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos((t_0 + (t_1 * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (acos(Float64(t_0 + Float64(t_1 * cos(Float64(lambda1 - lambda2))))) <= 0.0)
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(t_1 * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2)))))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcCos[N[(t$95$0 + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(t$95$1 * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) < 0.0

   1. Initial program 24.5%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 24.5

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

   5. Applied rewrites 24.5%

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

   6. Taylor expanded in phi2 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 24.5

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

   8. Applied rewrites 24.5%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. cos-diff N/A

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

      13. lift--.f64 N/A

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

      14. acos-cos-s 23.6

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

   10. Applied rewrites 23.6%

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

   if 0.0 < (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

   1. Initial program 79.8%

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

      1. cos-diff N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 98.7

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

   4. Applied rewrites 98.7%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<=
        (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
        0.0)
     (* (- lambda2 lambda1) R)
     (*
      R
      (acos
       (fma
        (cos phi2)
        (*
         (cos phi1)
         (fma (sin lambda1) (sin lambda2) (* (cos lambda1) (cos lambda2))))
        t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) <= 0.0) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos(fma(cos(phi2), (cos(phi1) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2)))), t_0));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) <= 0.0)
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(fma(cos(phi2), Float64(cos(phi1) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2)))), t_0)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) < 0.0

   1. Initial program 24.5%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 24.5

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

   5. Applied rewrites 24.5%

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

   6. Taylor expanded in phi2 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 24.5

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

   8. Applied rewrites 24.5%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. cos-diff N/A

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

      13. lift--.f64 N/A

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

      14. acos-cos-s 23.6

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

   10. Applied rewrites 23.6%

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

   if 0.0 < (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

   1. Initial program 79.8%

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

      1. cos-diff N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in phi1 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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 98.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, t\_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.35 \cdot 10^{-44}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t\_0\right)\right) \cdot \left(-R\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))
   (if (<= phi1 -4.5e-9)
     (* R (acos (fma (sin phi2) (sin phi1) t_0)))
     (if (<= phi1 1.35e-44)
       (*
        R
        (acos
         (*
          (cos phi2)
          (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))))
       (fma (* PI 0.5) R (* (asin (fma (sin phi1) (sin phi2) t_0)) (- R)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2)));
	double tmp;
	if (phi1 <= -4.5e-9) {
		tmp = R * acos(fma(sin(phi2), sin(phi1), t_0));
	} else if (phi1 <= 1.35e-44) {
		tmp = R * acos((cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = fma((((double) M_PI) * 0.5), R, (asin(fma(sin(phi1), sin(phi2), t_0)) * -R));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))
	tmp = 0.0
	if (phi1 <= -4.5e-9)
		tmp = Float64(R * acos(fma(sin(phi2), sin(phi1), t_0)));
	elseif (phi1 <= 1.35e-44)
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = fma(Float64(pi * 0.5), R, Float64(asin(fma(sin(phi1), sin(phi2), t_0)) * Float64(-R)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -4.5e-9], N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 1.35e-44], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] * R + N[(N[ArcSin[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] * (-R)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -4.49999999999999976e-9

   1. Initial program 84.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      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. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 84.3

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 84.3

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

   4. Applied rewrites 84.3%

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

   if -4.49999999999999976e-9 < phi1 < 1.35e-44

   1. Initial program 71.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 71.4

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

   5. Applied rewrites 71.4%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 90.4

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

   7. Applied rewrites 90.4%

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

   if 1.35e-44 < phi1

   1. Initial program 78.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-acos.f64 N/A

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

      12. *-commutative N/A

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

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi \cdot 0.5, R, \left(-\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right) \cdot R\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.35 \cdot 10^{-44}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(-R\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t\_0\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.35 \cdot 10^{-44}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= phi1 -4.5e-9)
     (* R (acos (fma (sin phi2) (sin phi1) (* (cos phi1) (* (cos phi2) t_0)))))
     (if (<= phi1 1.35e-44)
       (*
        R
        (acos
         (*
          (cos phi2)
          (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))))
       (*
        (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) t_0)))
        R)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -4.5e-9) {
		tmp = R * acos(fma(sin(phi2), sin(phi1), (cos(phi1) * (cos(phi2) * t_0))));
	} else if (phi1 <= 1.35e-44) {
		tmp = R * acos((cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * t_0))) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -4.5e-9)
		tmp = Float64(R * acos(fma(sin(phi2), sin(phi1), Float64(cos(phi1) * Float64(cos(phi2) * t_0)))));
	elseif (phi1 <= 1.35e-44)
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * t_0))) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -4.5e-9], N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 1.35e-44], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 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] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -4.49999999999999976e-9

   1. Initial program 84.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      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. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 84.3

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 84.3

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

   4. Applied rewrites 84.3%

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

   if -4.49999999999999976e-9 < phi1 < 1.35e-44

   1. Initial program 71.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 71.4

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

   5. Applied rewrites 71.4%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 90.4

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

   7. Applied rewrites 90.4%

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

   if 1.35e-44 < phi1

   1. Initial program 78.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.35 \cdot 10^{-44}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{if}\;\phi_1 \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 1.35 \cdot 10^{-44}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          R
          (acos
           (fma
            (sin phi2)
            (sin phi1)
            (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))))
   (if (<= phi1 -4.5e-9)
     t_0
     (if (<= phi1 1.35e-44)
       (*
        R
        (acos
         (*
          (cos phi2)
          (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))))
       t_0))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * acos(fma(sin(phi2), sin(phi1), (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))));
	double tmp;
	if (phi1 <= -4.5e-9) {
		tmp = t_0;
	} else if (phi1 <= 1.35e-44) {
		tmp = R * acos((cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * acos(fma(sin(phi2), sin(phi1), Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))))
	tmp = 0.0
	if (phi1 <= -4.5e-9)
		tmp = t_0;
	elseif (phi1 <= 1.35e-44)
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -4.5e-9], t$95$0, If[LessEqual[phi1, 1.35e-44], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.49999999999999976e-9 or 1.35e-44 < phi1

   1. Initial program 81.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      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. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 81.2

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 81.2

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

   4. Applied rewrites 81.2%

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

   if -4.49999999999999976e-9 < phi1 < 1.35e-44

   1. Initial program 71.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 71.4

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

   5. Applied rewrites 71.4%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 90.4

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

   7. Applied rewrites 90.4%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.35 \cdot 10^{-44}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 1:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \lambda_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -3.1e-6)
   (*
    R
    (acos
     (fma (sin phi2) (sin phi1) (* (* (cos phi1) (cos phi2)) (cos lambda1)))))
   (if (<= lambda1 1.0)
     (*
      R
      (acos
       (fma
        (cos phi1)
        (* (cos phi2) (cos lambda2))
        (* (sin phi1) (sin phi2)))))
     (*
      R
      (acos
       (*
        (cos phi2)
        (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1)))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3.1e-6) {
		tmp = R * acos(fma(sin(phi2), sin(phi1), ((cos(phi1) * cos(phi2)) * cos(lambda1))));
	} else if (lambda1 <= 1.0) {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * cos(lambda2)), (sin(phi1) * sin(phi2))));
	} else {
		tmp = R * acos((cos(phi2) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1)))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -3.1e-6)
		tmp = Float64(R * acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda1)))));
	elseif (lambda1 <= 1.0)
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * cos(lambda2)), Float64(sin(phi1) * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -3.1e-6], N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 1.0], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -3.1e-6

   1. Initial program 65.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 65.0

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

   5. Applied rewrites 65.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 65.0

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

   7. Applied rewrites 65.0%

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

   if -3.1e-6 < lambda1 < 1

   1. Initial program 89.7%

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

      1. cos-diff N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 90.0

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

   4. Applied rewrites 90.0%

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

   5. Taylor expanded in phi1 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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 90.0%

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

   8. Taylor expanded in lambda1 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 89.3

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

   10. Applied rewrites 89.3%

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

   if 1 < lambda1

   1. Initial program 61.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 36.4

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

   5. Applied rewrites 36.4%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 56.5

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

   7. Applied rewrites 56.5%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 1:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \lambda_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, t\_0\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 1:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \lambda_2, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= lambda1 -3.1e-6)
     (* R (acos (fma (cos phi2) (* (cos phi1) (cos lambda1)) t_0)))
     (if (<= lambda1 1.0)
       (* R (acos (fma (cos phi1) (* (cos phi2) (cos lambda2)) t_0)))
       (*
        R
        (acos
         (*
          (cos phi2)
          (fma
           (cos lambda2)
           (cos lambda1)
           (* (sin lambda2) (sin lambda1))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -3.1e-6) {
		tmp = R * acos(fma(cos(phi2), (cos(phi1) * cos(lambda1)), t_0));
	} else if (lambda1 <= 1.0) {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * cos(lambda2)), t_0));
	} else {
		tmp = R * acos((cos(phi2) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1)))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -3.1e-6)
		tmp = Float64(R * acos(fma(cos(phi2), Float64(cos(phi1) * cos(lambda1)), t_0)));
	elseif (lambda1 <= 1.0)
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * cos(lambda2)), t_0)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -3.1e-6], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 1.0], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -3.1e-6

   1. Initial program 65.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 65.0

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

   5. Applied rewrites 65.0%

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

   if -3.1e-6 < lambda1 < 1

   1. Initial program 89.7%

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

      1. cos-diff N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 90.0

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

   4. Applied rewrites 90.0%

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

   5. Taylor expanded in phi1 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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 90.0%

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

   8. Taylor expanded in lambda1 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 89.3

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

   10. Applied rewrites 89.3%

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

   if 1 < lambda1

   1. Initial program 61.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 36.4

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

   5. Applied rewrites 36.4%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 56.5

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

   7. Applied rewrites 56.5%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 1:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \lambda_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_2 \leq -1.65 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \lambda_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos lambda1) (cos lambda2))))
   (if (<= phi2 -1.65e-5)
     (*
      R
      (acos
       (fma
        (cos phi1)
        (* (cos phi2) (cos lambda2))
        (* (sin phi1) (sin phi2)))))
     (if (<= phi2 2e-8)
       (* R (acos (* (cos phi1) (fma (sin lambda1) (sin lambda2) t_0))))
       (* R (acos (* (cos phi2) (fma (sin lambda2) (sin lambda1) t_0))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(lambda1) * cos(lambda2);
	double tmp;
	if (phi2 <= -1.65e-5) {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * cos(lambda2)), (sin(phi1) * sin(phi2))));
	} else if (phi2 <= 2e-8) {
		tmp = R * acos((cos(phi1) * fma(sin(lambda1), sin(lambda2), t_0)));
	} else {
		tmp = R * acos((cos(phi2) * fma(sin(lambda2), sin(lambda1), t_0)));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(lambda1) * cos(lambda2))
	tmp = 0.0
	if (phi2 <= -1.65e-5)
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * cos(lambda2)), Float64(sin(phi1) * sin(phi2)))));
	elseif (phi2 <= 2e-8)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(sin(lambda1), sin(lambda2), t_0))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), t_0))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.65e-5], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2e-8], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.6500000000000001e-5

   1. Initial program 80.7%

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

      1. cos-diff N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in phi1 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. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in lambda1 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 58.6

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

   10. Applied rewrites 58.6%

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

   if -1.6500000000000001e-5 < phi2 < 2e-8

   1. Initial program 72.7%

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

      1. cos-diff N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 90.7

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

   4. Applied rewrites 90.7%

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

   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. lower-fma.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-cos.f64 89.8

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

   7. Applied rewrites 89.8%

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

   if 2e-8 < phi2

   1. Initial program 81.3%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 47.9

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

   5. Applied rewrites 47.9%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 59.0

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

   7. Applied rewrites 59.0%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.65 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \lambda_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_1 \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos lambda1) (cos lambda2))))
   (if (<= phi1 -2.9e-6)
     (* R (acos (* (cos phi1) (fma (sin lambda1) (sin lambda2) t_0))))
     (* R (acos (* (cos phi2) (fma (sin lambda2) (sin lambda1) t_0)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(lambda1) * cos(lambda2);
	double tmp;
	if (phi1 <= -2.9e-6) {
		tmp = R * acos((cos(phi1) * fma(sin(lambda1), sin(lambda2), t_0)));
	} else {
		tmp = R * acos((cos(phi2) * fma(sin(lambda2), sin(lambda1), t_0)));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(lambda1) * cos(lambda2))
	tmp = 0.0
	if (phi1 <= -2.9e-6)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(sin(lambda1), sin(lambda2), t_0))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), t_0))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2.9e-6], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.9000000000000002e-6

   1. Initial program 84.3%

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

      1. cos-diff N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 99.4

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

   4. Applied rewrites 99.4%

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

   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. lower-fma.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-cos.f64 52.4

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

   7. Applied rewrites 52.4%

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

   if -2.9000000000000002e-6 < phi1

   1. Initial program 74.3%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 52.5

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

   5. Applied rewrites 52.5%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 64.3

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

   7. Applied rewrites 64.3%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -7.5e-6)
   (*
    R
    (acos
     (+ (* (sin phi1) (sin phi2)) (* (cos phi1) (cos (- lambda2 lambda1))))))
   (*
    R
    (acos
     (*
      (cos phi2)
      (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -7.5e-6) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos((lambda2 - lambda1)))));
	} else {
		tmp = R * acos((cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -7.5e-6)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * cos(Float64(lambda2 - lambda1))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -7.5e-6], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -7.50000000000000019e-6

   1. Initial program 84.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 45.8

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

   5. Applied rewrites 45.8%

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

   if -7.50000000000000019e-6 < phi1

   1. Initial program 74.3%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 52.5

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

   5. Applied rewrites 52.5%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 64.3

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

   7. Applied rewrites 64.3%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -7.5e-6)
   (*
    R
    (acos
     (+ (* (sin phi1) (sin phi2)) (* (cos phi1) (cos (- lambda2 lambda1))))))
   (*
    R
    (acos
     (*
      (cos phi2)
      (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -7.5e-6) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos((lambda2 - lambda1)))));
	} else {
		tmp = R * acos((cos(phi2) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1)))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -7.5e-6)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * cos(Float64(lambda2 - lambda1))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -7.5e-6], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -7.50000000000000019e-6

   1. Initial program 84.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 45.8

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

   5. Applied rewrites 45.8%

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

   if -7.50000000000000019e-6 < phi1

   1. Initial program 74.3%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 52.5

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

   5. Applied rewrites 52.5%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 64.3

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

   7. Applied rewrites 64.3%

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi1 -3.7e-6)
     (* R (acos (+ t_1 (* (cos phi1) t_0))))
     (* R (acos (+ t_1 (* (cos phi2) t_0)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi1 <= -3.7e-6) {
		tmp = R * acos((t_1 + (cos(phi1) * t_0)));
	} else {
		tmp = R * acos((t_1 + (cos(phi2) * t_0)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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) :: t_1
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    t_1 = sin(phi1) * sin(phi2)
    if (phi1 <= (-3.7d-6)) then
        tmp = r * acos((t_1 + (cos(phi1) * t_0)))
    else
        tmp = r * acos((t_1 + (cos(phi2) * t_0)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double t_1 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (phi1 <= -3.7e-6) {
		tmp = R * Math.acos((t_1 + (Math.cos(phi1) * t_0)));
	} else {
		tmp = R * Math.acos((t_1 + (Math.cos(phi2) * t_0)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	t_1 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if phi1 <= -3.7e-6:
		tmp = R * math.acos((t_1 + (math.cos(phi1) * t_0)))
	else:
		tmp = R * math.acos((t_1 + (math.cos(phi2) * t_0)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi1 <= -3.7e-6)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi1) * t_0))));
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi2) * t_0))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	t_1 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (phi1 <= -3.7e-6)
		tmp = R * acos((t_1 + (cos(phi1) * t_0)));
	else
		tmp = R * acos((t_1 + (cos(phi2) * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -3.7e-6], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.7000000000000002e-6

   1. Initial program 84.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 45.8

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

   5. Applied rewrites 45.8%

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

   if -3.7000000000000002e-6 < phi1

   1. Initial program 74.3%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 52.4

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

   5. Applied rewrites 52.4%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi1 -3.7e-6)
     (* R (acos (* (cos phi1) t_0)))
     (* R (acos (+ (* (sin phi1) (sin phi2)) (* (cos phi2) t_0)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -3.7e-6) {
		tmp = R * acos((cos(phi1) * t_0));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi2) * t_0)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 <= (-3.7d-6)) then
        tmp = r * acos((cos(phi1) * t_0))
    else
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi2) * t_0)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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 <= -3.7e-6) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi2) * t_0)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi1 <= -3.7e-6:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi2) * t_0)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -3.7e-6)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi2) * t_0))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi1 <= -3.7e-6)
		tmp = R * acos((cos(phi1) * t_0));
	else
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi2) * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -3.7e-6], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.7000000000000002e-6

   1. Initial program 84.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 46.3

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

   5. Applied rewrites 46.3%

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

   if -3.7000000000000002e-6 < phi1

   1. Initial program 74.3%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 52.4

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

   5. Applied rewrites 52.4%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq -0.37:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 0.17:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(t\_0, \cos \phi_1 \cdot \mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), \sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot -0.16666666666666666, \phi_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi2 -0.37)
     (* R (acos (fma (cos phi2) (cos phi1) (* (sin phi1) (sin phi2)))))
     (if (<= phi2 0.17)
       (*
        R
        (acos
         (fma
          t_0
          (* (cos phi1) (fma -0.5 (* phi2 phi2) 1.0))
          (*
           (sin phi1)
           (fma phi2 (* (* phi2 phi2) -0.16666666666666666) phi2)))))
       (* R (acos (* (cos phi2) t_0)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= -0.37) {
		tmp = R * acos(fma(cos(phi2), cos(phi1), (sin(phi1) * sin(phi2))));
	} else if (phi2 <= 0.17) {
		tmp = R * acos(fma(t_0, (cos(phi1) * fma(-0.5, (phi2 * phi2), 1.0)), (sin(phi1) * fma(phi2, ((phi2 * phi2) * -0.16666666666666666), phi2))));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= -0.37)
		tmp = Float64(R * acos(fma(cos(phi2), cos(phi1), Float64(sin(phi1) * sin(phi2)))));
	elseif (phi2 <= 0.17)
		tmp = Float64(R * acos(fma(t_0, Float64(cos(phi1) * fma(-0.5, Float64(phi2 * phi2), 1.0)), Float64(sin(phi1) * fma(phi2, Float64(Float64(phi2 * phi2) * -0.16666666666666666), phi2)))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -0.37], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 0.17], N[(R * N[ArcCos[N[(t$95$0 * N[(N[Cos[phi1], $MachinePrecision] * N[(-0.5 * N[(phi2 * phi2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[(phi2 * N[(N[(phi2 * phi2), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.37

   1. Initial program 80.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 56.5

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

   5. Applied rewrites 56.5%

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

   6. Taylor expanded in lambda1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 36.0

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

   8. Applied rewrites 36.0%

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

   if -0.37 < phi2 < 0.170000000000000012

   1. Initial program 72.5%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\phi_2 \cdot \left(\sin \phi_1 + \phi_2 \cdot \left(\frac{-1}{2} \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \frac{-1}{6} \cdot \left(\phi_2 \cdot \sin \phi_1\right)\right)\right) + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   5. Applied rewrites 72.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right) \cdot \cos \phi_1, \sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, -0.16666666666666666 \cdot \left(\phi_2 \cdot \phi_2\right), \phi_2\right)\right)\right)} \cdot R \]

   if 0.170000000000000012 < phi2

   1. Initial program 82.3%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 48.3

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

   5. Applied rewrites 48.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.37:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 0.17:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_1 \cdot \mathsf{fma}\left(-0.5, \phi_2 \cdot \phi_2, 1\right), \sin \phi_1 \cdot \mathsf{fma}\left(\phi_2, \left(\phi_2 \cdot \phi_2\right) \cdot -0.16666666666666666, \phi_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi1 -3.7e-6)
     (* R (acos (* (cos phi1) t_0)))
     (* R (acos (* (cos phi2) t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -3.7e-6) {
		tmp = R * acos((cos(phi1) * t_0));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 <= (-3.7d-6)) then
        tmp = r * acos((cos(phi1) * t_0))
    else
        tmp = r * acos((cos(phi2) * t_0))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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 <= -3.7e-6) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * t_0));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi1 <= -3.7e-6:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R * math.acos((math.cos(phi2) * t_0))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -3.7e-6)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi1 <= -3.7e-6)
		tmp = R * acos((cos(phi1) * t_0));
	else
		tmp = R * acos((cos(phi2) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -3.7e-6], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.7000000000000002e-6

   1. Initial program 84.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 46.3

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

   5. Applied rewrites 46.3%

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

   if -3.7000000000000002e-6 < phi1

   1. Initial program 74.3%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 52.5

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

   5. Applied rewrites 52.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 48.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0025:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.0025)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (* (cos phi2) (cos (- lambda2 lambda1)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.0025) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 <= (-0.0025d0)) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos((cos(phi2) * cos((lambda2 - lambda1))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.0025) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos((lambda2 - lambda1))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.0025:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos((lambda2 - lambda1))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.0025)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -0.0025)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.0025], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.00250000000000000005

   1. Initial program 84.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 34.0

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

   8. Applied rewrites 34.0%

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

   if -0.00250000000000000005 < phi1

   1. Initial program 74.3%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 52.5

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

   5. Applied rewrites 52.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0025:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 29.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;t\_0 \leq 0.99995:\\ \;\;\;\;R \cdot \cos^{-1} t\_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= t_0 0.99995)
     (* R (acos t_0))
     (* R (fabs (remainder (- lambda1 lambda2) (* PI 2.0)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (t_0 <= 0.99995) {
		tmp = R * acos(t_0);
	} else {
		tmp = R * fabs(remainder((lambda1 - lambda2), (((double) M_PI) * 2.0)));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (t_0 <= 0.99995) {
		tmp = R * Math.acos(t_0);
	} else {
		tmp = R * Math.abs(Math.IEEEremainder((lambda1 - lambda2), (Math.PI * 2.0)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if t_0 <= 0.99995:
		tmp = R * math.acos(t_0)
	else:
		tmp = R * math.fabs(math.remainder((lambda1 - lambda2), (math.pi * 2.0)))
	return tmp

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.99995], N[(R * N[ArcCos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(R * N[Abs[N[With[{TMP1 = N[(lambda1 - lambda2), $MachinePrecision], TMP2 = N[(Pi * 2.0), $MachinePrecision]}, TMP1 - Round[TMP1 / TMP2] * TMP2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 lambda1 lambda2)) < 0.999950000000000006

   1. Initial program 76.3%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 47.1

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in phi2 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 33.9

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

   8. Applied rewrites 33.9%

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

   if 0.999950000000000006 < (cos.f64 (-.f64 lambda1 lambda2))

   1. Initial program 78.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 36.1

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

   5. Applied rewrites 36.1%

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

   6. Taylor expanded in phi2 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 9.4

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

   8. Applied rewrites 9.4%

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

      1. lift--.f64 N/A

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

      2. acos-cos-s N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-acos.f64 N/A

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

      5. acos-cos N/A

         \[\leadsto \color{blue}{\left|\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

      6. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

      7. lower-remainder.f64 N/A

         \[\leadsto \left|\color{blue}{\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}\right| \cdot R \]

      8. lift-acos.f64 N/A

         \[\leadsto \left|\left(\color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)} \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right| \cdot R \]

      9. lift-cos.f64 N/A

         \[\leadsto \left|\left(\cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right| \cdot R \]

      10. acos-cos-s N/A

          \[\leadsto \left|\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)} \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right| \cdot R \]

      11. lift-PI.f64 N/A

          \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right| \cdot R \]

      12. *-commutative N/A

          \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)\right| \cdot R \]

      13. lower-*.f64 20.6

          \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \color{blue}{\left(\pi \cdot 2\right)}\right)\right| \cdot R \]

   10. Applied rewrites 20.6%

       \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\lambda_1 - \lambda_2\right) \leq 0.99995:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.4 \cdot 10^{-22}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -2.4e-22)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (* (cos phi2) (cos lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.4e-22) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos((cos(phi2) * cos(lambda2)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (lambda1 <= (-2.4d-22)) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos((cos(phi2) * cos(lambda2)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.4e-22) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -2.4e-22:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -2.4e-22)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda2))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -2.4e-22)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos((cos(phi2) * cos(lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -2.4e-22], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.40000000000000002e-22

   1. Initial program 68.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 62.9

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-cos.f64 42.1

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

   8. Applied rewrites 42.1%

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

   if -2.40000000000000002e-22 < lambda1

   1. Initial program 80.2%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in lambda1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-cos.f64 38.2

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

   8. Applied rewrites 38.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_2\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.4 \cdot 10^{-22}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.7e-6)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (* (cos phi2) (cos lambda1))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.7e-6) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 <= (-3.7d-6)) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos((cos(phi2) * cos(lambda1)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.7e-6) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda1)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.7e-6:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda1)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.7e-6)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda1))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3.7e-6)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3.7e-6], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.7000000000000002e-6

   1. Initial program 84.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. associate-*r* N/A

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

      2. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \phi_1\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      3. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \lambda_1 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2}, \cos \lambda_1 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1 \cdot \cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1 \cdot \cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1} \cdot \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      9. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      12. lower-sin.f64 63.5

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 63.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_1\right) \cdot R \]

      3. lower-cos.f64 34.0

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   8. Applied rewrites 34.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_1\right)} \cdot R \]

   if -3.7000000000000002e-6 < phi1

   1. Initial program 74.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 52.5

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 52.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \lambda_1}\right) \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_1\right)} \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \lambda_1\right) \cdot R \]

      4. lower-cos.f64 35.4

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \lambda_1}\right) \cdot R \]

   8. Applied rewrites 35.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_1\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -3.1e-6)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (* (cos phi1) (cos lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3.1e-6) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (lambda1 <= (-3.1d-6)) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos((cos(phi1) * cos(lambda2)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3.1e-6) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -3.1e-6:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -3.1e-6)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda2))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -3.1e-6)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -3.1e-6], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -3.1e-6

   1. Initial program 65.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot \cos \phi_2} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \phi_1\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      3. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \lambda_1 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2}, \cos \lambda_1 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1 \cdot \cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1 \cdot \cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1} \cdot \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      9. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      12. lower-sin.f64 65.0

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 65.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_1\right) \cdot R \]

      3. lower-cos.f64 45.1

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   8. Applied rewrites 45.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_1\right)} \cdot R \]

   if -3.1e-6 < lambda1

   1. Initial program 80.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. 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. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 46.0

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Applied rewrites 46.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_2} \cdot \cos \phi_1\right) \cdot R \]

      3. lower-cos.f64 38.4

         \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   8. Applied rewrites 38.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_1\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 37.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 7.6 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 7.6e-25)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (cos (- lambda1 lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 7.6e-25) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos(cos((lambda1 - lambda2)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (lambda2 <= 7.6d-25) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos(cos((lambda1 - lambda2)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 7.6e-25) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos(Math.cos((lambda1 - lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 7.6e-25:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos(math.cos((lambda1 - lambda2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 7.6e-25)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(cos(Float64(lambda1 - lambda2))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 7.6e-25)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos(cos((lambda1 - lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 7.6e-25], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 7.5999999999999996e-25

   1. Initial program 82.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_1\right) \cdot \cos \phi_2} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      2. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \phi_1\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      3. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \lambda_1 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_2}, \cos \lambda_1 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1 \cdot \cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1 \cdot \cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \color{blue}{\cos \phi_1} \cdot \cos \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      9. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      10. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      11. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) \cdot R \]

      12. lower-sin.f64 64.1

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Applied rewrites 64.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \lambda_1, \sin \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_1\right) \cdot R \]

      3. lower-cos.f64 36.5

         \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   8. Applied rewrites 36.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_1\right)} \cdot R \]

   if 7.5999999999999996e-25 < lambda2

   1. Initial program 60.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 35.5

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 35.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 29.0

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Applied rewrites 29.0%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 7.6 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 26.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.35 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -2.35e-6)
   (* R (acos (cos lambda1)))
   (* R (acos (cos lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.35e-6) {
		tmp = R * acos(cos(lambda1));
	} else {
		tmp = R * acos(cos(lambda2));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (lambda1 <= (-2.35d-6)) then
        tmp = r * acos(cos(lambda1))
    else
        tmp = r * acos(cos(lambda2))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.35e-6) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else {
		tmp = R * Math.acos(Math.cos(lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -2.35e-6:
		tmp = R * math.acos(math.cos(lambda1))
	else:
		tmp = R * math.acos(math.cos(lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -2.35e-6)
		tmp = Float64(R * acos(cos(lambda1)));
	else
		tmp = Float64(R * acos(cos(lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -2.35e-6)
		tmp = R * acos(cos(lambda1));
	else
		tmp = R * acos(cos(lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -2.35e-6], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.34999999999999995e-6

   1. Initial program 65.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 44.1

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 44.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 33.8

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Applied rewrites 33.8%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]
   10. Step-by-step derivation

       1. lower-cos.f64 33.3

          \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   11. Applied rewrites 33.3%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   if -2.34999999999999995e-6 < lambda1

   1. Initial program 80.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 44.9

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 44.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 26.6

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Applied rewrites 26.6%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot R \]
   10. Step-by-step derivation

       1. cos-neg N/A

          \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_2} \cdot R \]

       2. lower-cos.f64 21.3

          \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_2} \cdot R \]

   11. Applied rewrites 21.3%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_2} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.35 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 24.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.8:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -0.8)
   (* R (acos (cos lambda1)))
   (* R (fabs (remainder (- lambda1 lambda2) (* PI 2.0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -0.8) {
		tmp = R * acos(cos(lambda1));
	} else {
		tmp = R * fabs(remainder((lambda1 - lambda2), (((double) M_PI) * 2.0)));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -0.8) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else {
		tmp = R * Math.abs(Math.IEEEremainder((lambda1 - lambda2), (Math.PI * 2.0)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -0.8:
		tmp = R * math.acos(math.cos(lambda1))
	else:
		tmp = R * math.fabs(math.remainder((lambda1 - lambda2), (math.pi * 2.0)))
	return tmp

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -0.8], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[Abs[N[With[{TMP1 = N[(lambda1 - lambda2), $MachinePrecision], TMP2 = N[(Pi * 2.0), $MachinePrecision]}, TMP1 - Round[TMP1 / TMP2] * TMP2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -0.80000000000000004

   1. Initial program 65.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 44.1

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 44.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 33.8

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Applied rewrites 33.8%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]
   10. Step-by-step derivation

       1. lower-cos.f64 33.3

          \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   11. Applied rewrites 33.3%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   if -0.80000000000000004 < lambda1

   1. Initial program 80.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. 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. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      12. lower--.f64 44.9

          \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   5. Applied rewrites 44.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 26.6

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Applied rewrites 26.6%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      2. acos-cos-s N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

      3. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \cos \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

      4. lift-acos.f64 N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

      5. acos-cos N/A

         \[\leadsto \color{blue}{\left|\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

      6. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

      7. lower-remainder.f64 N/A

         \[\leadsto \left|\color{blue}{\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}\right| \cdot R \]

      8. lift-acos.f64 N/A

         \[\leadsto \left|\left(\color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)} \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right| \cdot R \]

      9. lift-cos.f64 N/A

         \[\leadsto \left|\left(\cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right| \cdot R \]

      10. acos-cos-s N/A

          \[\leadsto \left|\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)} \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right| \cdot R \]

      11. lift-PI.f64 N/A

          \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right| \cdot R \]

      12. *-commutative N/A

          \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)\right| \cdot R \]

      13. lower-*.f64 19.8

          \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \color{blue}{\left(\pi \cdot 2\right)}\right)\right| \cdot R \]

   10. Applied rewrites 19.8%

       \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.8:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 19.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right| \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (fabs (remainder (- lambda1 lambda2) (* PI 2.0)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * fabs(remainder((lambda1 - lambda2), (((double) M_PI) * 2.0)));
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.abs(Math.IEEEremainder((lambda1 - lambda2), (Math.PI * 2.0)));
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.fabs(math.remainder((lambda1 - lambda2), (math.pi * 2.0)))

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Abs[N[With[{TMP1 = N[(lambda1 - lambda2), $MachinePrecision], TMP2 = N[(Pi * 2.0), $MachinePrecision]}, TMP1 - Round[TMP1 / TMP2] * TMP2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
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. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   2. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   3. sub-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

   4. remove-double-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

   5. mul-1-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

   6. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

   8. cos-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

   9. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

   10. mul-1-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

   11. unsub-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   12. lower--.f64 44.7

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

5. Applied rewrites 44.7%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

   2. remove-double-neg N/A

      \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

   3. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

   4. +-commutative N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

   5. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

   6. cos-neg N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   7. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   8. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

   9. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   10. lower--.f64 28.4

       \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

8. Applied rewrites 28.4%

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
9. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   2. acos-cos-s N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   3. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \cos \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   4. lift-acos.f64 N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   5. acos-cos N/A

      \[\leadsto \color{blue}{\left|\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

   6. lower-fabs.f64 N/A

      \[\leadsto \color{blue}{\left|\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

   7. lower-remainder.f64 N/A

      \[\leadsto \left|\color{blue}{\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}\right| \cdot R \]

   8. lift-acos.f64 N/A

      \[\leadsto \left|\left(\color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)} \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right| \cdot R \]

   9. lift-cos.f64 N/A

      \[\leadsto \left|\left(\cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right| \cdot R \]

   10. acos-cos-s N/A

       \[\leadsto \left|\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)} \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right| \cdot R \]

   11. lift-PI.f64 N/A

       \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right| \cdot R \]

   12. *-commutative N/A

       \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)\right| \cdot R \]

   13. lower-*.f64 19.6

       \[\leadsto \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \color{blue}{\left(\pi \cdot 2\right)}\right)\right| \cdot R \]

10. Applied rewrites 19.6%

    \[\leadsto \color{blue}{\left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|} \cdot R \]

11. Final simplification 19.6%

    \[\leadsto R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right| \]
12. Add Preprocessing

Alternative 25: 9.1% accurate, 69.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \left(\lambda_2 - \lambda_1\right) \cdot R \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* (- lambda2 lambda1) R))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return (lambda2 - lambda1) * R;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 = (lambda2 - lambda1) * r
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return (lambda2 - lambda1) * R;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return (lambda2 - lambda1) * R

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(Float64(lambda2 - lambda1) * R)
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = (lambda2 - lambda1) * R;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
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. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   2. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   3. sub-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

   4. remove-double-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

   5. mul-1-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

   6. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

   8. cos-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

   9. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

   10. mul-1-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

   11. unsub-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   12. lower--.f64 44.7

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

5. Applied rewrites 44.7%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

   2. remove-double-neg N/A

      \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

   3. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

   4. +-commutative N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

   5. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

   6. cos-neg N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   7. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   8. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

   9. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   10. lower--.f64 28.4

       \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

8. Applied rewrites 28.4%

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
9. Step-by-step derivation

   1. cos-diff N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)} \cdot R \]

   2. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot R \]

   3. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \lambda_2} + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot R \]

   4. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot R \]

   5. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_2} \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot R \]

   6. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot R \]

   7. lift-sin.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right) \cdot R \]

   8. lift-sin.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right) \cdot R \]

   9. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right) \cdot R \]

   10. lift-sin.f64 N/A

       \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right) \cdot R \]

   11. lift-sin.f64 N/A

       \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right) \cdot R \]

   12. cos-diff N/A

       \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]

   13. lift--.f64 N/A

       \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]

   14. acos-cos-s 4.7

       \[\leadsto \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]

10. Applied rewrites 4.7%

    \[\leadsto \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]
11. Add Preprocessing

Alternative 26: 3.6% accurate, 78.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \lambda_2 \cdot \left(-R\right) \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* lambda2 (- R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return lambda2 * -R;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 = lambda2 * -r
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return lambda2 * -R;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return lambda2 * -R

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(lambda2 * Float64(-R))
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = lambda2 * -R;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(lambda2 * (-R)), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
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. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   2. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   3. sub-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

   4. remove-double-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

   5. mul-1-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

   6. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

   8. cos-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

   9. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

   10. mul-1-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

   11. unsub-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   12. lower--.f64 44.7

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

5. Applied rewrites 44.7%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

   2. remove-double-neg N/A

      \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

   3. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

   4. +-commutative N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

   5. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

   6. cos-neg N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   7. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   8. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

   9. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   10. lower--.f64 28.4

       \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

8. Applied rewrites 28.4%

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

9. Taylor expanded in lambda1 around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \lambda_2\right)} \cdot R \]
10. Step-by-step derivation

    1. neg-mul-1 N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot R \]

    2. lower-neg.f64 6.5

       \[\leadsto \color{blue}{\left(-\lambda_2\right)} \cdot R \]

11. Applied rewrites 6.5%

    \[\leadsto \color{blue}{\left(-\lambda_2\right)} \cdot R \]

12. Final simplification 6.5%

    \[\leadsto \lambda_2 \cdot \left(-R\right) \]
13. Add Preprocessing

Alternative 27: 3.6% accurate, 104.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \lambda_1 \cdot R \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* lambda1 R))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 * R;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 = lambda1 * r
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 * R;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return lambda1 * R

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 * R)
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = lambda1 * R;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 * R), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
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. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   2. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   3. sub-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

   4. remove-double-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

   5. mul-1-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

   6. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}\right) \cdot R \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)\right) \cdot R \]

   8. cos-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

   9. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) \cdot R \]

   10. mul-1-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \cdot R \]

   11. unsub-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

   12. lower--.f64 44.7

       \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

5. Applied rewrites 44.7%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

6. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

   2. remove-double-neg N/A

      \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

   3. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

   4. +-commutative N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

   5. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

   6. cos-neg N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   7. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   8. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

   9. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   10. lower--.f64 28.4

       \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

8. Applied rewrites 28.4%

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

9. Taylor expanded in lambda1 around inf

   \[\leadsto \color{blue}{R \cdot \lambda_1} \]
10. Step-by-step derivation

    1. lower-*.f64 6.9

       \[\leadsto \color{blue}{R \cdot \lambda_1} \]

11. Applied rewrites 6.9%

    \[\leadsto \color{blue}{R \cdot \lambda_1} \]

12. Final simplification 6.9%

    \[\leadsto \lambda_1 \cdot R \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))