Spherical law of cosines

Percentage Accurate: 74.2% → 94.2%

Time: 28.2s

Alternatives: 29

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 74.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.2% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (*
     (* (cos phi1) (cos phi2))
     (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos 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)) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(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)) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(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[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 74.3%

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

   1. cos-diff 93.4%

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

   2. +-commutative 93.4%

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

   3. *-commutative 93.4%

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

   4. fma-def 93.4%

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

3. Applied egg-rr 93.4%

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

4. Final simplification 93.4%

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

Alternative 2: 94.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. cos-diff 93.4%

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

   2. +-commutative 93.4%

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

3. Applied egg-rr 93.4%

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

4. Final simplification 93.4%

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

Alternative 3: 68.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot e^{\log \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3.4e-5)
   (*
    R
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (*
       (cos phi1)
       (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2)))))))
   (*
    R
    (exp
     (log
      (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 (phi2 <= 3.4e-5) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))))));
	} else {
		tmp = R * exp(log(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 (phi2 <= 3.4e-5)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2)))))));
	else
		tmp = Float64(R * exp(log(acos(fma(sin(phi1), sin(phi2), Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3.4e-5], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[Exp[N[Log[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[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.4e-5

   1. Initial program 72.1%

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

   2. Taylor expanded in phi2 around 0 59.3%

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

      1. sub-neg 59.3%

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

      2. +-commutative 59.3%

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

      3. neg-mul-1 59.3%

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

      4. neg-mul-1 59.3%

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

      5. remove-double-neg 59.3%

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

      6. mul-1-neg 59.3%

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

      7. distribute-neg-in 59.3%

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

      8. +-commutative 59.3%

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

      9. cos-neg 59.3%

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

      10. +-commutative 59.3%

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

      11. mul-1-neg 59.3%

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

      12. unsub-neg 59.3%

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

   4. Simplified 59.3%

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

      1. cos-diff 46.9%

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

      2. *-commutative 46.9%

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

   6. Applied egg-rr 72.8%

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

      1. +-commutative 46.9%

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

      2. *-commutative 46.9%

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

      3. fma-def 47.0%

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

   8. Simplified 72.9%

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

   if 3.4e-5 < phi2

   1. Initial program 80.9%

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

      1. log1p-expm1-u 80.8%

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

      2. *-commutative 80.8%

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

   3. Applied egg-rr 80.8%

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

      1. add-exp-log 80.8%

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

      2. fma-def 80.9%

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

      3. log1p-expm1-u 81.0%

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

      4. *-commutative 81.0%

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

   5. Applied egg-rr 81.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot e^{\log \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \end{array} \]

Alternative 4: 68.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot e^{\log \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 6.5e-7)
   (*
    R
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (*
       (cos phi1)
       (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda2) (sin lambda1)))))))
   (*
    R
    (exp
     (log
      (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 (phi2 <= 6.5e-7) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1))))));
	} else {
		tmp = R * exp(log(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 (phi2 <= 6.5e-7)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda2) * sin(lambda1)))))));
	else
		tmp = Float64(R * exp(log(acos(fma(sin(phi1), sin(phi2), Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 6.5e-7], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[Exp[N[Log[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[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 6.50000000000000024e-7

   1. Initial program 72.1%

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

   2. Taylor expanded in phi2 around 0 59.3%

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

      1. sub-neg 59.3%

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

      2. +-commutative 59.3%

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

      3. neg-mul-1 59.3%

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

      4. neg-mul-1 59.3%

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

      5. remove-double-neg 59.3%

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

      6. mul-1-neg 59.3%

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

      7. distribute-neg-in 59.3%

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

      8. +-commutative 59.3%

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

      9. cos-neg 59.3%

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

      10. +-commutative 59.3%

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

      11. mul-1-neg 59.3%

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

      12. unsub-neg 59.3%

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

   4. Simplified 59.3%

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

      1. cos-diff 46.9%

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

      2. *-commutative 46.9%

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

   6. Applied egg-rr 72.8%

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

   if 6.50000000000000024e-7 < phi2

   1. Initial program 80.9%

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

      1. log1p-expm1-u 80.8%

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

      2. *-commutative 80.8%

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

   3. Applied egg-rr 80.8%

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

      1. add-exp-log 80.8%

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

      2. fma-def 80.9%

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

      3. log1p-expm1-u 81.0%

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

      4. *-commutative 81.0%

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

   5. Applied egg-rr 81.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot e^{\log \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \end{array} \]

Alternative 5: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + t_0 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + \log \left(1 + \mathsf{expm1}\left(t_0 \cdot t_1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (* (sin phi1) (sin phi2))))
   (if (<= phi1 -3.1e-6)
     (* R (acos (+ t_2 (* t_0 (log1p (expm1 t_1))))))
     (if (<= phi1 3.6e-6)
       (*
        R
        (acos
         (+
          t_2
          (*
           (cos phi2)
           (+
            (* (cos lambda1) (cos lambda2))
            (* (sin lambda2) (sin lambda1)))))))
       (* R (acos (+ t_2 (log (+ 1.0 (expm1 (* t_0 t_1)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi1 <= -3.1e-6) {
		tmp = R * acos((t_2 + (t_0 * log1p(expm1(t_1)))));
	} else if (phi1 <= 3.6e-6) {
		tmp = R * acos((t_2 + (cos(phi2) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1))))));
	} else {
		tmp = R * acos((t_2 + log((1.0 + expm1((t_0 * t_1))))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.cos(phi2);
	double t_1 = Math.cos((lambda1 - lambda2));
	double t_2 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (phi1 <= -3.1e-6) {
		tmp = R * Math.acos((t_2 + (t_0 * Math.log1p(Math.expm1(t_1)))));
	} else if (phi1 <= 3.6e-6) {
		tmp = R * Math.acos((t_2 + (Math.cos(phi2) * ((Math.cos(lambda1) * Math.cos(lambda2)) + (Math.sin(lambda2) * Math.sin(lambda1))))));
	} else {
		tmp = R * Math.acos((t_2 + Math.log((1.0 + Math.expm1((t_0 * t_1))))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.cos(phi2)
	t_1 = math.cos((lambda1 - lambda2))
	t_2 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if phi1 <= -3.1e-6:
		tmp = R * math.acos((t_2 + (t_0 * math.log1p(math.expm1(t_1)))))
	elif phi1 <= 3.6e-6:
		tmp = R * math.acos((t_2 + (math.cos(phi2) * ((math.cos(lambda1) * math.cos(lambda2)) + (math.sin(lambda2) * math.sin(lambda1))))))
	else:
		tmp = R * math.acos((t_2 + math.log((1.0 + math.expm1((t_0 * t_1))))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi1 <= -3.1e-6)
		tmp = Float64(R * acos(Float64(t_2 + Float64(t_0 * log1p(expm1(t_1))))));
	elseif (phi1 <= 3.6e-6)
		tmp = Float64(R * acos(Float64(t_2 + Float64(cos(phi2) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda2) * sin(lambda1)))))));
	else
		tmp = Float64(R * acos(Float64(t_2 + log(Float64(1.0 + expm1(Float64(t_0 * t_1)))))));
	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[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -3.1e-6], N[(R * N[ArcCos[N[(t$95$2 + N[(t$95$0 * N[Log[1 + N[(Exp[t$95$1] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 3.6e-6], N[(R * N[ArcCos[N[(t$95$2 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$2 + N[Log[N[(1.0 + N[(Exp[N[(t$95$0 * t$95$1), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.1e-6

   1. Initial program 80.5%

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

      1. log1p-expm1-u 80.6%

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

   3. Applied egg-rr 80.6%

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

   if -3.1e-6 < phi1 < 3.59999999999999984e-6

   1. Initial program 66.4%

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

      1. cos-diff 87.5%

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

      2. +-commutative 87.5%

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

      3. *-commutative 87.5%

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

      4. fma-def 87.5%

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

   3. Applied egg-rr 87.5%

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

   4. Taylor expanded in phi1 around 0 87.5%

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

   if 3.59999999999999984e-6 < phi1

   1. Initial program 83.1%

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

      1. log1p-expm1-u 83.0%

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

      2. log1p-udef 83.0%

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

      3. *-commutative 83.0%

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

   3. Applied egg-rr 83.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \log \left(1 + \mathsf{expm1}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]

Alternative 6: 68.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq 7 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= phi2 7e-7)
     (*
      R
      (acos
       (+
        t_0
        (*
         (cos phi1)
         (+
          (* (cos lambda1) (cos lambda2))
          (* (sin lambda2) (sin lambda1)))))))
     (*
      R
      (acos
       (+
        t_0
        (*
         (* (cos phi1) (cos phi2))
         (expm1 (log1p (cos (- lambda1 lambda2)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= 7e-7) {
		tmp = R * acos((t_0 + (cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1))))));
	} else {
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * expm1(log1p(cos((lambda1 - lambda2)))))));
	}
	return tmp;
}

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 (phi2 <= 7e-7) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * ((Math.cos(lambda1) * Math.cos(lambda2)) + (Math.sin(lambda2) * Math.sin(lambda1))))));
	} else {
		tmp = R * Math.acos((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * Math.expm1(Math.log1p(Math.cos((lambda1 - lambda2)))))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if phi2 <= 7e-7:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * ((math.cos(lambda1) * math.cos(lambda2)) + (math.sin(lambda2) * math.sin(lambda1))))))
	else:
		tmp = R * math.acos((t_0 + ((math.cos(phi1) * math.cos(phi2)) * math.expm1(math.log1p(math.cos((lambda1 - lambda2)))))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= 7e-7)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda2) * sin(lambda1)))))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * expm1(log1p(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]}, If[LessEqual[phi2, 7e-7], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $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[(Exp[N[Log[1 + N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 6.99999999999999968e-7

   1. Initial program 72.1%

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

   2. Taylor expanded in phi2 around 0 59.3%

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

      1. sub-neg 59.3%

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

      2. +-commutative 59.3%

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

      3. neg-mul-1 59.3%

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

      4. neg-mul-1 59.3%

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

      5. remove-double-neg 59.3%

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

      6. mul-1-neg 59.3%

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

      7. distribute-neg-in 59.3%

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

      8. +-commutative 59.3%

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

      9. cos-neg 59.3%

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

      10. +-commutative 59.3%

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

      11. mul-1-neg 59.3%

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

      12. unsub-neg 59.3%

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

   4. Simplified 59.3%

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

      1. cos-diff 46.9%

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

      2. *-commutative 46.9%

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

   6. Applied egg-rr 72.8%

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

   if 6.99999999999999968e-7 < phi2

   1. Initial program 80.9%

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

      1. expm1-log1p-u 80.9%

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

   3. Applied egg-rr 80.9%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 70.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 2900000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 2900000.0)
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))
   (*
    R
    (acos
     (+
      (*
       (cos phi1)
       (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))
      (* phi1 (sin phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= 2900000.0) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = R * acos(((cos(phi1) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))) + (phi1 * sin(phi2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= 2900000.0)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2)))) + Float64(phi1 * sin(phi2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, 2900000.0], N[(R * 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[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < 2.9e6

   1. Initial program 80.3%

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

      1. add-cube-cbrt 79.3%

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

      2. pow3 79.2%

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

      3. +-commutative 79.2%

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

      4. *-commutative 79.2%

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

      5. fma-def 79.2%

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

   3. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R}\right)}^{3}} \]
   4. Step-by-step derivation

      1. rem-cube-cbrt 80.3%

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

      2. fma-udef 80.3%

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

      3. log1p-expm1-u 80.3%

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

      4. +-commutative 80.3%

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

      5. fma-def 80.3%

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

      6. log1p-expm1-u 80.4%

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

      7. *-commutative 80.4%

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

   5. Applied egg-rr 80.4%

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

   if 2.9e6 < lambda1

   1. Initial program 55.4%

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

   2. Taylor expanded in phi2 around 0 38.4%

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

      1. sub-neg 38.4%

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

      2. +-commutative 38.4%

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

      3. neg-mul-1 38.4%

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

      4. neg-mul-1 38.4%

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

      5. remove-double-neg 38.4%

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

      6. mul-1-neg 38.4%

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

      7. distribute-neg-in 38.4%

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

      8. +-commutative 38.4%

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

      9. cos-neg 38.4%

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

      10. +-commutative 38.4%

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

      11. mul-1-neg 38.4%

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

      12. unsub-neg 38.4%

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

   4. Simplified 38.4%

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

   5. Taylor expanded in phi1 around 0 25.5%

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

      1. cos-diff 40.0%

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

      2. *-commutative 40.0%

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

   7. Applied egg-rr 43.1%

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

      1. +-commutative 40.0%

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

      2. *-commutative 40.0%

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

      3. fma-def 40.1%

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

   9. Simplified 43.2%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 2900000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 8: 70.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 1450000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 1450000.0)
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))
   (*
    R
    (acos
     (+
      (*
       (cos phi1)
       (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda2) (sin lambda1))))
      (* phi1 (sin phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= 1450000.0) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = R * acos(((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1)))) + (phi1 * sin(phi2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= 1450000.0)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda2) * sin(lambda1)))) + Float64(phi1 * sin(phi2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, 1450000.0], N[(R * 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[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < 1.45e6

   1. Initial program 80.3%

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

      1. add-cube-cbrt 79.3%

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

      2. pow3 79.2%

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

      3. +-commutative 79.2%

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

      4. *-commutative 79.2%

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

      5. fma-def 79.2%

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

   3. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R}\right)}^{3}} \]
   4. Step-by-step derivation

      1. rem-cube-cbrt 80.3%

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

      2. fma-udef 80.3%

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

      3. log1p-expm1-u 80.3%

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

      4. +-commutative 80.3%

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

      5. fma-def 80.3%

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

      6. log1p-expm1-u 80.4%

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

      7. *-commutative 80.4%

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

   5. Applied egg-rr 80.4%

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

   if 1.45e6 < lambda1

   1. Initial program 55.4%

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

   2. Taylor expanded in phi2 around 0 38.4%

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

      1. sub-neg 38.4%

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

      2. +-commutative 38.4%

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

      3. neg-mul-1 38.4%

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

      4. neg-mul-1 38.4%

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

      5. remove-double-neg 38.4%

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

      6. mul-1-neg 38.4%

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

      7. distribute-neg-in 38.4%

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

      8. +-commutative 38.4%

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

      9. cos-neg 38.4%

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

      10. +-commutative 38.4%

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

      11. mul-1-neg 38.4%

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

      12. unsub-neg 38.4%

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

   4. Simplified 38.4%

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

   5. Taylor expanded in phi1 around 0 25.5%

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

      1. cos-diff 40.0%

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

      2. *-commutative 40.0%

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

   7. Applied egg-rr 43.1%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 1450000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 9: 69.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 2900000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 2900000.0)
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))
   (*
    R
    (acos
     (+
      (*
       (cos phi1)
       (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))
      (* phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= 2900000.0) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = R * acos(((cos(phi1) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))) + (phi1 * phi2)));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= 2900000.0)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2)))) + Float64(phi1 * phi2))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, 2900000.0], N[(R * 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[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < 2.9e6

   1. Initial program 80.3%

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

      1. add-cube-cbrt 79.3%

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

      2. pow3 79.2%

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

      3. +-commutative 79.2%

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

      4. *-commutative 79.2%

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

      5. fma-def 79.2%

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

   3. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R}\right)}^{3}} \]
   4. Step-by-step derivation

      1. rem-cube-cbrt 80.3%

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

      2. fma-udef 80.3%

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

      3. log1p-expm1-u 80.3%

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

      4. +-commutative 80.3%

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

      5. fma-def 80.3%

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

      6. log1p-expm1-u 80.4%

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

      7. *-commutative 80.4%

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

   5. Applied egg-rr 80.4%

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

   if 2.9e6 < lambda1

   1. Initial program 55.4%

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

   2. Taylor expanded in phi2 around 0 38.4%

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

      1. sub-neg 38.4%

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

      2. +-commutative 38.4%

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

      3. neg-mul-1 38.4%

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

      4. neg-mul-1 38.4%

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

      5. remove-double-neg 38.4%

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

      6. mul-1-neg 38.4%

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

      7. distribute-neg-in 38.4%

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

      8. +-commutative 38.4%

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

      9. cos-neg 38.4%

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

      10. +-commutative 38.4%

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

      11. mul-1-neg 38.4%

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

      12. unsub-neg 38.4%

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

   4. Simplified 38.4%

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

   5. Taylor expanded in phi1 around 0 25.5%

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

   6. Taylor expanded in phi2 around 0 22.4%

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

      1. *-commutative 22.4%

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

   8. Simplified 22.4%

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

      1. cos-diff 40.0%

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

      2. *-commutative 40.0%

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

   10. Applied egg-rr 40.0%

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

       1. +-commutative 40.0%

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

       2. *-commutative 40.0%

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

       3. fma-def 40.1%

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

   12. Simplified 40.1%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 2900000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \]

Alternative 10: 69.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 750000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 750000.0)
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))
   (*
    R
    (acos
     (+
      (*
       (cos phi1)
       (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda2) (sin lambda1))))
      (* phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= 750000.0) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = R * acos(((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1)))) + (phi1 * phi2)));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= 750000.0)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda2) * sin(lambda1)))) + Float64(phi1 * phi2))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, 750000.0], N[(R * 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[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < 7.5e5

   1. Initial program 80.3%

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

      1. add-cube-cbrt 79.3%

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

      2. pow3 79.2%

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

      3. +-commutative 79.2%

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

      4. *-commutative 79.2%

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

      5. fma-def 79.2%

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

   3. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R}\right)}^{3}} \]
   4. Step-by-step derivation

      1. rem-cube-cbrt 80.3%

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

      2. fma-udef 80.3%

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

      3. log1p-expm1-u 80.3%

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

      4. +-commutative 80.3%

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

      5. fma-def 80.3%

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

      6. log1p-expm1-u 80.4%

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

      7. *-commutative 80.4%

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

   5. Applied egg-rr 80.4%

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

   if 7.5e5 < lambda1

   1. Initial program 55.4%

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

   2. Taylor expanded in phi2 around 0 38.4%

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

      1. sub-neg 38.4%

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

      2. +-commutative 38.4%

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

      3. neg-mul-1 38.4%

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

      4. neg-mul-1 38.4%

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

      5. remove-double-neg 38.4%

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

      6. mul-1-neg 38.4%

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

      7. distribute-neg-in 38.4%

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

      8. +-commutative 38.4%

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

      9. cos-neg 38.4%

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

      10. +-commutative 38.4%

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

      11. mul-1-neg 38.4%

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

      12. unsub-neg 38.4%

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

   4. Simplified 38.4%

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

   5. Taylor expanded in phi1 around 0 25.5%

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

   6. Taylor expanded in phi2 around 0 22.4%

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

      1. *-commutative 22.4%

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

   8. Simplified 22.4%

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

      1. cos-diff 40.0%

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

      2. *-commutative 40.0%

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

   10. Applied egg-rr 40.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 750000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \]

Alternative 11: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 2900000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 2900000.0)
   (*
    R
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))
   (*
    R
    (acos
     (+
      (*
       (cos phi1)
       (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda2) (sin lambda1))))
      (* phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= 2900000.0) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = R * acos(((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1)))) + (phi1 * phi2)));
	}
	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 <= 2900000.0d0) then
        tmp = r * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
    else
        tmp = r * acos(((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1)))) + (phi1 * phi2)))
    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 <= 2900000.0) {
		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(phi1) * ((Math.cos(lambda1) * Math.cos(lambda2)) + (Math.sin(lambda2) * Math.sin(lambda1)))) + (phi1 * phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= 2900000.0:
		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(phi1) * ((math.cos(lambda1) * math.cos(lambda2)) + (math.sin(lambda2) * math.sin(lambda1)))) + (phi1 * phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= 2900000.0)
		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(Float64(cos(phi1) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda2) * sin(lambda1)))) + Float64(phi1 * phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= 2900000.0)
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	else
		tmp = R * acos(((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1)))) + (phi1 * phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, 2900000.0], 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[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < 2.9e6

   1. Initial program 80.3%

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

   if 2.9e6 < lambda1

   1. Initial program 55.4%

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

   2. Taylor expanded in phi2 around 0 38.4%

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

      1. sub-neg 38.4%

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

      2. +-commutative 38.4%

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

      3. neg-mul-1 38.4%

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

      4. neg-mul-1 38.4%

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

      5. remove-double-neg 38.4%

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

      6. mul-1-neg 38.4%

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

      7. distribute-neg-in 38.4%

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

      8. +-commutative 38.4%

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

      9. cos-neg 38.4%

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

      10. +-commutative 38.4%

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

      11. mul-1-neg 38.4%

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

      12. unsub-neg 38.4%

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

   4. Simplified 38.4%

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

   5. Taylor expanded in phi1 around 0 25.5%

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

   6. Taylor expanded in phi2 around 0 22.4%

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

      1. *-commutative 22.4%

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

   8. Simplified 22.4%

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

      1. cos-diff 40.0%

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

      2. *-commutative 40.0%

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

   10. Applied egg-rr 40.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 2900000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \]

Alternative 12: 59.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -120000000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= lambda1 -120000000000.0)
     (* R (acos (+ t_0 (* (cos phi2) (cos (- lambda2 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 (lambda1 <= -120000000000.0) {
		tmp = R * acos((t_0 + (cos(phi2) * cos((lambda2 - 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 (lambda1 <= (-120000000000.0d0)) then
        tmp = r * acos((t_0 + (cos(phi2) * cos((lambda2 - 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 (lambda1 <= -120000000000.0) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi2) * Math.cos((lambda2 - 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 lambda1 <= -120000000000.0:
		tmp = R * math.acos((t_0 + (math.cos(phi2) * math.cos((lambda2 - 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 (lambda1 <= -120000000000.0)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi2) * cos(Float64(lambda2 - lambda1))))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(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 (lambda1 <= -120000000000.0)
		tmp = R * acos((t_0 + (cos(phi2) * cos((lambda2 - 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[lambda1, -120000000000.0], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.2e11

   1. Initial program 57.6%

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

   2. Taylor expanded in phi1 around 0 35.8%

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

      1. sub-neg 35.8%

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

      2. +-commutative 35.8%

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

      3. neg-mul-1 35.8%

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

      4. neg-mul-1 35.8%

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

      5. remove-double-neg 35.8%

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

      6. mul-1-neg 35.8%

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

      7. distribute-neg-in 35.8%

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

      8. +-commutative 35.8%

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

      9. cos-neg 35.8%

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

      10. +-commutative 35.8%

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

      11. mul-1-neg 35.8%

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

      12. unsub-neg 35.8%

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

   4. Simplified 35.8%

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

   if -1.2e11 < lambda1

   1. Initial program 78.3%

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

   2. Taylor expanded in lambda1 around 0 66.6%

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

      1. associate-*r* 66.6%

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

      2. *-commutative 66.6%

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

      3. associate-*l* 66.6%

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

      4. cos-neg 66.6%

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

   4. Simplified 66.6%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -120000000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternative 13: 63.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -2.8 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= lambda1 -2.8e-7)
     (* R (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (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 (lambda1 <= -2.8e-7) {
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * 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 (lambda1 <= (-2.8d-7)) then
        tmp = r * acos((t_0 + ((cos(phi1) * cos(phi2)) * 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 (lambda1 <= -2.8e-7) {
		tmp = R * Math.acos((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * 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 lambda1 <= -2.8e-7:
		tmp = R * math.acos((t_0 + ((math.cos(phi1) * math.cos(phi2)) * 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 (lambda1 <= -2.8e-7)
		tmp = Float64(R * acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda1)))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(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 (lambda1 <= -2.8e-7)
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * 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[lambda1, -2.8e-7], N[(R * N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.80000000000000019e-7

   1. Initial program 57.0%

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

   2. Taylor expanded in lambda2 around 0 57.4%

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

   if -2.80000000000000019e-7 < lambda1

   1. Initial program 78.6%

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

   2. Taylor expanded in lambda1 around 0 66.8%

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

      1. associate-*r* 66.8%

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

      2. *-commutative 66.8%

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

      3. associate-*l* 66.8%

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

      4. cos-neg 66.8%

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

   4. Simplified 66.8%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.8 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternative 14: 74.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := 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]

Derivation

1. Initial program 74.3%

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

2. Final simplification 74.3%

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

Alternative 15: 57.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -0.2:\\ \;\;\;\;R \cdot \cos^{-1} \left(\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2} + \cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\phi_1 \leq 0.059:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \left(\cos \phi_2 \cdot t_1\right) \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (cos (- lambda2 lambda1))))
   (if (<= phi1 -0.2)
     (*
      R
      (acos
       (+
        (/ (- (cos (- phi1 phi2)) (cos (+ phi1 phi2))) 2.0)
        (* (cos phi1) t_1))))
     (if (<= phi1 0.059)
       (*
        R
        (acos (+ t_0 (* (* (cos phi2) t_1) (+ 1.0 (* (* phi1 phi1) -0.5))))))
       (* R (acos (+ t_0 (* (cos phi1) (cos phi2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double t_1 = cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -0.2) {
		tmp = R * acos((((cos((phi1 - phi2)) - cos((phi1 + phi2))) / 2.0) + (cos(phi1) * t_1)));
	} else if (phi1 <= 0.059) {
		tmp = R * acos((t_0 + ((cos(phi2) * t_1) * (1.0 + ((phi1 * phi1) * -0.5)))));
	} else {
		tmp = R * acos((t_0 + (cos(phi1) * cos(phi2))));
	}
	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((lambda2 - lambda1))
    if (phi1 <= (-0.2d0)) then
        tmp = r * acos((((cos((phi1 - phi2)) - cos((phi1 + phi2))) / 2.0d0) + (cos(phi1) * t_1)))
    else if (phi1 <= 0.059d0) then
        tmp = r * acos((t_0 + ((cos(phi2) * t_1) * (1.0d0 + ((phi1 * phi1) * (-0.5d0))))))
    else
        tmp = r * acos((t_0 + (cos(phi1) * cos(phi2))))
    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((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -0.2) {
		tmp = R * Math.acos((((Math.cos((phi1 - phi2)) - Math.cos((phi1 + phi2))) / 2.0) + (Math.cos(phi1) * t_1)));
	} else if (phi1 <= 0.059) {
		tmp = R * Math.acos((t_0 + ((Math.cos(phi2) * t_1) * (1.0 + ((phi1 * phi1) * -0.5)))));
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * Math.cos(phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	t_1 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi1 <= -0.2:
		tmp = R * math.acos((((math.cos((phi1 - phi2)) - math.cos((phi1 + phi2))) / 2.0) + (math.cos(phi1) * t_1)))
	elif phi1 <= 0.059:
		tmp = R * math.acos((t_0 + ((math.cos(phi2) * t_1) * (1.0 + ((phi1 * phi1) * -0.5)))))
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * math.cos(phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	t_1 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -0.2)
		tmp = Float64(R * acos(Float64(Float64(Float64(cos(Float64(phi1 - phi2)) - cos(Float64(phi1 + phi2))) / 2.0) + Float64(cos(phi1) * t_1))));
	elseif (phi1 <= 0.059)
		tmp = Float64(R * acos(Float64(t_0 + Float64(Float64(cos(phi2) * t_1) * Float64(1.0 + Float64(Float64(phi1 * phi1) * -0.5))))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * cos(phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	t_1 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi1 <= -0.2)
		tmp = R * acos((((cos((phi1 - phi2)) - cos((phi1 + phi2))) / 2.0) + (cos(phi1) * t_1)));
	elseif (phi1 <= 0.059)
		tmp = R * acos((t_0 + ((cos(phi2) * t_1) * (1.0 + ((phi1 * phi1) * -0.5)))));
	else
		tmp = R * acos((t_0 + (cos(phi1) * cos(phi2))));
	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[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -0.2], N[(R * N[ArcCos[N[(N[(N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] - N[Cos[N[(phi1 + phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.059], N[(R * N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 + N[(N[(phi1 * phi1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.20000000000000001

   1. Initial program 80.4%

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

   2. Taylor expanded in phi2 around 0 59.2%

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

      1. sub-neg 59.2%

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

      2. +-commutative 59.2%

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

      3. neg-mul-1 59.2%

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

      4. neg-mul-1 59.2%

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

      5. remove-double-neg 59.2%

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

      6. mul-1-neg 59.2%

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

      7. distribute-neg-in 59.2%

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

      8. +-commutative 59.2%

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

      9. cos-neg 59.2%

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

      10. +-commutative 59.2%

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

      11. mul-1-neg 59.2%

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

      12. unsub-neg 59.2%

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

   4. Simplified 59.2%

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

      1. sin-mult 60.5%

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

   6. Applied egg-rr 60.5%

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

   if -0.20000000000000001 < phi1 < 0.058999999999999997

   1. Initial program 66.5%

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

   2. Taylor expanded in phi1 around 0 66.2%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2 + -0.5 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 \cdot {\phi_1}^{2}\right)\right)\right)}\right) \cdot R \]
   3. Step-by-step derivation

      1. *-rgt-identity 66.2%

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

      2. *-commutative 66.2%

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

      3. associate-*r* 66.2%

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

      4. associate-*l* 66.2%

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

      5. distribute-lft-out 66.2%

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

   4. Simplified 66.2%

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

   if 0.058999999999999997 < phi1

   1. Initial program 83.9%

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

      1. log1p-expm1-u 83.8%

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

      2. *-commutative 83.8%

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

   3. Applied egg-rr 83.8%

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

   4. Taylor expanded in lambda1 around 0 68.7%

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

   5. Taylor expanded in lambda2 around 0 46.7%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.2:\\ \;\;\;\;R \cdot \cos^{-1} \left(\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2} + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.059:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 16: 56.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -1.46 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2} + \cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\phi_1 \leq 26000:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (cos (- lambda2 lambda1))))
   (if (<= phi1 -1.46e-5)
     (*
      R
      (acos
       (+
        (/ (- (cos (- phi1 phi2)) (cos (+ phi1 phi2))) 2.0)
        (* (cos phi1) t_1))))
     (if (<= phi1 26000.0)
       (* R (acos (+ t_0 (* (cos phi2) t_1))))
       (* R (acos (+ t_0 (* (cos phi1) (cos phi2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double t_1 = cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -1.46e-5) {
		tmp = R * acos((((cos((phi1 - phi2)) - cos((phi1 + phi2))) / 2.0) + (cos(phi1) * t_1)));
	} else if (phi1 <= 26000.0) {
		tmp = R * acos((t_0 + (cos(phi2) * t_1)));
	} else {
		tmp = R * acos((t_0 + (cos(phi1) * cos(phi2))));
	}
	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((lambda2 - lambda1))
    if (phi1 <= (-1.46d-5)) then
        tmp = r * acos((((cos((phi1 - phi2)) - cos((phi1 + phi2))) / 2.0d0) + (cos(phi1) * t_1)))
    else if (phi1 <= 26000.0d0) then
        tmp = r * acos((t_0 + (cos(phi2) * t_1)))
    else
        tmp = r * acos((t_0 + (cos(phi1) * cos(phi2))))
    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((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -1.46e-5) {
		tmp = R * Math.acos((((Math.cos((phi1 - phi2)) - Math.cos((phi1 + phi2))) / 2.0) + (Math.cos(phi1) * t_1)));
	} else if (phi1 <= 26000.0) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi2) * t_1)));
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * Math.cos(phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	t_1 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi1 <= -1.46e-5:
		tmp = R * math.acos((((math.cos((phi1 - phi2)) - math.cos((phi1 + phi2))) / 2.0) + (math.cos(phi1) * t_1)))
	elif phi1 <= 26000.0:
		tmp = R * math.acos((t_0 + (math.cos(phi2) * t_1)))
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * math.cos(phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	t_1 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -1.46e-5)
		tmp = Float64(R * acos(Float64(Float64(Float64(cos(Float64(phi1 - phi2)) - cos(Float64(phi1 + phi2))) / 2.0) + Float64(cos(phi1) * t_1))));
	elseif (phi1 <= 26000.0)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi2) * t_1))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * cos(phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	t_1 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi1 <= -1.46e-5)
		tmp = R * acos((((cos((phi1 - phi2)) - cos((phi1 + phi2))) / 2.0) + (cos(phi1) * t_1)));
	elseif (phi1 <= 26000.0)
		tmp = R * acos((t_0 + (cos(phi2) * t_1)));
	else
		tmp = R * acos((t_0 + (cos(phi1) * cos(phi2))));
	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[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -1.46e-5], N[(R * N[ArcCos[N[(N[(N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] - N[Cos[N[(phi1 + phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 26000.0], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.46000000000000008e-5

   1. Initial program 80.5%

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

   2. Taylor expanded in phi2 around 0 58.8%

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

      1. sub-neg 58.8%

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

      2. +-commutative 58.8%

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

      3. neg-mul-1 58.8%

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

      4. neg-mul-1 58.8%

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

      5. remove-double-neg 58.8%

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

      6. mul-1-neg 58.8%

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

      7. distribute-neg-in 58.8%

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

      8. +-commutative 58.8%

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

      9. cos-neg 58.8%

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

      10. +-commutative 58.8%

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

      11. mul-1-neg 58.8%

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

      12. unsub-neg 58.8%

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

   4. Simplified 58.8%

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

      1. sin-mult 59.9%

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

   6. Applied egg-rr 59.9%

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

   if -1.46000000000000008e-5 < phi1 < 26000

   1. Initial program 66.4%

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

   2. Taylor expanded in phi1 around 0 64.6%

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

      1. sub-neg 64.6%

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

      2. +-commutative 64.6%

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

      3. neg-mul-1 64.6%

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

      4. neg-mul-1 64.6%

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

      5. remove-double-neg 64.6%

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

      6. mul-1-neg 64.6%

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

      7. distribute-neg-in 64.6%

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

      8. +-commutative 64.6%

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

      9. cos-neg 64.6%

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

      10. +-commutative 64.6%

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

      11. mul-1-neg 64.6%

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

      12. unsub-neg 64.6%

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

   4. Simplified 64.6%

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

   if 26000 < phi1

   1. Initial program 84.4%

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

      1. log1p-expm1-u 84.3%

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

      2. *-commutative 84.3%

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

   3. Applied egg-rr 84.3%

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

   4. Taylor expanded in lambda1 around 0 69.6%

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

   5. Taylor expanded in lambda2 around 0 46.6%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.46 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2} + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 26000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 17: 56.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -25000:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq 26000:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi1 -25000.0)
     (* R (acos (+ t_1 (* (cos phi1) t_0))))
     (if (<= phi1 26000.0)
       (* R (acos (+ t_1 (* (cos phi2) t_0))))
       (* R (acos (+ t_1 (* (cos phi1) (cos phi2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi1 <= -25000.0) {
		tmp = R * acos((t_1 + (cos(phi1) * t_0)));
	} else if (phi1 <= 26000.0) {
		tmp = R * acos((t_1 + (cos(phi2) * t_0)));
	} else {
		tmp = R * acos((t_1 + (cos(phi1) * cos(phi2))));
	}
	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 = cos((lambda2 - lambda1))
    t_1 = sin(phi1) * sin(phi2)
    if (phi1 <= (-25000.0d0)) then
        tmp = r * acos((t_1 + (cos(phi1) * t_0)))
    else if (phi1 <= 26000.0d0) then
        tmp = r * acos((t_1 + (cos(phi2) * t_0)))
    else
        tmp = r * acos((t_1 + (cos(phi1) * cos(phi2))))
    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 t_1 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (phi1 <= -25000.0) {
		tmp = R * Math.acos((t_1 + (Math.cos(phi1) * t_0)));
	} else if (phi1 <= 26000.0) {
		tmp = R * Math.acos((t_1 + (Math.cos(phi2) * t_0)));
	} else {
		tmp = R * Math.acos((t_1 + (Math.cos(phi1) * Math.cos(phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	t_1 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if phi1 <= -25000.0:
		tmp = R * math.acos((t_1 + (math.cos(phi1) * t_0)))
	elif phi1 <= 26000.0:
		tmp = R * math.acos((t_1 + (math.cos(phi2) * t_0)))
	else:
		tmp = R * math.acos((t_1 + (math.cos(phi1) * math.cos(phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi1 <= -25000.0)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi1) * t_0))));
	elseif (phi1 <= 26000.0)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi2) * t_0))));
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi1) * cos(phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	t_1 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (phi1 <= -25000.0)
		tmp = R * acos((t_1 + (cos(phi1) * t_0)));
	elseif (phi1 <= 26000.0)
		tmp = R * acos((t_1 + (cos(phi2) * t_0)));
	else
		tmp = R * acos((t_1 + (cos(phi1) * cos(phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -25000.0], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 26000.0], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -25000

   1. Initial program 81.0%

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

   2. Taylor expanded in phi2 around 0 60.9%

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

      1. sub-neg 60.9%

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

      2. +-commutative 60.9%

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

      3. neg-mul-1 60.9%

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

      4. neg-mul-1 60.9%

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

      5. remove-double-neg 60.9%

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

      6. mul-1-neg 60.9%

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

      7. distribute-neg-in 60.9%

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

      8. +-commutative 60.9%

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

      9. cos-neg 60.9%

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

      10. +-commutative 60.9%

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

      11. mul-1-neg 60.9%

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

      12. unsub-neg 60.9%

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

   4. Simplified 60.9%

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

   if -25000 < phi1 < 26000

   1. Initial program 66.6%

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

   2. Taylor expanded in phi1 around 0 63.4%

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

      1. sub-neg 63.4%

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

      2. +-commutative 63.4%

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

      3. neg-mul-1 63.4%

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

      4. neg-mul-1 63.4%

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

      5. remove-double-neg 63.4%

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

      6. mul-1-neg 63.4%

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

      7. distribute-neg-in 63.4%

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

      8. +-commutative 63.4%

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

      9. cos-neg 63.4%

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

      10. +-commutative 63.4%

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

      11. mul-1-neg 63.4%

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

      12. unsub-neg 63.4%

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

   4. Simplified 63.4%

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

   if 26000 < phi1

   1. Initial program 84.4%

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

      1. log1p-expm1-u 84.3%

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

      2. *-commutative 84.3%

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

   3. Applied egg-rr 84.3%

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

   4. Taylor expanded in lambda1 around 0 69.6%

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

   5. Taylor expanded in lambda2 around 0 46.6%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -25000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 26000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 18: 48.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq 0.000115:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= phi2 0.000115)
     (* R (acos (+ t_0 (* (cos phi1) (cos (- lambda2 lambda1))))))
     (* R (acos (+ t_0 (* (cos phi1) (cos phi2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= 0.000115) {
		tmp = R * acos((t_0 + (cos(phi1) * cos((lambda2 - lambda1)))));
	} else {
		tmp = R * acos((t_0 + (cos(phi1) * cos(phi2))));
	}
	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 (phi2 <= 0.000115d0) then
        tmp = r * acos((t_0 + (cos(phi1) * cos((lambda2 - lambda1)))))
    else
        tmp = r * acos((t_0 + (cos(phi1) * cos(phi2))))
    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 (phi2 <= 0.000115) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * Math.cos((lambda2 - lambda1)))));
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * Math.cos(phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if phi2 <= 0.000115:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * math.cos((lambda2 - lambda1)))))
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * math.cos(phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= 0.000115)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * cos(Float64(lambda2 - lambda1))))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * cos(phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (phi2 <= 0.000115)
		tmp = R * acos((t_0 + (cos(phi1) * cos((lambda2 - lambda1)))));
	else
		tmp = R * acos((t_0 + (cos(phi1) * cos(phi2))));
	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[phi2, 0.000115], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.15e-4

   1. Initial program 72.1%

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

   2. Taylor expanded in phi2 around 0 59.3%

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

      1. sub-neg 59.3%

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

      2. +-commutative 59.3%

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

      3. neg-mul-1 59.3%

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

      4. neg-mul-1 59.3%

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

      5. remove-double-neg 59.3%

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

      6. mul-1-neg 59.3%

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

      7. distribute-neg-in 59.3%

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

      8. +-commutative 59.3%

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

      9. cos-neg 59.3%

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

      10. +-commutative 59.3%

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

      11. mul-1-neg 59.3%

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

      12. unsub-neg 59.3%

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

   4. Simplified 59.3%

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

   if 1.15e-4 < phi2

   1. Initial program 80.9%

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

      1. log1p-expm1-u 80.8%

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

      2. *-commutative 80.8%

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

   3. Applied egg-rr 80.8%

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

   4. Taylor expanded in lambda1 around 0 60.0%

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

   5. Taylor expanded in lambda2 around 0 42.9%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.000115:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 19: 38.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.5:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 7.5)
   (*
    R
    (acos (+ (* (cos phi1) (cos (- lambda2 lambda1))) (* (sin phi1) phi2))))
   (* R (acos (+ (* (sin phi1) (sin phi2)) (* (cos phi1) (cos lambda1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 7.5) {
		tmp = R * acos(((cos(phi1) * cos((lambda2 - lambda1))) + (sin(phi1) * phi2)));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(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 <= 7.5d0) then
        tmp = r * acos(((cos(phi1) * cos((lambda2 - lambda1))) + (sin(phi1) * phi2)))
    else
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(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 <= 7.5) {
		tmp = R * Math.acos(((Math.cos(phi1) * Math.cos((lambda2 - lambda1))) + (Math.sin(phi1) * phi2)));
	} else {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi1) * Math.cos(lambda1))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 7.5:
		tmp = R * math.acos(((math.cos(phi1) * math.cos((lambda2 - lambda1))) + (math.sin(phi1) * phi2)))
	else:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi1) * math.cos(lambda1))))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 7.5)
		tmp = R * acos(((cos(phi1) * cos((lambda2 - lambda1))) + (sin(phi1) * phi2)));
	else
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if phi2 < 7.5

   1. Initial program 72.2%

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

   2. Taylor expanded in phi2 around 0 58.9%

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

      1. sub-neg 58.9%

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

      2. +-commutative 58.9%

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

      3. neg-mul-1 58.9%

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

      4. neg-mul-1 58.9%

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

      5. remove-double-neg 58.9%

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

      6. mul-1-neg 58.9%

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

      7. distribute-neg-in 58.9%

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

      8. +-commutative 58.9%

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

      9. cos-neg 58.9%

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

      10. +-commutative 58.9%

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

      11. mul-1-neg 58.9%

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

      12. unsub-neg 58.9%

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

   4. Simplified 58.9%

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

   5. Taylor expanded in phi2 around 0 56.0%

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

   if 7.5 < phi2

   1. Initial program 80.8%

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

   2. Taylor expanded in phi2 around 0 16.9%

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

      1. sub-neg 16.9%

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

      2. +-commutative 16.9%

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

      3. neg-mul-1 16.9%

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

      4. neg-mul-1 16.9%

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

      5. remove-double-neg 16.9%

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

      6. mul-1-neg 16.9%

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

      7. distribute-neg-in 16.9%

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

      8. +-commutative 16.9%

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

      9. cos-neg 16.9%

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

      10. +-commutative 16.9%

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

      11. mul-1-neg 16.9%

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

      12. unsub-neg 16.9%

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

   4. Simplified 16.9%

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

   5. Taylor expanded in lambda2 around 0 14.9%

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

      1. cos-neg 4.9%

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

   7. Simplified 14.9%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.5:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \end{array} \]

Alternative 20: 35.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 1.9 \cdot 10^{-37}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= lambda2 1.9e-37)
     (* R (acos (+ t_0 (* (cos phi1) (cos lambda1)))))
     (* R (acos (+ t_0 (* (cos phi1) (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 <= 1.9e-37) {
		tmp = R * acos((t_0 + (cos(phi1) * cos(lambda1))));
	} else {
		tmp = R * acos((t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    if (lambda2 <= 1.9d-37) then
        tmp = r * acos((t_0 + (cos(phi1) * cos(lambda1))))
    else
        tmp = r * acos((t_0 + (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 t_0 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda2 <= 1.9e-37) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * Math.cos(lambda1))));
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * 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 <= 1.9e-37:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * math.cos(lambda1))))
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * 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 <= 1.9e-37)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * cos(lambda1)))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * 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 <= 1.9e-37)
		tmp = R * acos((t_0 + (cos(phi1) * cos(lambda1))));
	else
		tmp = R * acos((t_0 + (cos(phi1) * 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, 1.9e-37], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.9000000000000002e-37

   1. Initial program 79.0%

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

   2. Taylor expanded in phi2 around 0 48.7%

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

      1. sub-neg 48.7%

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

      2. +-commutative 48.7%

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

      3. neg-mul-1 48.7%

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

      4. neg-mul-1 48.7%

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

      5. remove-double-neg 48.7%

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

      6. mul-1-neg 48.7%

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

      7. distribute-neg-in 48.7%

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

      8. +-commutative 48.7%

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

      9. cos-neg 48.7%

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

      10. +-commutative 48.7%

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

      11. mul-1-neg 48.7%

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

      12. unsub-neg 48.7%

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

   4. Simplified 48.7%

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

   5. Taylor expanded in lambda2 around 0 37.9%

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

      1. cos-neg 19.8%

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

   7. Simplified 37.9%

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

   if 1.9000000000000002e-37 < lambda2

   1. Initial program 63.7%

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

   2. Taylor expanded in phi2 around 0 48.8%

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

      1. sub-neg 48.8%

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

      2. +-commutative 48.8%

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

      3. neg-mul-1 48.8%

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

      4. neg-mul-1 48.8%

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

      5. remove-double-neg 48.8%

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

      6. mul-1-neg 48.8%

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

      7. distribute-neg-in 48.8%

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

      8. +-commutative 48.8%

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

      9. cos-neg 48.8%

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

      10. +-commutative 48.8%

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

      11. mul-1-neg 48.8%

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

      12. unsub-neg 48.8%

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

   4. Simplified 48.8%

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

   5. Taylor expanded in lambda1 around 0 47.4%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.9 \cdot 10^{-37}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 21: 44.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 4.6e-9)
   (*
    R
    (acos (+ (* (cos phi1) (cos (- lambda2 lambda1))) (* (sin phi1) phi2))))
   (* R (acos (+ (* (sin phi1) (sin phi2)) (* (cos phi2) (cos lambda2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4.6e-9) {
		tmp = R * acos(((cos(phi1) * cos((lambda2 - lambda1))) + (sin(phi1) * phi2)));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (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) :: tmp
    if (phi2 <= 4.6d-9) then
        tmp = r * acos(((cos(phi1) * cos((lambda2 - lambda1))) + (sin(phi1) * phi2)))
    else
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (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 tmp;
	if (phi2 <= 4.6e-9) {
		tmp = R * Math.acos(((Math.cos(phi1) * Math.cos((lambda2 - lambda1))) + (Math.sin(phi1) * phi2)));
	} else {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi2) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 4.6e-9:
		tmp = R * math.acos(((math.cos(phi1) * math.cos((lambda2 - lambda1))) + (math.sin(phi1) * phi2)))
	else:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi2) * math.cos(lambda2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 4.6e-9)
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1))) + Float64(sin(phi1) * phi2))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi2) * cos(lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 4.6e-9)
		tmp = R * acos(((cos(phi1) * cos((lambda2 - lambda1))) + (sin(phi1) * phi2)));
	else
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi2) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 4.6e-9], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.5999999999999998e-9

   1. Initial program 72.1%

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

   2. Taylor expanded in phi2 around 0 59.3%

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

      1. sub-neg 59.3%

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

      2. +-commutative 59.3%

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

      3. neg-mul-1 59.3%

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

      4. neg-mul-1 59.3%

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

      5. remove-double-neg 59.3%

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

      6. mul-1-neg 59.3%

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

      7. distribute-neg-in 59.3%

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

      8. +-commutative 59.3%

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

      9. cos-neg 59.3%

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

      10. +-commutative 59.3%

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

      11. mul-1-neg 59.3%

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

      12. unsub-neg 59.3%

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

   4. Simplified 59.3%

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

   5. Taylor expanded in phi2 around 0 56.5%

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

   if 4.5999999999999998e-9 < phi2

   1. Initial program 80.9%

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

      1. log1p-expm1-u 80.8%

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

      2. *-commutative 80.8%

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

   3. Applied egg-rr 80.8%

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

   4. Taylor expanded in lambda1 around 0 60.0%

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

   5. Taylor expanded in phi1 around 0 33.0%

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

      1. cos-neg 33.0%

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

   7. Simplified 33.0%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 22: 45.1% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 7.5e-5)
   (*
    R
    (acos (+ (* (cos phi1) (cos (- lambda2 lambda1))) (* (sin phi1) phi2))))
   (* R (acos (+ (* (sin phi1) (sin phi2)) (* (cos phi1) (cos phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 7.5e-5) {
		tmp = R * acos(((cos(phi1) * cos((lambda2 - lambda1))) + (sin(phi1) * phi2)));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))));
	}
	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 <= 7.5d-5) then
        tmp = r * acos(((cos(phi1) * cos((lambda2 - lambda1))) + (sin(phi1) * phi2)))
    else
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))))
    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 <= 7.5e-5) {
		tmp = R * Math.acos(((Math.cos(phi1) * Math.cos((lambda2 - lambda1))) + (Math.sin(phi1) * phi2)));
	} else {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi1) * Math.cos(phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 7.5e-5:
		tmp = R * math.acos(((math.cos(phi1) * math.cos((lambda2 - lambda1))) + (math.sin(phi1) * phi2)))
	else:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi1) * math.cos(phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 7.5e-5)
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1))) + Float64(sin(phi1) * phi2))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * cos(phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 7.5e-5)
		tmp = R * acos(((cos(phi1) * cos((lambda2 - lambda1))) + (sin(phi1) * phi2)));
	else
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 7.5e-5], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 7.49999999999999934e-5

   1. Initial program 72.1%

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

   2. Taylor expanded in phi2 around 0 59.3%

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

      1. sub-neg 59.3%

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

      2. +-commutative 59.3%

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

      3. neg-mul-1 59.3%

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

      4. neg-mul-1 59.3%

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

      5. remove-double-neg 59.3%

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

      6. mul-1-neg 59.3%

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

      7. distribute-neg-in 59.3%

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

      8. +-commutative 59.3%

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

      9. cos-neg 59.3%

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

      10. +-commutative 59.3%

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

      11. mul-1-neg 59.3%

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

      12. unsub-neg 59.3%

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

   4. Simplified 59.3%

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

   5. Taylor expanded in phi2 around 0 56.5%

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

   if 7.49999999999999934e-5 < phi2

   1. Initial program 80.9%

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

      1. log1p-expm1-u 80.8%

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

      2. *-commutative 80.8%

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

   3. Applied egg-rr 80.8%

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

   4. Taylor expanded in lambda1 around 0 60.0%

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

   5. Taylor expanded in lambda2 around 0 42.9%

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

4. Final simplification 53.1%

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

Alternative 23: 32.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -3 \cdot 10^{+17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos (- lambda2 lambda1)))))
   (if (<= phi1 -3e+17)
     (* R (acos (+ t_0 (* (sin phi1) phi2))))
     (* R (acos (+ (* phi1 (sin phi2)) t_0))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -3e+17) {
		tmp = R * acos((t_0 + (sin(phi1) * phi2)));
	} else {
		tmp = R * acos(((phi1 * sin(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(phi1) * cos((lambda2 - lambda1))
    if (phi1 <= (-3d+17)) then
        tmp = r * acos((t_0 + (sin(phi1) * phi2)))
    else
        tmp = r * acos(((phi1 * sin(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(phi1) * Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -3e+17) {
		tmp = R * Math.acos((t_0 + (Math.sin(phi1) * phi2)));
	} else {
		tmp = R * Math.acos(((phi1 * Math.sin(phi2)) + t_0));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.cos((lambda2 - lambda1))
	tmp = 0
	if phi1 <= -3e+17:
		tmp = R * math.acos((t_0 + (math.sin(phi1) * phi2)))
	else:
		tmp = R * math.acos(((phi1 * math.sin(phi2)) + t_0))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))
	tmp = 0.0
	if (phi1 <= -3e+17)
		tmp = Float64(R * acos(Float64(t_0 + Float64(sin(phi1) * phi2))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi1 <= -3e+17)
		tmp = R * acos((t_0 + (sin(phi1) * phi2)));
	else
		tmp = R * acos(((phi1 * sin(phi2)) + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -3e+17], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3e17

   1. Initial program 81.8%

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

   2. Taylor expanded in phi2 around 0 60.9%

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

      1. sub-neg 60.9%

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

      2. +-commutative 60.9%

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

      3. neg-mul-1 60.9%

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

      4. neg-mul-1 60.9%

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

      5. remove-double-neg 60.9%

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

      6. mul-1-neg 60.9%

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

      7. distribute-neg-in 60.9%

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

      8. +-commutative 60.9%

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

      9. cos-neg 60.9%

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

      10. +-commutative 60.9%

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

      11. mul-1-neg 60.9%

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

      12. unsub-neg 60.9%

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

   4. Simplified 60.9%

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

   5. Taylor expanded in phi2 around 0 55.6%

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

   if -3e17 < phi1

   1. Initial program 72.1%

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

   2. Taylor expanded in phi2 around 0 45.1%

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

      1. sub-neg 45.1%

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

      2. +-commutative 45.1%

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

      3. neg-mul-1 45.1%

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

      4. neg-mul-1 45.1%

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

      5. remove-double-neg 45.1%

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

      6. mul-1-neg 45.1%

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

      7. distribute-neg-in 45.1%

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

      8. +-commutative 45.1%

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

      9. cos-neg 45.1%

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

      10. +-commutative 45.1%

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

      11. mul-1-neg 45.1%

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

      12. unsub-neg 45.1%

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

   4. Simplified 45.1%

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

   5. Taylor expanded in phi1 around 0 35.1%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3 \cdot 10^{+17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]

Alternative 24: 23.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -12:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* phi1 (sin phi2))))
   (if (<= lambda1 -12.0)
     (* R (acos (+ t_0 (* (cos phi1) (cos lambda1)))))
     (* R (acos (+ t_0 (* (cos phi1) (cos lambda2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = phi1 * sin(phi2);
	double tmp;
	if (lambda1 <= -12.0) {
		tmp = R * acos((t_0 + (cos(phi1) * cos(lambda1))));
	} else {
		tmp = R * acos((t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = phi1 * sin(phi2)
    if (lambda1 <= (-12.0d0)) then
        tmp = r * acos((t_0 + (cos(phi1) * cos(lambda1))))
    else
        tmp = r * acos((t_0 + (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 t_0 = phi1 * Math.sin(phi2);
	double tmp;
	if (lambda1 <= -12.0) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * Math.cos(lambda1))));
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = phi1 * math.sin(phi2)
	tmp = 0
	if lambda1 <= -12.0:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * math.cos(lambda1))))
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * math.cos(lambda2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(phi1 * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -12.0)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * cos(lambda1)))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * cos(lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = phi1 * sin(phi2);
	tmp = 0.0;
	if (lambda1 <= -12.0)
		tmp = R * acos((t_0 + (cos(phi1) * cos(lambda1))));
	else
		tmp = R * acos((t_0 + (cos(phi1) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -12.0], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -12

   1. Initial program 57.0%

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

   2. Taylor expanded in phi2 around 0 39.8%

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

      1. sub-neg 39.8%

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

      2. +-commutative 39.8%

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

      3. neg-mul-1 39.8%

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

      4. neg-mul-1 39.8%

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

      5. remove-double-neg 39.8%

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

      6. mul-1-neg 39.8%

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

      7. distribute-neg-in 39.8%

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

      8. +-commutative 39.8%

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

      9. cos-neg 39.8%

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

      10. +-commutative 39.8%

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

      11. mul-1-neg 39.8%

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

      12. unsub-neg 39.8%

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

   4. Simplified 39.8%

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

   5. Taylor expanded in phi1 around 0 24.1%

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

   6. Taylor expanded in lambda2 around 0 24.5%

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

      1. cos-neg 24.5%

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

   8. Simplified 24.5%

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

   if -12 < lambda1

   1. Initial program 78.6%

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

   2. Taylor expanded in phi2 around 0 50.9%

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

      1. sub-neg 50.9%

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

      2. +-commutative 50.9%

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

      3. neg-mul-1 50.9%

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

      4. neg-mul-1 50.9%

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

      5. remove-double-neg 50.9%

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

      6. mul-1-neg 50.9%

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

      7. distribute-neg-in 50.9%

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

      8. +-commutative 50.9%

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

      9. cos-neg 50.9%

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

      10. +-commutative 50.9%

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

      11. mul-1-neg 50.9%

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

      12. unsub-neg 50.9%

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

   4. Simplified 50.9%

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

   5. Taylor expanded in phi1 around 0 31.5%

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

   6. Taylor expanded in lambda1 around 0 26.4%

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

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -12:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 25: 27.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

2. Taylor expanded in phi2 around 0 48.7%

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

   1. sub-neg 48.7%

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

   2. +-commutative 48.7%

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

   3. neg-mul-1 48.7%

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

   4. neg-mul-1 48.7%

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

   5. remove-double-neg 48.7%

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

   6. mul-1-neg 48.7%

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

   7. distribute-neg-in 48.7%

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

   8. +-commutative 48.7%

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

   9. cos-neg 48.7%

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

   10. +-commutative 48.7%

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

   11. mul-1-neg 48.7%

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

   12. unsub-neg 48.7%

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

4. Simplified 48.7%

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

5. Taylor expanded in phi1 around 0 30.1%

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

6. Final simplification 30.1%

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

Alternative 26: 19.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.6 \cdot 10^{-90}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 3.6e-90)
   (* R (acos (+ (* phi1 phi2) (* (cos phi1) (cos lambda1)))))
   (* R (acos (+ (* phi1 phi2) (cos (- lambda2 lambda1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 3.6e-90) {
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))));
	} else {
		tmp = R * acos(((phi1 * 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 (lambda2 <= 3.6d-90) then
        tmp = r * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))))
    else
        tmp = r * acos(((phi1 * 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 (lambda2 <= 3.6e-90) {
		tmp = R * Math.acos(((phi1 * phi2) + (Math.cos(phi1) * Math.cos(lambda1))));
	} else {
		tmp = R * Math.acos(((phi1 * phi2) + Math.cos((lambda2 - lambda1))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 3.6e-90:
		tmp = R * math.acos(((phi1 * phi2) + (math.cos(phi1) * math.cos(lambda1))))
	else:
		tmp = R * math.acos(((phi1 * phi2) + math.cos((lambda2 - lambda1))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 3.6e-90)
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(phi1) * cos(lambda1)))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 3.6e-90)
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))));
	else
		tmp = R * acos(((phi1 * phi2) + cos((lambda2 - lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 3.6e-90], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 3.59999999999999981e-90

   1. Initial program 78.4%

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

   2. Taylor expanded in phi2 around 0 49.9%

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

      1. sub-neg 49.9%

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

      2. +-commutative 49.9%

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

      3. neg-mul-1 49.9%

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

      4. neg-mul-1 49.9%

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

      5. remove-double-neg 49.9%

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

      6. mul-1-neg 49.9%

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

      7. distribute-neg-in 49.9%

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

      8. +-commutative 49.9%

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

      9. cos-neg 49.9%

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

      10. +-commutative 49.9%

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

      11. mul-1-neg 49.9%

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

      12. unsub-neg 49.9%

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

   4. Simplified 49.9%

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

   5. Taylor expanded in phi1 around 0 27.8%

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

   6. Taylor expanded in phi2 around 0 26.1%

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

      1. *-commutative 26.1%

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

   8. Simplified 26.1%

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

   9. Taylor expanded in lambda2 around 0 18.1%

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

       1. cos-neg 18.1%

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

       2. *-commutative 18.1%

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

   11. Simplified 18.1%

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

   if 3.59999999999999981e-90 < lambda2

   1. Initial program 66.5%

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

   2. Taylor expanded in phi2 around 0 46.5%

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

      1. sub-neg 46.5%

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

      2. +-commutative 46.5%

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

      3. neg-mul-1 46.5%

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

      4. neg-mul-1 46.5%

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

      5. remove-double-neg 46.5%

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

      6. mul-1-neg 46.5%

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

      7. distribute-neg-in 46.5%

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

      8. +-commutative 46.5%

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

      9. cos-neg 46.5%

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

      10. +-commutative 46.5%

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

      11. mul-1-neg 46.5%

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

      12. unsub-neg 46.5%

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

   4. Simplified 46.5%

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

   5. Taylor expanded in phi1 around 0 34.3%

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

   6. Taylor expanded in phi2 around 0 32.2%

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

      1. *-commutative 32.2%

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

   8. Simplified 32.2%

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

   9. Taylor expanded in phi1 around 0 24.4%

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

4. Final simplification 20.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.6 \cdot 10^{-90}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]

Alternative 27: 21.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -12:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -12.0)
   (* R (acos (+ (* phi1 phi2) (* (cos phi1) (cos lambda1)))))
   (* R (acos (+ (* phi1 phi2) (* (cos phi1) (cos lambda2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -12.0) {
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))));
	} else {
		tmp = R * acos(((phi1 * phi2) + (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 <= (-12.0d0)) then
        tmp = r * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))))
    else
        tmp = r * acos(((phi1 * phi2) + (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 <= -12.0) {
		tmp = R * Math.acos(((phi1 * phi2) + (Math.cos(phi1) * Math.cos(lambda1))));
	} else {
		tmp = R * Math.acos(((phi1 * phi2) + (Math.cos(phi1) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -12.0:
		tmp = R * math.acos(((phi1 * phi2) + (math.cos(phi1) * math.cos(lambda1))))
	else:
		tmp = R * math.acos(((phi1 * phi2) + (math.cos(phi1) * math.cos(lambda2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -12.0)
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(phi1) * cos(lambda1)))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(phi1) * cos(lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -12.0)
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))));
	else
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -12.0], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -12

   1. Initial program 57.0%

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

   2. Taylor expanded in phi2 around 0 39.8%

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

      1. sub-neg 39.8%

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

      2. +-commutative 39.8%

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

      3. neg-mul-1 39.8%

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

      4. neg-mul-1 39.8%

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

      5. remove-double-neg 39.8%

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

      6. mul-1-neg 39.8%

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

      7. distribute-neg-in 39.8%

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

      8. +-commutative 39.8%

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

      9. cos-neg 39.8%

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

      10. +-commutative 39.8%

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

      11. mul-1-neg 39.8%

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

      12. unsub-neg 39.8%

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

   4. Simplified 39.8%

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

   5. Taylor expanded in phi1 around 0 24.1%

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

   6. Taylor expanded in phi2 around 0 20.2%

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

      1. *-commutative 20.2%

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

   8. Simplified 20.2%

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

   9. Taylor expanded in lambda2 around 0 20.5%

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

       1. cos-neg 20.5%

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

       2. *-commutative 20.5%

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

   11. Simplified 20.5%

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

   if -12 < lambda1

   1. Initial program 78.6%

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

   2. Taylor expanded in phi2 around 0 50.9%

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

      1. sub-neg 50.9%

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

      2. +-commutative 50.9%

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

      3. neg-mul-1 50.9%

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

      4. neg-mul-1 50.9%

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

      5. remove-double-neg 50.9%

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

      6. mul-1-neg 50.9%

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

      7. distribute-neg-in 50.9%

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

      8. +-commutative 50.9%

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

      9. cos-neg 50.9%

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

      10. +-commutative 50.9%

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

      11. mul-1-neg 50.9%

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

      12. unsub-neg 50.9%

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

   4. Simplified 50.9%

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

   5. Taylor expanded in phi1 around 0 31.5%

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

   6. Taylor expanded in phi2 around 0 30.2%

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

      1. *-commutative 30.2%

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

   8. Simplified 30.2%

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

   9. Taylor expanded in lambda1 around 0 25.3%

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

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -12:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 28: 25.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

2. Taylor expanded in phi2 around 0 48.7%

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

   1. sub-neg 48.7%

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

   2. +-commutative 48.7%

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

   3. neg-mul-1 48.7%

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

   4. neg-mul-1 48.7%

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

   5. remove-double-neg 48.7%

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

   6. mul-1-neg 48.7%

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

   7. distribute-neg-in 48.7%

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

   8. +-commutative 48.7%

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

   9. cos-neg 48.7%

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

   10. +-commutative 48.7%

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

   11. mul-1-neg 48.7%

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

   12. unsub-neg 48.7%

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

4. Simplified 48.7%

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

5. Taylor expanded in phi1 around 0 30.1%

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

6. Taylor expanded in phi2 around 0 28.2%

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

   1. *-commutative 28.2%

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

8. Simplified 28.2%

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

9. Final simplification 28.2%

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

Alternative 29: 17.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

2. Taylor expanded in phi2 around 0 48.7%

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

   1. sub-neg 48.7%

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

   2. +-commutative 48.7%

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

   3. neg-mul-1 48.7%

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

   4. neg-mul-1 48.7%

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

   5. remove-double-neg 48.7%

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

   6. mul-1-neg 48.7%

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

   7. distribute-neg-in 48.7%

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

   8. +-commutative 48.7%

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

   9. cos-neg 48.7%

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

   10. +-commutative 48.7%

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

   11. mul-1-neg 48.7%

       \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_2 + \color{blue}{\left(-\lambda_1\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

   12. unsub-neg 48.7%

       \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

4. Simplified 48.7%

   \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1}\right) \cdot R \]

5. Taylor expanded in phi1 around 0 30.1%

   \[\leadsto \cos^{-1} \left(\color{blue}{\phi_1 \cdot \sin \phi_2} + \cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

6. Taylor expanded in phi2 around 0 28.2%

   \[\leadsto \cos^{-1} \left(\color{blue}{\phi_1 \cdot \phi_2} + \cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]
7. Step-by-step derivation

   1. *-commutative 28.2%

      \[\leadsto \cos^{-1} \left(\color{blue}{\phi_2 \cdot \phi_1} + \cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

8. Simplified 28.2%

   \[\leadsto \cos^{-1} \left(\color{blue}{\phi_2 \cdot \phi_1} + \cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right) \cdot R \]

9. Taylor expanded in phi1 around 0 20.7%

   \[\leadsto \cos^{-1} \left(\phi_2 \cdot \phi_1 + \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

10. Final simplification 20.7%

    \[\leadsto R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \left(\lambda_2 - \lambda_1\right)\right) \]

Reproduce

herbie shell --seed 2023187 
(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))