Spherical law of cosines

Percentage Accurate: 74.1% → 93.8%

Time: 28.2s

Alternatives: 29

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: 74.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;t\_0 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right) \leq 1:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(t\_1, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \mathsf{log1p}\left(\mathsf{expm1}\left(t\_0\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (* (cos phi1) (cos phi2))))
   (if (<= (+ t_0 (* t_1 (cos (- lambda1 lambda2)))) 1.0)
     (*
      R
      (acos
       (fma
        t_1
        (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2)))
        (log1p (expm1 t_0)))))
     (* R (* lambda2 (- 1.0 (/ lambda1 lambda2)))))))

C

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 ((t_0 + (t_1 * cos((lambda1 - lambda2)))) <= 1.0) {
		tmp = R * acos(fma(t_1, ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))), log1p(expm1(t_0))));
	} else {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	}
	return tmp;
}

Julia

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 (Float64(t_0 + Float64(t_1 * cos(Float64(lambda1 - lambda2)))) <= 1.0)
		tmp = Float64(R * acos(fma(t_1, Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))), log1p(expm1(t_0)))));
	else
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	end
	return tmp
end

Wolfram

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[(t$95$0 + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(R * N[ArcCos[N[(t$95$1 * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))) < 1

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

      1. cos-diff 91.9%

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

      2. +-commutative 91.9%

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

      3. *-commutative 91.9%

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

      4. *-commutative 91.9%

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

   6. Applied egg-rr 91.9%

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

      1. log1p-expm1-u 91.9%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   8. Applied egg-rr 91.9%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   if 1 < (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in phi2 around 0 42.1%

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

   6. Taylor expanded in phi1 around 0 22.3%

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

   7. Taylor expanded in lambda2 around inf 6.9%

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

      1. mul-1-neg 6.9%

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

      2. unsub-neg 6.9%

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

   9. Simplified 6.9%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 93.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;t\_0 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right) \leq 1:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \left(t\_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + t\_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (* (cos phi1) (cos phi2))))
   (if (<= (+ t_0 (* t_1 (cos (- lambda1 lambda2)))) 1.0)
     (*
      R
      (acos
       (+
        t_0
        (+
         (* t_1 (* (cos lambda1) (cos lambda2)))
         (* t_1 (* (sin lambda1) (sin lambda2)))))))
     (* R (* lambda2 (- 1.0 (/ lambda1 lambda2)))))))

C

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 ((t_0 + (t_1 * cos((lambda1 - lambda2)))) <= 1.0) {
		tmp = R * acos((t_0 + ((t_1 * (cos(lambda1) * cos(lambda2))) + (t_1 * (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    t_1 = cos(phi1) * cos(phi2)
    if ((t_0 + (t_1 * cos((lambda1 - lambda2)))) <= 1.0d0) then
        tmp = r * acos((t_0 + ((t_1 * (cos(lambda1) * cos(lambda2))) + (t_1 * (sin(lambda1) * sin(lambda2))))))
    else
        tmp = r * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.sin(phi2);
	double t_1 = Math.cos(phi1) * Math.cos(phi2);
	double tmp;
	if ((t_0 + (t_1 * Math.cos((lambda1 - lambda2)))) <= 1.0) {
		tmp = R * Math.acos((t_0 + ((t_1 * (Math.cos(lambda1) * Math.cos(lambda2))) + (t_1 * (Math.sin(lambda1) * Math.sin(lambda2))))));
	} else {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	t_1 = math.cos(phi1) * math.cos(phi2)
	tmp = 0
	if (t_0 + (t_1 * math.cos((lambda1 - lambda2)))) <= 1.0:
		tmp = R * math.acos((t_0 + ((t_1 * (math.cos(lambda1) * math.cos(lambda2))) + (t_1 * (math.sin(lambda1) * math.sin(lambda2))))))
	else:
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)))
	return tmp

Julia

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 (Float64(t_0 + Float64(t_1 * cos(Float64(lambda1 - lambda2)))) <= 1.0)
		tmp = Float64(R * acos(Float64(t_0 + Float64(Float64(t_1 * Float64(cos(lambda1) * cos(lambda2))) + Float64(t_1 * Float64(sin(lambda1) * sin(lambda2)))))));
	else
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	t_1 = cos(phi1) * cos(phi2);
	tmp = 0.0;
	if ((t_0 + (t_1 * cos((lambda1 - lambda2)))) <= 1.0)
		tmp = R * acos((t_0 + ((t_1 * (cos(lambda1) * cos(lambda2))) + (t_1 * (sin(lambda1) * sin(lambda2))))));
	else
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

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[(t$95$0 + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(R * N[ArcCos[N[(t$95$0 + N[(N[(t$95$1 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))) < 1

   1. Initial program 71.6%

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

      1. cos-diff 91.9%

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

      2. distribute-lft-in 91.9%

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

   4. Applied egg-rr 91.9%

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

   if 1 < (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in phi2 around 0 42.1%

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

   6. Taylor expanded in phi1 around 0 22.3%

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

   7. Taylor expanded in lambda2 around inf 6.9%

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

      1. mul-1-neg 6.9%

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

      2. unsub-neg 6.9%

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

   9. Simplified 6.9%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

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

Alternative 3: 93.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (* (cos phi1) (cos phi2))))
   (if (<= (+ t_0 (* t_1 (cos (- lambda1 lambda2)))) 1.0)
     (*
      R
      (acos
       (fma
        t_1
        (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2)))
        t_0)))
     (* R (* lambda2 (- 1.0 (/ lambda1 lambda2)))))))

C

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 ((t_0 + (t_1 * cos((lambda1 - lambda2)))) <= 1.0) {
		tmp = R * acos(fma(t_1, ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))), t_0));
	} else {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	}
	return tmp;
}

Julia

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 (Float64(t_0 + Float64(t_1 * cos(Float64(lambda1 - lambda2)))) <= 1.0)
		tmp = Float64(R * acos(fma(t_1, Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))), t_0)));
	else
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	end
	return tmp
end

Wolfram

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[(t$95$0 + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(R * N[ArcCos[N[(t$95$1 * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))) < 1

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

      1. cos-diff 91.9%

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

      2. +-commutative 91.9%

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

      3. *-commutative 91.9%

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

      4. *-commutative 91.9%

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

   6. Applied egg-rr 91.9%

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

   if 1 < (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in phi2 around 0 42.1%

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

   6. Taylor expanded in phi1 around 0 22.3%

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

   7. Taylor expanded in lambda2 around inf 6.9%

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

      1. mul-1-neg 6.9%

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

      2. unsub-neg 6.9%

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

   9. Simplified 6.9%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;t\_0 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right) \leq 1:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + t\_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (* (cos phi1) (cos phi2))))
   (if (<= (+ t_0 (* t_1 (cos (- lambda1 lambda2)))) 1.0)
     (*
      R
      (acos
       (+
        t_0
        (*
         t_1
         (+
          (* (sin lambda1) (sin lambda2))
          (* (cos lambda1) (cos lambda2)))))))
     (* R (* lambda2 (- 1.0 (/ lambda1 lambda2)))))))

C

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 ((t_0 + (t_1 * cos((lambda1 - lambda2)))) <= 1.0) {
		tmp = R * acos((t_0 + (t_1 * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))));
	} else {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    t_1 = cos(phi1) * cos(phi2)
    if ((t_0 + (t_1 * cos((lambda1 - lambda2)))) <= 1.0d0) then
        tmp = r * acos((t_0 + (t_1 * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))))
    else
        tmp = r * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.sin(phi2);
	double t_1 = Math.cos(phi1) * Math.cos(phi2);
	double tmp;
	if ((t_0 + (t_1 * Math.cos((lambda1 - lambda2)))) <= 1.0) {
		tmp = R * Math.acos((t_0 + (t_1 * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2))))));
	} else {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	t_1 = math.cos(phi1) * math.cos(phi2)
	tmp = 0
	if (t_0 + (t_1 * math.cos((lambda1 - lambda2)))) <= 1.0:
		tmp = R * math.acos((t_0 + (t_1 * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2))))))
	else:
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)))
	return tmp

Julia

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 (Float64(t_0 + Float64(t_1 * cos(Float64(lambda1 - lambda2)))) <= 1.0)
		tmp = Float64(R * acos(Float64(t_0 + Float64(t_1 * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))))));
	else
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	t_1 = cos(phi1) * cos(phi2);
	tmp = 0.0;
	if ((t_0 + (t_1 * cos((lambda1 - lambda2)))) <= 1.0)
		tmp = R * acos((t_0 + (t_1 * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))));
	else
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

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[(t$95$0 + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(R * N[ArcCos[N[(t$95$0 + N[(t$95$1 * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))) < 1

   1. Initial program 71.6%

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

      1. cos-diff 91.9%

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

      2. distribute-lft-in 91.9%

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

   4. Applied egg-rr 91.9%

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

      1. +-commutative 91.9%

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

      2. associate-*l* 91.9%

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

      3. fma-define 91.9%

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

      4. associate-*r* 91.9%

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

   6. Applied egg-rr 91.9%

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

      1. fma-define 91.9%

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

      2. associate-*r* 91.9%

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

      3. associate-*l* 91.9%

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

      4. distribute-lft-out 91.9%

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

   8. Simplified 91.9%

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

   if 1 < (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in phi2 around 0 42.1%

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

   6. Taylor expanded in phi1 around 0 22.3%

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

   7. Taylor expanded in lambda2 around inf 6.9%

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

      1. mul-1-neg 6.9%

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

      2. unsub-neg 6.9%

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

   9. Simplified 6.9%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

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

Alternative 5: 83.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t\_0 \cdot t\_1\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt[3]{{\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(t\_0, t\_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2))) (t_1 (cos (- lambda2 lambda1))))
   (if (<= phi1 -3.7e-18)
     (*
      R
      (-
       (/ PI 2.0)
       (- (* PI 0.5) (acos (fma (sin phi1) (sin phi2) (* t_0 t_1))))))
     (if (<= phi1 1.15e-15)
       (*
        R
        (acos
         (*
          (cos phi2)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))
       (*
        R
        (cbrt
         (pow
          (- (* PI 0.5) (asin (fma t_0 t_1 (* (sin phi1) (sin phi2)))))
          3.0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double t_1 = cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -3.7e-18) {
		tmp = R * ((((double) M_PI) / 2.0) - ((((double) M_PI) * 0.5) - acos(fma(sin(phi1), sin(phi2), (t_0 * t_1)))));
	} else if (phi1 <= 1.15e-15) {
		tmp = R * acos((cos(phi2) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * cbrt(pow(((((double) M_PI) * 0.5) - asin(fma(t_0, t_1, (sin(phi1) * sin(phi2))))), 3.0));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -3.7e-18)
		tmp = Float64(R * Float64(Float64(pi / 2.0) - Float64(Float64(pi * 0.5) - acos(fma(sin(phi1), sin(phi2), Float64(t_0 * t_1))))));
	elseif (phi1 <= 1.15e-15)
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * cbrt((Float64(Float64(pi * 0.5) - asin(fma(t_0, t_1, Float64(sin(phi1) * sin(phi2))))) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -3.7e-18], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 1.15e-15], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[Power[N[Power[N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(t$95$0 * t$95$1 + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.7000000000000003e-18

   1. Initial program 74.5%

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

   3. Simplified 74.5%

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

      1. cos-diff 97.4%

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

      2. +-commutative 97.4%

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

      3. *-commutative 97.4%

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

      4. *-commutative 97.4%

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

   6. Applied egg-rr 97.4%

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

      1. +-commutative 97.4%

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

      2. cos-diff 74.5%

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

      3. fma-define 74.5%

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

      4. +-commutative 74.5%

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

      5. log1p-expm1-u 74.5%

         \[\leadsto R \cdot \cos^{-1} \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

      6. acos-asin 74.3%

         \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \]

      7. log1p-expm1-u 74.3%

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

      8. +-commutative 74.3%

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

      9. fma-define 74.3%

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

   8. Applied egg-rr 74.3%

      \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \]
   9. Step-by-step derivation

      1. fma-undefine 74.3%

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

      2. +-commutative 74.3%

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

      3. fma-undefine 74.3%

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

      4. *-commutative 74.3%

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

   10. Simplified 74.3%

       \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)} \]
   11. Step-by-step derivation

       1. asin-acos 74.5%

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

       2. div-inv 74.5%

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

       3. metadata-eval 74.5%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)\right) \]

       4. *-commutative 74.5%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)\right) \]

       5. cos-diff 97.4%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right)\right)\right)\right) \]

       6. *-commutative 97.4%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right) \]

       7. *-commutative 97.4%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right)\right)\right)\right) \]

       8. cos-diff 74.5%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right)\right)\right)\right) \]

   12. Applied egg-rr 74.5%

       \[\leadsto R \cdot \left(\frac{\pi}{2} - \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\right) \]

   if -3.7000000000000003e-18 < phi1 < 1.14999999999999995e-15

   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. Step-by-step derivation

   3. Simplified 65.1%

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

      1. cos-diff 82.9%

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

      2. +-commutative 82.9%

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

      3. *-commutative 82.9%

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

      4. *-commutative 82.9%

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

   6. Applied egg-rr 82.9%

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

      1. log1p-expm1-u 82.9%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   8. Applied egg-rr 82.9%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   9. Taylor expanded in phi1 around 0 82.9%

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

       1. cos-neg 82.9%

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

       2. *-commutative 82.9%

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

       3. cos-neg 82.9%

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

       4. fma-undefine 83.0%

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

   11. Simplified 83.0%

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

   if 1.14999999999999995e-15 < phi1

   1. Initial program 77.7%

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

   3. Simplified 77.7%

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

      1. cos-diff 98.9%

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

      2. *-commutative 98.9%

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

      3. *-commutative 98.9%

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

      4. cos-diff 77.7%

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

      5. fma-define 77.7%

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

      6. +-commutative 77.7%

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

      7. acos-asin 77.5%

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

      8. sub-neg 77.5%

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

      9. div-inv 77.5%

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

      10. metadata-eval 77.5%

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

      11. +-commutative 77.5%

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

   6. Applied egg-rr 77.5%

      \[\leadsto R \cdot \color{blue}{\left(\pi \cdot 0.5 + \left(-\sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. sub-neg 77.5%

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

      2. *-commutative 77.5%

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

      3. fma-define 77.5%

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

      4. associate-*r* 77.5%

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

      5. fma-define 77.5%

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

   8. Simplified 77.5%

      \[\leadsto R \cdot \color{blue}{\left(0.5 \cdot \pi - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 77.3%

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

      2. pow3 77.3%

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

   10. Applied egg-rr 77.3%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt[3]{{\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t\_0, \sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi1 -3.7e-18)
     (*
      R
      (-
       (/ PI 2.0)
       (-
        (* PI 0.5)
        (acos (fma (sin phi1) (sin phi2) (* (* (cos phi1) (cos phi2)) t_0))))))
     (if (<= phi1 1.15e-15)
       (*
        R
        (acos
         (*
          (cos phi2)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))
       (*
        R
        (log
         (exp
          (acos
           (fma
            (cos phi1)
            (* (cos phi2) t_0)
            (* (sin phi1) (sin phi2)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -3.7e-18) {
		tmp = R * ((((double) M_PI) / 2.0) - ((((double) M_PI) * 0.5) - acos(fma(sin(phi1), sin(phi2), ((cos(phi1) * cos(phi2)) * t_0)))));
	} else if (phi1 <= 1.15e-15) {
		tmp = R * acos((cos(phi2) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * log(exp(acos(fma(cos(phi1), (cos(phi2) * t_0), (sin(phi1) * sin(phi2))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -3.7e-18)
		tmp = Float64(R * Float64(Float64(pi / 2.0) - Float64(Float64(pi * 0.5) - acos(fma(sin(phi1), sin(phi2), Float64(Float64(cos(phi1) * cos(phi2)) * t_0))))));
	elseif (phi1 <= 1.15e-15)
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * log(exp(acos(fma(cos(phi1), Float64(cos(phi2) * t_0), Float64(sin(phi1) * sin(phi2)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -3.7e-18], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 1.15e-15], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[Log[N[Exp[N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.7000000000000003e-18

   1. Initial program 74.5%

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

   3. Simplified 74.5%

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

      1. cos-diff 97.4%

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

      2. +-commutative 97.4%

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

      3. *-commutative 97.4%

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

      4. *-commutative 97.4%

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

   6. Applied egg-rr 97.4%

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

      1. +-commutative 97.4%

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

      2. cos-diff 74.5%

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

      3. fma-define 74.5%

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

      4. +-commutative 74.5%

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

      5. log1p-expm1-u 74.5%

         \[\leadsto R \cdot \cos^{-1} \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

      6. acos-asin 74.3%

         \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \]

      7. log1p-expm1-u 74.3%

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

      8. +-commutative 74.3%

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

      9. fma-define 74.3%

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

   8. Applied egg-rr 74.3%

      \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \]
   9. Step-by-step derivation

      1. fma-undefine 74.3%

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

      2. +-commutative 74.3%

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

      3. fma-undefine 74.3%

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

      4. *-commutative 74.3%

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

   10. Simplified 74.3%

       \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)} \]
   11. Step-by-step derivation

       1. asin-acos 74.5%

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

       2. div-inv 74.5%

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

       3. metadata-eval 74.5%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)\right) \]

       4. *-commutative 74.5%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)\right) \]

       5. cos-diff 97.4%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right)\right)\right)\right) \]

       6. *-commutative 97.4%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right) \]

       7. *-commutative 97.4%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right)\right)\right)\right) \]

       8. cos-diff 74.5%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right)\right)\right)\right) \]

   12. Applied egg-rr 74.5%

       \[\leadsto R \cdot \left(\frac{\pi}{2} - \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\right) \]

   if -3.7000000000000003e-18 < phi1 < 1.14999999999999995e-15

   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. Step-by-step derivation

   3. Simplified 65.1%

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

      1. cos-diff 82.9%

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

      2. +-commutative 82.9%

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

      3. *-commutative 82.9%

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

      4. *-commutative 82.9%

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

   6. Applied egg-rr 82.9%

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

      1. log1p-expm1-u 82.9%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   8. Applied egg-rr 82.9%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   9. Taylor expanded in phi1 around 0 82.9%

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

       1. cos-neg 82.9%

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

       2. *-commutative 82.9%

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

       3. cos-neg 82.9%

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

       4. fma-undefine 83.0%

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

   11. Simplified 83.0%

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

   if 1.14999999999999995e-15 < phi1

   1. Initial program 77.7%

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

   3. Simplified 77.7%

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

      1. add-log-exp 77.7%

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

      2. cos-diff 98.9%

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

      3. *-commutative 98.9%

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

      4. *-commutative 98.9%

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

      5. cos-diff 77.7%

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

      6. fma-define 77.7%

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

      7. associate-*r* 77.7%

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

      8. fma-undefine 77.7%

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

      9. cos-diff 98.9%

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

      10. *-commutative 98.9%

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

      11. *-commutative 98.9%

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

      12. cos-diff 77.7%

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

   6. Applied egg-rr 77.7%

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

Alternative 7: 83.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t\_0 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_2 \cdot \sqrt[3]{{\sin \phi_1}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2))))
   (if (<= phi1 -3.7e-18)
     (*
      R
      (-
       (/ PI 2.0)
       (-
        (* PI 0.5)
        (acos (fma (sin phi1) (sin phi2) (* t_0 (cos (- lambda2 lambda1))))))))
     (if (<= phi1 1.15e-15)
       (*
        R
        (acos
         (*
          (cos phi2)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))
       (*
        R
        (acos
         (+
          (* t_0 (cos (- lambda1 lambda2)))
          (* (sin phi2) (cbrt (pow (sin phi1) 3.0))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double tmp;
	if (phi1 <= -3.7e-18) {
		tmp = R * ((((double) M_PI) / 2.0) - ((((double) M_PI) * 0.5) - acos(fma(sin(phi1), sin(phi2), (t_0 * cos((lambda2 - lambda1)))))));
	} else if (phi1 <= 1.15e-15) {
		tmp = R * acos((cos(phi2) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos(((t_0 * cos((lambda1 - lambda2))) + (sin(phi2) * cbrt(pow(sin(phi1), 3.0)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -3.7e-18)
		tmp = Float64(R * Float64(Float64(pi / 2.0) - Float64(Float64(pi * 0.5) - acos(fma(sin(phi1), sin(phi2), Float64(t_0 * cos(Float64(lambda2 - lambda1))))))));
	elseif (phi1 <= 1.15e-15)
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(Float64(t_0 * cos(Float64(lambda1 - lambda2))) + Float64(sin(phi2) * cbrt((sin(phi1) ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -3.7e-18], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(t$95$0 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 1.15e-15], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Power[N[Power[N[Sin[phi1], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.7000000000000003e-18

   1. Initial program 74.5%

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

   3. Simplified 74.5%

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

      1. cos-diff 97.4%

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

      2. +-commutative 97.4%

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

      3. *-commutative 97.4%

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

      4. *-commutative 97.4%

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

   6. Applied egg-rr 97.4%

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

      1. +-commutative 97.4%

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

      2. cos-diff 74.5%

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

      3. fma-define 74.5%

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

      4. +-commutative 74.5%

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

      5. log1p-expm1-u 74.5%

         \[\leadsto R \cdot \cos^{-1} \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

      6. acos-asin 74.3%

         \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \]

      7. log1p-expm1-u 74.3%

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

      8. +-commutative 74.3%

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

      9. fma-define 74.3%

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

   8. Applied egg-rr 74.3%

      \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \]
   9. Step-by-step derivation

      1. fma-undefine 74.3%

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

      2. +-commutative 74.3%

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

      3. fma-undefine 74.3%

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

      4. *-commutative 74.3%

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

   10. Simplified 74.3%

       \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)} \]
   11. Step-by-step derivation

       1. asin-acos 74.5%

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

       2. div-inv 74.5%

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

       3. metadata-eval 74.5%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)\right) \]

       4. *-commutative 74.5%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)\right) \]

       5. cos-diff 97.4%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right)\right)\right)\right) \]

       6. *-commutative 97.4%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right) \]

       7. *-commutative 97.4%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right)\right)\right)\right) \]

       8. cos-diff 74.5%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right)\right)\right)\right) \]

   12. Applied egg-rr 74.5%

       \[\leadsto R \cdot \left(\frac{\pi}{2} - \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\right) \]

   if -3.7000000000000003e-18 < phi1 < 1.14999999999999995e-15

   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. Step-by-step derivation

   3. Simplified 65.1%

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

      1. cos-diff 82.9%

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

      2. +-commutative 82.9%

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

      3. *-commutative 82.9%

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

      4. *-commutative 82.9%

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

   6. Applied egg-rr 82.9%

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

      1. log1p-expm1-u 82.9%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   8. Applied egg-rr 82.9%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   9. Taylor expanded in phi1 around 0 82.9%

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

       1. cos-neg 82.9%

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

       2. *-commutative 82.9%

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

       3. cos-neg 82.9%

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

       4. fma-undefine 83.0%

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

   11. Simplified 83.0%

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

   if 1.14999999999999995e-15 < phi1

   1. Initial program 77.7%

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

      1. add-cbrt-cube 77.7%

         \[\leadsto \cos^{-1} \left(\color{blue}{\sqrt[3]{\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1}} \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. pow3 77.7%

         \[\leadsto \cos^{-1} \left(\sqrt[3]{\color{blue}{{\sin \phi_1}^{3}}} \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied egg-rr 77.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_2 \cdot \sqrt[3]{{\sin \phi_1}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t\_0 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2))))
   (if (<= phi1 -3.7e-18)
     (*
      R
      (-
       (/ PI 2.0)
       (-
        (* PI 0.5)
        (acos (fma (sin phi1) (sin phi2) (* t_0 (cos (- lambda2 lambda1))))))))
     (if (<= phi1 1.15e-15)
       (*
        R
        (acos
         (*
          (cos phi2)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))
       (*
        R
        (acos
         (+
          (* t_0 (cos (- lambda1 lambda2)))
          (expm1 (log1p (* (sin phi1) (sin phi2)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double tmp;
	if (phi1 <= -3.7e-18) {
		tmp = R * ((((double) M_PI) / 2.0) - ((((double) M_PI) * 0.5) - acos(fma(sin(phi1), sin(phi2), (t_0 * cos((lambda2 - lambda1)))))));
	} else if (phi1 <= 1.15e-15) {
		tmp = R * acos((cos(phi2) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos(((t_0 * cos((lambda1 - lambda2))) + expm1(log1p((sin(phi1) * sin(phi2))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -3.7e-18)
		tmp = Float64(R * Float64(Float64(pi / 2.0) - Float64(Float64(pi * 0.5) - acos(fma(sin(phi1), sin(phi2), Float64(t_0 * cos(Float64(lambda2 - lambda1))))))));
	elseif (phi1 <= 1.15e-15)
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(Float64(t_0 * cos(Float64(lambda1 - lambda2))) + expm1(log1p(Float64(sin(phi1) * sin(phi2)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -3.7e-18], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(t$95$0 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 1.15e-15], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(Exp[N[Log[1 + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.7000000000000003e-18

   1. Initial program 74.5%

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

   3. Simplified 74.5%

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

      1. cos-diff 97.4%

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

      2. +-commutative 97.4%

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

      3. *-commutative 97.4%

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

      4. *-commutative 97.4%

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

   6. Applied egg-rr 97.4%

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

      1. +-commutative 97.4%

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

      2. cos-diff 74.5%

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

      3. fma-define 74.5%

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

      4. +-commutative 74.5%

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

      5. log1p-expm1-u 74.5%

         \[\leadsto R \cdot \cos^{-1} \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

      6. acos-asin 74.3%

         \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \]

      7. log1p-expm1-u 74.3%

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

      8. +-commutative 74.3%

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

      9. fma-define 74.3%

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

   8. Applied egg-rr 74.3%

      \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \]
   9. Step-by-step derivation

      1. fma-undefine 74.3%

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

      2. +-commutative 74.3%

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

      3. fma-undefine 74.3%

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

      4. *-commutative 74.3%

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

   10. Simplified 74.3%

       \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)} \]
   11. Step-by-step derivation

       1. asin-acos 74.5%

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

       2. div-inv 74.5%

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

       3. metadata-eval 74.5%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)\right) \]

       4. *-commutative 74.5%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)\right) \]

       5. cos-diff 97.4%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right)\right)\right)\right) \]

       6. *-commutative 97.4%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right) \]

       7. *-commutative 97.4%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right)\right)\right)\right) \]

       8. cos-diff 74.5%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right)\right)\right)\right) \]

   12. Applied egg-rr 74.5%

       \[\leadsto R \cdot \left(\frac{\pi}{2} - \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\right) \]

   if -3.7000000000000003e-18 < phi1 < 1.14999999999999995e-15

   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. Step-by-step derivation

   3. Simplified 65.1%

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

      1. cos-diff 82.9%

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

      2. +-commutative 82.9%

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

      3. *-commutative 82.9%

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

      4. *-commutative 82.9%

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

   6. Applied egg-rr 82.9%

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

      1. log1p-expm1-u 82.9%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   8. Applied egg-rr 82.9%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   9. Taylor expanded in phi1 around 0 82.9%

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

       1. cos-neg 82.9%

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

       2. *-commutative 82.9%

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

       3. cos-neg 82.9%

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

       4. fma-undefine 83.0%

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

   11. Simplified 83.0%

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

   if 1.14999999999999995e-15 < phi1

   1. Initial program 77.7%

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

      1. expm1-log1p-u 77.7%

         \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. expm1-undefine 77.6%

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \phi_2\right)} - 1\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied egg-rr 77.6%

      \[\leadsto \cos^{-1} \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \phi_2\right)} - 1\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   5. Step-by-step derivation

      1. expm1-define 77.7%

         \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   6. Simplified 77.7%

      \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.7e-18)
   (*
    R
    (-
     (/ PI 2.0)
     (-
      (* PI 0.5)
      (acos
       (fma
        (sin phi1)
        (sin phi2)
        (* (* (cos phi1) (cos phi2)) (cos (- lambda2 lambda1))))))))
   (if (<= phi1 1.15e-15)
     (*
      R
      (acos
       (*
        (cos phi2)
        (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))
     (*
      R
      (acos
       (fma
        (sin phi1)
        (sin phi2)
        (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.7e-18) {
		tmp = R * ((((double) M_PI) / 2.0) - ((((double) M_PI) * 0.5) - acos(fma(sin(phi1), sin(phi2), ((cos(phi1) * cos(phi2)) * cos((lambda2 - lambda1)))))));
	} else if (phi1 <= 1.15e-15) {
		tmp = R * acos((cos(phi2) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.7e-18)
		tmp = Float64(R * Float64(Float64(pi / 2.0) - Float64(Float64(pi * 0.5) - acos(fma(sin(phi1), sin(phi2), Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda2 - lambda1))))))));
	elseif (phi1 <= 1.15e-15)
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3.7e-18], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 1.15e-15], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.7000000000000003e-18

   1. Initial program 74.5%

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

   3. Simplified 74.5%

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

      1. cos-diff 97.4%

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

      2. +-commutative 97.4%

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

      3. *-commutative 97.4%

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

      4. *-commutative 97.4%

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

   6. Applied egg-rr 97.4%

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

      1. +-commutative 97.4%

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

      2. cos-diff 74.5%

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

      3. fma-define 74.5%

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

      4. +-commutative 74.5%

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

      5. log1p-expm1-u 74.5%

         \[\leadsto R \cdot \cos^{-1} \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

      6. acos-asin 74.3%

         \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \]

      7. log1p-expm1-u 74.3%

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

      8. +-commutative 74.3%

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

      9. fma-define 74.3%

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

   8. Applied egg-rr 74.3%

      \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \]
   9. Step-by-step derivation

      1. fma-undefine 74.3%

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

      2. +-commutative 74.3%

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

      3. fma-undefine 74.3%

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

      4. *-commutative 74.3%

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

   10. Simplified 74.3%

       \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)} \]
   11. Step-by-step derivation

       1. asin-acos 74.5%

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

       2. div-inv 74.5%

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

       3. metadata-eval 74.5%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)\right) \]

       4. *-commutative 74.5%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)\right) \]

       5. cos-diff 97.4%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right)\right)\right)\right) \]

       6. *-commutative 97.4%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right) \]

       7. *-commutative 97.4%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right)\right)\right)\right) \]

       8. cos-diff 74.5%

          \[\leadsto R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right)\right)\right)\right) \]

   12. Applied egg-rr 74.5%

       \[\leadsto R \cdot \left(\frac{\pi}{2} - \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\right) \]

   if -3.7000000000000003e-18 < phi1 < 1.14999999999999995e-15

   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. Step-by-step derivation

   3. Simplified 65.1%

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

      1. cos-diff 82.9%

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

      2. +-commutative 82.9%

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

      3. *-commutative 82.9%

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

      4. *-commutative 82.9%

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

   6. Applied egg-rr 82.9%

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

      1. log1p-expm1-u 82.9%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   8. Applied egg-rr 82.9%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   9. Taylor expanded in phi1 around 0 82.9%

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

       1. cos-neg 82.9%

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

       2. *-commutative 82.9%

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

       3. cos-neg 82.9%

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

       4. fma-undefine 83.0%

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

   11. Simplified 83.0%

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

   if 1.14999999999999995e-15 < phi1

   1. Initial program 77.7%

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

   3. Simplified 77.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \left(\pi \cdot 0.5 - \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.3 \cdot 10^{-12} \lor \neg \left(\phi_1 \leq 1.15 \cdot 10^{-15}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -3.3e-12) (not (<= phi1 1.15e-15)))
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))
   (*
    R
    (acos
     (*
      (cos phi2)
      (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -3.3e-12) || !(phi1 <= 1.15e-15)) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))));
	} else {
		tmp = R * acos((cos(phi2) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -3.3e-12) || !(phi1 <= 1.15e-15))
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -3.3e-12], N[Not[LessEqual[phi1, 1.15e-15]], $MachinePrecision]], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $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[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.3000000000000001e-12 or 1.14999999999999995e-15 < phi1

   1. Initial program 76.7%

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

   3. Simplified 76.7%

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

   if -3.3000000000000001e-12 < phi1 < 1.14999999999999995e-15

   1. Initial program 64.5%

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

   3. Simplified 64.5%

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

      1. cos-diff 82.2%

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

      2. +-commutative 82.2%

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

      3. *-commutative 82.2%

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

      4. *-commutative 82.2%

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

   6. Applied egg-rr 82.2%

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

      1. log1p-expm1-u 82.2%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   8. Applied egg-rr 82.2%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   9. Taylor expanded in phi1 around 0 82.2%

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

       1. cos-neg 82.2%

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

       2. *-commutative 82.2%

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

       3. cos-neg 82.2%

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

       4. fma-undefine 82.3%

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

   11. Simplified 82.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.3 \cdot 10^{-12} \lor \neg \left(\phi_1 \leq 1.15 \cdot 10^{-15}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.3 \cdot 10^{-12} \lor \neg \left(\phi_1 \leq 1.15 \cdot 10^{-15}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -3.3e-12) (not (<= phi1 1.15e-15)))
   (*
    R
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))
   (*
    R
    (acos
     (*
      (cos phi2)
      (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -3.3e-12) || !(phi1 <= 1.15e-15)) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = R * acos((cos(phi2) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -3.3e-12) || !(phi1 <= 1.15e-15))
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -3.3e-12], N[Not[LessEqual[phi1, 1.15e-15]], $MachinePrecision]], N[(R * 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]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.3000000000000001e-12 or 1.14999999999999995e-15 < phi1

   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

   if -3.3000000000000001e-12 < phi1 < 1.14999999999999995e-15

   1. Initial program 64.5%

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

   3. Simplified 64.5%

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

      1. cos-diff 82.2%

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

      2. +-commutative 82.2%

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

      3. *-commutative 82.2%

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

      4. *-commutative 82.2%

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

   6. Applied egg-rr 82.2%

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

      1. log1p-expm1-u 82.2%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   8. Applied egg-rr 82.2%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   9. Taylor expanded in phi1 around 0 82.2%

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

       1. cos-neg 82.2%

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

       2. *-commutative 82.2%

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

       3. cos-neg 82.2%

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

       4. fma-undefine 82.3%

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

   11. Simplified 82.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.3 \cdot 10^{-12} \lor \neg \left(\phi_1 \leq 1.15 \cdot 10^{-15}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(t\_1, \cos \left(\lambda_2 - \lambda_1\right), t\_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (* (cos phi1) (cos phi2))))
   (if (<= phi1 -3.1e-12)
     (* R (acos (fma t_1 (cos (- lambda2 lambda1)) t_0)))
     (if (<= phi1 1.15e-15)
       (*
        R
        (acos
         (*
          (cos phi2)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))
       (* R (acos (+ t_0 (* t_1 (cos (- lambda1 lambda2))))))))))

C

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 (phi1 <= -3.1e-12) {
		tmp = R * acos(fma(t_1, cos((lambda2 - lambda1)), t_0));
	} else if (phi1 <= 1.15e-15) {
		tmp = R * acos((cos(phi2) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos((t_0 + (t_1 * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Julia

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 (phi1 <= -3.1e-12)
		tmp = Float64(R * acos(fma(t_1, cos(Float64(lambda2 - lambda1)), t_0)));
	elseif (phi1 <= 1.15e-15)
		tmp = Float64(R * acos(Float64(cos(phi2) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(t_1 * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

Wolfram

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[phi1, -3.1e-12], N[(R * N[ArcCos[N[(t$95$1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 1.15e-15], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.1000000000000001e-12

   1. Initial program 75.5%

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

   3. Simplified 75.5%

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

   if -3.1000000000000001e-12 < phi1 < 1.14999999999999995e-15

   1. Initial program 64.5%

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

   3. Simplified 64.5%

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

      1. cos-diff 82.2%

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

      2. +-commutative 82.2%

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

      3. *-commutative 82.2%

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

      4. *-commutative 82.2%

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

   6. Applied egg-rr 82.2%

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

      1. log1p-expm1-u 82.2%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   8. Applied egg-rr 82.2%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   9. Taylor expanded in phi1 around 0 82.2%

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

       1. cos-neg 82.2%

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

       2. *-commutative 82.2%

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

       3. cos-neg 82.2%

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

       4. fma-undefine 82.3%

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

   11. Simplified 82.3%

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

   if 1.14999999999999995e-15 < phi1

   1. Initial program 77.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.15 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.8 \cdot 10^{-9} \lor \neg \left(\phi_1 \leq 1.15 \cdot 10^{-15}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -6.8e-9) (not (<= phi1 1.15e-15)))
   (*
    R
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))
   (*
    R
    (acos
     (*
      (cos phi2)
      (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -6.8e-9) || !(phi1 <= 1.15e-15)) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = R * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((phi1 <= (-6.8d-9)) .or. (.not. (phi1 <= 1.15d-15))) then
        tmp = r * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
    else
        tmp = r * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -6.8e-9) || !(phi1 <= 1.15e-15)) {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi1 <= -6.8e-9) or not (phi1 <= 1.15e-15):
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = R * math.acos((math.cos(phi2) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -6.8e-9) || !(phi1 <= 1.15e-15))
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi1 <= -6.8e-9) || ~((phi1 <= 1.15e-15)))
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	else
		tmp = R * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -6.8e-9], N[Not[LessEqual[phi1, 1.15e-15]], $MachinePrecision]], N[(R * 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]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $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 < -6.7999999999999997e-9 or 1.14999999999999995e-15 < phi1

   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

   if -6.7999999999999997e-9 < phi1 < 1.14999999999999995e-15

   1. Initial program 64.5%

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

   3. Simplified 64.5%

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

      1. cos-diff 82.2%

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

      2. +-commutative 82.2%

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

      3. *-commutative 82.2%

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

      4. *-commutative 82.2%

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

   6. Applied egg-rr 82.2%

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

      1. log1p-expm1-u 82.2%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   8. Applied egg-rr 82.2%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   9. Taylor expanded in phi1 around 0 82.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.8 \cdot 10^{-9} \lor \neg \left(\phi_1 \leq 1.15 \cdot 10^{-15}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.2 \cdot 10^{-9} \lor \neg \left(\phi_1 \leq 1.15 \cdot 10^{-15}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -7.2e-9) (not (<= phi1 1.15e-15)))
   (*
    R
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (* (* (cos phi1) (cos phi2)) (cos lambda2)))))
   (*
    R
    (acos
     (*
      (cos phi2)
      (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -7.2e-9) || !(phi1 <= 1.15e-15)) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	} else {
		tmp = R * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((phi1 <= (-7.2d-9)) .or. (.not. (phi1 <= 1.15d-15))) then
        tmp = r * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda2))))
    else
        tmp = r * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -7.2e-9) || !(phi1 <= 1.15e-15)) {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos(lambda2))));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi1 <= -7.2e-9) or not (phi1 <= 1.15e-15):
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos(lambda2))))
	else:
		tmp = R * math.acos((math.cos(phi2) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -7.2e-9) || !(phi1 <= 1.15e-15))
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda2)))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi1 <= -7.2e-9) || ~((phi1 <= 1.15e-15)))
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	else
		tmp = R * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -7.2e-9], N[Not[LessEqual[phi1, 1.15e-15]], $MachinePrecision]], N[(R * 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[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $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 < -7.2e-9 or 1.14999999999999995e-15 < phi1

   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 lambda1 around 0 57.5%

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

      1. cos-neg 57.5%

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

      2. associate-*r* 57.5%

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

      3. *-commutative 57.5%

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

   5. Simplified 57.5%

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

   if -7.2e-9 < phi1 < 1.14999999999999995e-15

   1. Initial program 64.5%

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

   3. Simplified 64.5%

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

      1. cos-diff 82.2%

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

      2. +-commutative 82.2%

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

      3. *-commutative 82.2%

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

      4. *-commutative 82.2%

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

   6. Applied egg-rr 82.2%

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

      1. log1p-expm1-u 82.2%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   8. Applied egg-rr 82.2%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   9. Taylor expanded in phi1 around 0 82.2%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.2 \cdot 10^{-9} \lor \neg \left(\phi_1 \leq 1.15 \cdot 10^{-15}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -4.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 9.8 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= lambda2 -4.8e-8)
     (*
      R
      (acos
       (*
        (cos phi2)
        (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))))))
     (if (<= lambda2 9.8e-5)
       (* R (acos (+ t_0 (* (cos phi2) (* (cos phi1) (cos lambda1))))))
       (* R (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos lambda2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda2 <= -4.8e-8) {
		tmp = R * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	} else if (lambda2 <= 9.8e-5) {
		tmp = R * acos((t_0 + (cos(phi2) * (cos(phi1) * cos(lambda1)))));
	} else {
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    if (lambda2 <= (-4.8d-8)) then
        tmp = r * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))))
    else if (lambda2 <= 9.8d-5) then
        tmp = r * acos((t_0 + (cos(phi2) * (cos(phi1) * cos(lambda1)))))
    else
        tmp = r * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda2))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda2 <= -4.8e-8) {
		tmp = R * Math.acos((Math.cos(phi2) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))));
	} else if (lambda2 <= 9.8e-5) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi2) * (Math.cos(phi1) * Math.cos(lambda1)))));
	} else {
		tmp = R * Math.acos((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if lambda2 <= -4.8e-8:
		tmp = R * math.acos((math.cos(phi2) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2)))))
	elif lambda2 <= 9.8e-5:
		tmp = R * math.acos((t_0 + (math.cos(phi2) * (math.cos(phi1) * math.cos(lambda1)))))
	else:
		tmp = R * math.acos((t_0 + ((math.cos(phi1) * math.cos(phi2)) * math.cos(lambda2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda2 <= -4.8e-8)
		tmp = Float64(R * acos(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))));
	elseif (lambda2 <= 9.8e-5)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi2) * Float64(cos(phi1) * cos(lambda1))))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda2 <= -4.8e-8)
		tmp = R * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	elseif (lambda2 <= 9.8e-5)
		tmp = R * acos((t_0 + (cos(phi2) * (cos(phi1) * cos(lambda1)))));
	else
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -4.8e-8], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 9.8e-5], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -4.79999999999999997e-8

   1. Initial program 51.1%

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

   3. Simplified 51.1%

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

      1. cos-diff 99.2%

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

      2. +-commutative 99.2%

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

      3. *-commutative 99.2%

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

      4. *-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. log1p-expm1-u 99.1%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   8. Applied egg-rr 99.1%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   9. Taylor expanded in phi1 around 0 50.6%

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

   if -4.79999999999999997e-8 < lambda2 < 9.8e-5

   1. Initial program 85.3%

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

   3. Taylor expanded in lambda2 around 0 85.3%

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

      1. associate-*r* 85.4%

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

      2. *-commutative 85.4%

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

      3. *-commutative 85.4%

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

   5. Simplified 85.4%

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

   if 9.8e-5 < lambda2

   1. Initial program 56.3%

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

   3. Taylor expanded in lambda1 around 0 56.0%

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

      1. cos-neg 56.0%

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

      2. associate-*r* 56.0%

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

      3. *-commutative 56.0%

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

   5. Simplified 56.0%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -4.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 9.8 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_1 \leq -0.00032:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2)))))
   (if (<= phi1 -0.00032)
     (* R (acos (* (cos phi1) t_0)))
     (* R (acos (* (cos phi2) t_0))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2));
	double tmp;
	if (phi1 <= -0.00032) {
		tmp = R * acos((cos(phi1) * t_0));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))
    if (phi1 <= (-0.00032d0)) then
        tmp = r * acos((cos(phi1) * t_0))
    else
        tmp = r * acos((cos(phi2) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2));
	double tmp;
	if (phi1 <= -0.00032) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * t_0));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2))
	tmp = 0
	if phi1 <= -0.00032:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R * math.acos((math.cos(phi2) * t_0))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))
	tmp = 0.0
	if (phi1 <= -0.00032)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2));
	tmp = 0.0;
	if (phi1 <= -0.00032)
		tmp = R * acos((cos(phi1) * t_0));
	else
		tmp = R * acos((cos(phi2) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -0.00032], 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 < -3.20000000000000026e-4

   1. Initial program 75.5%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in phi2 around 0 43.5%

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

      1. cos-diff 98.7%

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

      2. +-commutative 98.7%

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

      3. *-commutative 98.7%

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

      4. *-commutative 98.7%

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

   7. Applied egg-rr 52.6%

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

   if -3.20000000000000026e-4 < phi1

   1. Initial program 70.2%

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

   3. Simplified 70.2%

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

      1. cos-diff 89.5%

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

      2. +-commutative 89.5%

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

      3. *-commutative 89.5%

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

      4. *-commutative 89.5%

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

   6. Applied egg-rr 89.5%

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

      1. log1p-expm1-u 89.5%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   8. Applied egg-rr 89.5%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\right) \]

   9. Taylor expanded in phi1 around 0 54.8%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00032:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0175:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.0175)
   (*
    R
    (acos
     (*
      (cos phi1)
      (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))))))
   (* R (acos (* (cos phi2) (cos (- lambda2 lambda1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.0175) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 0.0175d0) then
        tmp = r * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))))
    else
        tmp = r * acos((cos(phi2) * cos((lambda2 - lambda1))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.0175) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos((lambda2 - lambda1))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.0175:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2)))))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos((lambda2 - lambda1))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.0175)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 0.0175)
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	else
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.0175], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $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 phi2 < 0.017500000000000002

   1. Initial program 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in phi2 around 0 49.8%

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

      1. cos-diff 89.8%

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

      2. +-commutative 89.8%

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

      3. *-commutative 89.8%

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

      4. *-commutative 89.8%

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

   7. Applied egg-rr 64.2%

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

   if 0.017500000000000002 < phi2

   1. Initial program 74.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in phi1 around 0 46.3%

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

Alternative 18: 50.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0175:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.0175)
   (*
    R
    (acos
     (+ (* (sin phi1) (sin phi2)) (* (cos phi1) (cos (- lambda1 lambda2))))))
   (* R (acos (* (cos phi2) (cos (- lambda2 lambda1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.0175) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos((lambda1 - lambda2)))));
	} else {
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 0.0175d0) then
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos((lambda1 - lambda2)))))
    else
        tmp = r * acos((cos(phi2) * cos((lambda2 - lambda1))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.0175) {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi1) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos((lambda2 - lambda1))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.0175:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi1) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos((lambda2 - lambda1))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.0175)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 0.0175)
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos((lambda1 - lambda2)))));
	else
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.0175], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $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 phi2 < 0.017500000000000002

   1. Initial program 70.7%

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

   3. Taylor expanded in phi2 around 0 49.5%

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

   if 0.017500000000000002 < phi2

   1. Initial program 74.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in phi1 around 0 46.3%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0175:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 50.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 0.0175:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi2 0.0175)
     (* R (acos (* (cos phi1) t_0)))
     (* R (acos (* (cos phi2) t_0))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 0.0175) {
		tmp = R * acos((cos(phi1) * t_0));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (phi2 <= 0.0175d0) then
        tmp = r * acos((cos(phi1) * t_0))
    else
        tmp = r * acos((cos(phi2) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 0.0175) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * t_0));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi2 <= 0.0175:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R * math.acos((math.cos(phi2) * t_0))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= 0.0175)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi2 <= 0.0175)
		tmp = R * acos((cos(phi1) * t_0));
	else
		tmp = R * acos((cos(phi2) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 0.0175], 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 phi2 < 0.017500000000000002

   1. Initial program 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in phi2 around 0 49.8%

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

   if 0.017500000000000002 < phi2

   1. Initial program 74.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in phi1 around 0 46.3%

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

Alternative 20: 37.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.0085:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -0.0085)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (* (cos phi1) (cos lambda2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -0.0085) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-0.0085d0)) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos((cos(phi1) * cos(lambda2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -0.0085) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -0.0085:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -0.0085)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -0.0085)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -0.0085], 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 < -0.0085000000000000006

   1. Initial program 58.7%

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

   3. Simplified 58.7%

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

   5. Taylor expanded in phi2 around 0 40.7%

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

   6. Taylor expanded in lambda2 around 0 40.5%

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

      1. cos-neg 40.5%

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

   8. Simplified 40.5%

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

   if -0.0085000000000000006 < lambda1

   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. Step-by-step derivation

   3. Simplified 76.4%

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

   5. Taylor expanded in phi2 around 0 42.7%

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

   6. Taylor expanded in lambda1 around 0 36.2%

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

Alternative 21: 34.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.0052:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 0.0052)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (- (/ PI 2.0) (asin (cos (- lambda2 lambda1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 0.0052) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * ((((double) M_PI) / 2.0) - asin(cos((lambda2 - lambda1))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 0.0052) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * ((Math.PI / 2.0) - Math.asin(Math.cos((lambda2 - lambda1))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 0.0052:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * ((math.pi / 2.0) - math.asin(math.cos((lambda2 - lambda1))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 0.0052)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * Float64(Float64(pi / 2.0) - asin(cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 0.0052)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * ((pi / 2.0) - asin(cos((lambda2 - lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 0.0052], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 0.0051999999999999998

   1. Initial program 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in phi2 around 0 43.6%

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

   6. Taylor expanded in lambda2 around 0 38.4%

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

      1. cos-neg 38.4%

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

   8. Simplified 38.4%

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

   if 0.0051999999999999998 < lambda2

   1. Initial program 56.3%

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

   3. Simplified 56.3%

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

   5. Taylor expanded in phi2 around 0 36.8%

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

   6. Taylor expanded in phi1 around 0 30.6%

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

      1. acos-asin 30.5%

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

      2. cos-diff 40.3%

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

      3. *-commutative 40.3%

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

      4. *-commutative 40.3%

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

      5. cos-diff 30.5%

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

   8. Applied egg-rr 30.5%

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

      1. cos-neg 30.5%

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

      2. sub-neg 30.5%

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

      3. +-commutative 30.5%

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

      4. distribute-neg-in 30.5%

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

      5. remove-double-neg 30.5%

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

      6. sub-neg 30.5%

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

   10. Simplified 30.5%

       \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 42.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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 = r * acos((cos(phi1) * cos((lambda2 - lambda1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Simplified 71.6%

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

5. Taylor expanded in phi2 around 0 42.1%

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

Alternative 23: 16.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -10:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 \cdot \left(R \cdot \left(-1 + \frac{\lambda_2}{\lambda_1}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -10.0)
   (* R (- (/ PI 2.0) (asin (cos (- lambda2 lambda1)))))
   (* lambda1 (* R (+ -1.0 (/ lambda2 lambda1))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -10.0) {
		tmp = R * ((((double) M_PI) / 2.0) - asin(cos((lambda2 - lambda1))));
	} else {
		tmp = lambda1 * (R * (-1.0 + (lambda2 / lambda1)));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -10.0) {
		tmp = R * ((Math.PI / 2.0) - Math.asin(Math.cos((lambda2 - lambda1))));
	} else {
		tmp = lambda1 * (R * (-1.0 + (lambda2 / lambda1)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda1 - lambda2) <= -10.0:
		tmp = R * ((math.pi / 2.0) - math.asin(math.cos((lambda2 - lambda1))))
	else:
		tmp = lambda1 * (R * (-1.0 + (lambda2 / lambda1)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -10.0)
		tmp = Float64(R * Float64(Float64(pi / 2.0) - asin(cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(lambda1 * Float64(R * Float64(-1.0 + Float64(lambda2 / lambda1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -10.0)
		tmp = R * ((pi / 2.0) - asin(cos((lambda2 - lambda1))));
	else
		tmp = lambda1 * (R * (-1.0 + (lambda2 / lambda1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -10.0], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(lambda1 * N[(R * N[(-1.0 + N[(lambda2 / lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -10

   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. Step-by-step derivation

   3. Simplified 71.4%

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

   5. Taylor expanded in phi2 around 0 45.8%

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

   6. Taylor expanded in phi1 around 0 32.0%

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

      1. acos-asin 32.0%

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

      2. cos-diff 39.8%

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

      3. *-commutative 39.8%

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

      4. *-commutative 39.8%

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

      5. cos-diff 32.0%

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

   8. Applied egg-rr 32.0%

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

      1. cos-neg 32.0%

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

      2. sub-neg 32.0%

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

      3. +-commutative 32.0%

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

      4. distribute-neg-in 32.0%

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

      5. remove-double-neg 32.0%

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

      6. sub-neg 32.0%

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

   10. Simplified 32.0%

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

   if -10 < (-.f64 lambda1 lambda2)

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in phi2 around 0 40.1%

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

   6. Taylor expanded in phi1 around 0 17.1%

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

   7. Taylor expanded in lambda1 around inf 7.6%

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

      1. *-commutative 7.6%

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

      2. associate-/l* 7.6%

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

      3. distribute-lft-out 7.6%

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

   9. Simplified 7.6%

      \[\leadsto \color{blue}{\lambda_1 \cdot \left(R \cdot \left(-1 + \frac{\lambda_2}{\lambda_1}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 16.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -10:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 \cdot \left(R \cdot \left(-1 + \frac{\lambda_2}{\lambda_1}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -10.0)
   (* R (acos (cos (- lambda2 lambda1))))
   (* lambda1 (* R (+ -1.0 (/ lambda2 lambda1))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -10.0) {
		tmp = R * acos(cos((lambda2 - lambda1)));
	} else {
		tmp = lambda1 * (R * (-1.0 + (lambda2 / lambda1)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((lambda1 - lambda2) <= (-10.0d0)) then
        tmp = r * acos(cos((lambda2 - lambda1)))
    else
        tmp = lambda1 * (r * ((-1.0d0) + (lambda2 / lambda1)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -10.0) {
		tmp = R * Math.acos(Math.cos((lambda2 - lambda1)));
	} else {
		tmp = lambda1 * (R * (-1.0 + (lambda2 / lambda1)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda1 - lambda2) <= -10.0:
		tmp = R * math.acos(math.cos((lambda2 - lambda1)))
	else:
		tmp = lambda1 * (R * (-1.0 + (lambda2 / lambda1)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -10.0)
		tmp = Float64(R * acos(cos(Float64(lambda2 - lambda1))));
	else
		tmp = Float64(lambda1 * Float64(R * Float64(-1.0 + Float64(lambda2 / lambda1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -10.0)
		tmp = R * acos(cos((lambda2 - lambda1)));
	else
		tmp = lambda1 * (R * (-1.0 + (lambda2 / lambda1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -10.0], N[(R * N[ArcCos[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 * N[(R * N[(-1.0 + N[(lambda2 / lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -10

   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. Step-by-step derivation

   3. Simplified 71.4%

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

   5. Taylor expanded in phi2 around 0 45.8%

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

   6. Taylor expanded in phi1 around 0 32.0%

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

   if -10 < (-.f64 lambda1 lambda2)

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in phi2 around 0 40.1%

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

   6. Taylor expanded in phi1 around 0 17.1%

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

   7. Taylor expanded in lambda1 around inf 7.6%

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

      1. *-commutative 7.6%

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

      2. associate-/l* 7.6%

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

      3. distribute-lft-out 7.6%

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

   9. Simplified 7.6%

      \[\leadsto \color{blue}{\lambda_1 \cdot \left(R \cdot \left(-1 + \frac{\lambda_2}{\lambda_1}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 21.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 6 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 6e-6) (* R (acos (cos lambda1))) (* R (acos (cos lambda2)))))

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 6d-6) then
        tmp = r * acos(cos(lambda1))
    else
        tmp = r * acos(cos(lambda2))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 6e-6)
		tmp = R * acos(cos(lambda1));
	else
		tmp = R * acos(cos(lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 6e-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 lambda2 < 6.0000000000000002e-6

   1. Initial program 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in phi2 around 0 43.6%

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

   6. Taylor expanded in phi1 around 0 20.1%

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

   7. Taylor expanded in lambda2 around 0 17.1%

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

      1. cos-neg 38.4%

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

   9. Simplified 17.1%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \lambda_1} \]

   if 6.0000000000000002e-6 < lambda2

   1. Initial program 56.3%

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

   3. Simplified 56.3%

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

   5. Taylor expanded in phi2 around 0 36.8%

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

   6. Taylor expanded in phi1 around 0 30.6%

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

   7. Taylor expanded in lambda1 around 0 30.1%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \lambda_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 12.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.5e-6)
   (* R (acos (cos lambda1)))
   (* R (* lambda2 (- 1.0 (/ lambda1 lambda2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.5e-6) {
		tmp = R * acos(cos(lambda1));
	} else {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-1.5d-6)) then
        tmp = r * acos(cos(lambda1))
    else
        tmp = r * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.5e-6) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.5e-6:
		tmp = R * math.acos(math.cos(lambda1))
	else:
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.5e-6)
		tmp = Float64(R * acos(cos(lambda1)));
	else
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.5e-6)
		tmp = R * acos(cos(lambda1));
	else
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.5e-6], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.5e-6

   1. Initial program 58.4%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in phi2 around 0 40.2%

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

   6. Taylor expanded in phi1 around 0 27.1%

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

   7. Taylor expanded in lambda2 around 0 26.6%

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

      1. cos-neg 39.7%

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

   9. Simplified 26.6%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \lambda_1} \]

   if -1.5e-6 < lambda1

   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. Step-by-step derivation

   3. Simplified 76.7%

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

   5. Taylor expanded in phi2 around 0 42.8%

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

   6. Taylor expanded in phi1 around 0 20.5%

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

   7. Taylor expanded in lambda2 around inf 7.9%

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

      1. mul-1-neg 7.9%

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

      2. unsub-neg 7.9%

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

   9. Simplified 7.9%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 6.7% accurate, 68.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5 \cdot 10^{-145}:\\ \;\;\;\;-\lambda_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -5e-145) (- (* lambda1 R)) (* lambda2 R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -5e-145) {
		tmp = -(lambda1 * R);
	} else {
		tmp = lambda2 * R;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-5d-145)) then
        tmp = -(lambda1 * r)
    else
        tmp = lambda2 * r
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -5e-145) {
		tmp = -(lambda1 * R);
	} else {
		tmp = lambda2 * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -5e-145:
		tmp = -(lambda1 * R)
	else:
		tmp = lambda2 * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -5e-145)
		tmp = Float64(-Float64(lambda1 * R));
	else
		tmp = Float64(lambda2 * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -5e-145)
		tmp = -(lambda1 * R);
	else
		tmp = lambda2 * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -5e-145], (-N[(lambda1 * R), $MachinePrecision]), N[(lambda2 * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.9999999999999998e-145

   1. Initial program 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in phi2 around 0 41.7%

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

   6. Taylor expanded in phi1 around 0 24.5%

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

   7. Taylor expanded in lambda2 around 0 8.4%

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

      1. associate-*r* 8.4%

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

      2. mul-1-neg 8.4%

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

   9. Simplified 8.4%

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

   if -4.9999999999999998e-145 < lambda1

   1. Initial program 75.2%

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

   3. Simplified 75.2%

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

   5. Taylor expanded in phi2 around 0 42.4%

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

   6. Taylor expanded in phi1 around 0 21.1%

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

   7. Taylor expanded in lambda2 around inf 7.7%

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   8. Step-by-step derivation

      1. *-commutative 7.7%

         \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

   9. Simplified 7.7%

      \[\leadsto \color{blue}{\lambda_2 \cdot R} \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5 \cdot 10^{-145}:\\ \;\;\;\;-\lambda_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 5.8% accurate, 122.6× speedup

Math

\[\begin{array}{l} \\ R \cdot \left(\lambda_2 - \lambda_1\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (- lambda2 lambda1)))

C

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

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 = r * (lambda2 - lambda1)
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (lambda2 - lambda1);
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * (lambda2 - lambda1)

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * Float64(lambda2 - lambda1))
end

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.6%

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

3. Simplified 71.6%

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

5. Taylor expanded in phi2 around 0 42.1%

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

6. Taylor expanded in phi1 around 0 22.3%

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

7. Taylor expanded in lambda2 around 0 7.0%

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

Alternative 29: 5.2% accurate, 204.3× speedup

Math

\[\begin{array}{l} \\ \lambda_2 \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* lambda2 R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return 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 = lambda2 * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return lambda2 * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return lambda2 * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(lambda2 * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = lambda2 * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(lambda2 * R), $MachinePrecision]

Derivation

1. Initial program 71.6%

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

3. Simplified 71.6%

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

5. Taylor expanded in phi2 around 0 42.1%

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

6. Taylor expanded in phi1 around 0 22.3%

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

7. Taylor expanded in lambda2 around inf 6.5%

   \[\leadsto \color{blue}{R \cdot \lambda_2} \]
8. Step-by-step derivation

   1. *-commutative 6.5%

      \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

9. Simplified 6.5%

   \[\leadsto \color{blue}{\lambda_2 \cdot R} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024113 
(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))