Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.1% → 99.9%

Time: 23.7s

Alternatives: 15

Speedup: 3.0×

• 31.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t_0 \cdot t_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (* R (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	return R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

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
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	return R * Math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	return R * math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	return Float64(R * sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(R * N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t_0 \cdot t_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (* R (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	return R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

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
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	return R * Math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	return R * math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	return Float64(R * sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(R * N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right) + \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right) \cdot \left(\lambda_2 - \lambda_1\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (+
    (* (- lambda1 lambda2) (* (cos (* 0.5 phi1)) (cos (* phi2 0.5))))
    (* (* (sin (* 0.5 phi1)) (sin (* phi2 0.5))) (- lambda2 lambda1)))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot((((lambda1 - lambda2) * (cos((0.5 * phi1)) * cos((phi2 * 0.5)))) + ((sin((0.5 * phi1)) * sin((phi2 * 0.5))) * (lambda2 - lambda1))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot((((lambda1 - lambda2) * (Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5)))) + ((Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5))) * (lambda2 - lambda1))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot((((lambda1 - lambda2) * (math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5)))) + ((math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))) * (lambda2 - lambda1))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(Float64(lambda1 - lambda2) * Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5)))) + Float64(Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5))) * Float64(lambda2 - lambda1))), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot((((lambda1 - lambda2) * (cos((0.5 * phi1)) * cos((phi2 * 0.5)))) + ((sin((0.5 * phi1)) * sin((phi2 * 0.5))) * (lambda2 - lambda1))), (phi1 - phi2));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

   1. hypot-def 95.7%

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

3. Simplified 95.7%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Step-by-step derivation

   1. *-commutative 95.7%

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

   2. flip-- 72.6%

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

   3. associate-*r/ 72.6%

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

   4. associate-/l* 72.6%

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

   5. div-inv 72.6%

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

   6. metadata-eval 72.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}, \phi_1 - \phi_2\right) \]

   7. *-un-lft-identity 72.6%

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

   8. associate-/l* 72.6%

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

   9. flip-- 95.6%

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

5. Applied egg-rr 95.6%

   \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}{\frac{1}{\lambda_1 - \lambda_2}}}, \phi_1 - \phi_2\right) \]
6. Step-by-step derivation

   1. *-commutative 95.6%

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

   2. +-commutative 95.6%

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

   3. distribute-rgt-in 95.6%

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

   4. *-commutative 95.6%

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

   5. cos-sum 99.8%

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

   6. *-commutative 99.8%

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

   7. *-commutative 99.8%

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

7. Applied egg-rr 99.8%

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

   1. div-sub 99.8%

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

   2. div-inv 99.8%

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

   3. *-commutative 99.8%

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

   4. remove-double-div 99.9%

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

   5. div-inv 99.9%

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

   6. *-commutative 99.9%

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

   7. remove-double-div 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right) + \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right) \cdot \left(\lambda_2 - \lambda_1\right), \phi_1 - \phi_2\right) \]

Alternative 2: 88.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\\ t_1 := \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\\ \mathbf{if}\;\lambda_1 \leq -1.5 \cdot 10^{+186}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot t_1 - \lambda_1 \cdot t_0, \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_1 \leq -1 \cdot 10^{-73}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot t_0 - \lambda_2 \cdot t_1, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (* 0.5 phi1)) (sin (* phi2 0.5))))
        (t_1 (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))))
   (if (<= lambda1 -1.5e+186)
     (* R (hypot (- (* lambda1 t_1) (* lambda1 t_0)) (- phi1 phi2)))
     (if (<= lambda1 -1e-73)
       (*
        R
        (hypot
         (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0)))
         (- phi1 phi2)))
       (* R (hypot (- (* lambda2 t_0) (* lambda2 t_1)) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1)) * sin((phi2 * 0.5));
	double t_1 = cos((0.5 * phi1)) * cos((phi2 * 0.5));
	double tmp;
	if (lambda1 <= -1.5e+186) {
		tmp = R * hypot(((lambda1 * t_1) - (lambda1 * t_0)), (phi1 - phi2));
	} else if (lambda1 <= -1e-73) {
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda2 * t_0) - (lambda2 * t_1)), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5));
	double t_1 = Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5));
	double tmp;
	if (lambda1 <= -1.5e+186) {
		tmp = R * Math.hypot(((lambda1 * t_1) - (lambda1 * t_0)), (phi1 - phi2));
	} else if (lambda1 <= -1e-73) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda2 * t_0) - (lambda2 * t_1)), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))
	t_1 = math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))
	tmp = 0
	if lambda1 <= -1.5e+186:
		tmp = R * math.hypot(((lambda1 * t_1) - (lambda1 * t_0)), (phi1 - phi2))
	elif lambda1 <= -1e-73:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda2 * t_0) - (lambda2 * t_1)), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5)))
	t_1 = Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5)))
	tmp = 0.0
	if (lambda1 <= -1.5e+186)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 * t_1) - Float64(lambda1 * t_0)), Float64(phi1 - phi2)));
	elseif (lambda1 <= -1e-73)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda2 * t_0) - Float64(lambda2 * t_1)), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin((0.5 * phi1)) * sin((phi2 * 0.5));
	t_1 = cos((0.5 * phi1)) * cos((phi2 * 0.5));
	tmp = 0.0;
	if (lambda1 <= -1.5e+186)
		tmp = R * hypot(((lambda1 * t_1) - (lambda1 * t_0)), (phi1 - phi2));
	elseif (lambda1 <= -1e-73)
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda2 * t_0) - (lambda2 * t_1)), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -1.5e+186], N[(R * N[Sqrt[N[(N[(lambda1 * t$95$1), $MachinePrecision] - N[(lambda1 * t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, -1e-73], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda2 * t$95$0), $MachinePrecision] - N[(lambda2 * t$95$1), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.49999999999999991e186

   1. Initial program 52.3%

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

      1. hypot-def 88.2%

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

   3. Simplified 88.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. *-commutative 88.2%

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

      2. flip-- 44.7%

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

      3. associate-*r/ 44.7%

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

      4. associate-/l* 44.7%

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

      5. div-inv 44.7%

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

      6. metadata-eval 44.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}, \phi_1 - \phi_2\right) \]

      7. *-un-lft-identity 44.7%

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

      8. associate-/l* 44.7%

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

      9. flip-- 88.2%

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

   5. Applied egg-rr 88.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}{\frac{1}{\lambda_1 - \lambda_2}}}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 88.2%

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

      2. +-commutative 88.2%

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

      3. distribute-rgt-in 88.2%

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

      4. *-commutative 88.2%

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

      5. cos-sum 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. div-sub 99.8%

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

      2. div-inv 99.9%

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

      3. *-commutative 99.9%

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

      4. remove-double-div 99.8%

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

      5. div-inv 99.8%

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

      6. *-commutative 99.8%

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

      7. remove-double-div 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in lambda2 around 0 98.2%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right) - \lambda_1 \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}, \phi_1 - \phi_2\right) \]

   if -1.49999999999999991e186 < lambda1 < -9.99999999999999997e-74

   1. Initial program 61.1%

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

      1. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   if -9.99999999999999997e-74 < lambda1

   1. Initial program 64.7%

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

      1. hypot-def 95.5%

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

   3. Simplified 95.5%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. *-commutative 95.5%

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

      2. flip-- 75.0%

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

      3. associate-*r/ 75.0%

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

      4. associate-/l* 74.9%

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

      5. div-inv 74.9%

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

      6. metadata-eval 74.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}, \phi_1 - \phi_2\right) \]

      7. *-un-lft-identity 74.9%

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

      8. associate-/l* 74.9%

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

      9. flip-- 95.4%

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

   5. Applied egg-rr 95.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}{\frac{1}{\lambda_1 - \lambda_2}}}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 95.4%

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

      2. +-commutative 95.4%

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

      3. distribute-rgt-in 95.4%

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

      4. *-commutative 95.4%

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

      5. cos-sum 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. div-sub 99.8%

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

      2. div-inv 99.8%

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

      3. *-commutative 99.8%

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

      4. remove-double-div 99.9%

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

      5. div-inv 99.9%

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

      6. *-commutative 99.9%

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

      7. remove-double-div 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in lambda1 around 0 87.7%

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

       1. distribute-lft-out-- 87.7%

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

   12. Simplified 87.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.5 \cdot 10^{+186}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right) - \lambda_1 \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_1 \leq -1 \cdot 10^{-73}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right) - \lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 3: 88.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\\ t_1 := \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\\ \mathbf{if}\;\lambda_1 \leq -1.5 \cdot 10^{+186}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(t_1 - t_0\right), \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_1 \leq -4 \cdot 10^{-70}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(t_0 - t_1\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (* 0.5 phi1)) (sin (* phi2 0.5))))
        (t_1 (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))))
   (if (<= lambda1 -1.5e+186)
     (* R (hypot (* lambda1 (- t_1 t_0)) (- phi1 phi2)))
     (if (<= lambda1 -4e-70)
       (*
        R
        (hypot
         (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0)))
         (- phi1 phi2)))
       (* R (hypot (* lambda2 (- t_0 t_1)) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1)) * sin((phi2 * 0.5));
	double t_1 = cos((0.5 * phi1)) * cos((phi2 * 0.5));
	double tmp;
	if (lambda1 <= -1.5e+186) {
		tmp = R * hypot((lambda1 * (t_1 - t_0)), (phi1 - phi2));
	} else if (lambda1 <= -4e-70) {
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5));
	double t_1 = Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5));
	double tmp;
	if (lambda1 <= -1.5e+186) {
		tmp = R * Math.hypot((lambda1 * (t_1 - t_0)), (phi1 - phi2));
	} else if (lambda1 <= -4e-70) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))
	t_1 = math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))
	tmp = 0
	if lambda1 <= -1.5e+186:
		tmp = R * math.hypot((lambda1 * (t_1 - t_0)), (phi1 - phi2))
	elif lambda1 <= -4e-70:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))
	else:
		tmp = R * math.hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5)))
	t_1 = Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5)))
	tmp = 0.0
	if (lambda1 <= -1.5e+186)
		tmp = Float64(R * hypot(Float64(lambda1 * Float64(t_1 - t_0)), Float64(phi1 - phi2)));
	elseif (lambda1 <= -4e-70)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(lambda2 * Float64(t_0 - t_1)), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin((0.5 * phi1)) * sin((phi2 * 0.5));
	t_1 = cos((0.5 * phi1)) * cos((phi2 * 0.5));
	tmp = 0.0;
	if (lambda1 <= -1.5e+186)
		tmp = R * hypot((lambda1 * (t_1 - t_0)), (phi1 - phi2));
	elseif (lambda1 <= -4e-70)
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	else
		tmp = R * hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -1.5e+186], N[(R * N[Sqrt[N[(lambda1 * N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, -4e-70], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda2 * N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.49999999999999991e186

   1. Initial program 52.3%

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

      1. hypot-def 88.2%

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

   3. Simplified 88.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. *-commutative 88.2%

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

      2. flip-- 44.7%

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

      3. associate-*r/ 44.7%

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

      4. associate-/l* 44.7%

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

      5. div-inv 44.7%

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

      6. metadata-eval 44.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}, \phi_1 - \phi_2\right) \]

      7. *-un-lft-identity 44.7%

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

      8. associate-/l* 44.7%

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

      9. flip-- 88.2%

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

   5. Applied egg-rr 88.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}{\frac{1}{\lambda_1 - \lambda_2}}}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 88.2%

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

      2. +-commutative 88.2%

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

      3. distribute-rgt-in 88.2%

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

      4. *-commutative 88.2%

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

      5. cos-sum 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in lambda1 around inf 98.1%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}, \phi_1 - \phi_2\right) \]

   if -1.49999999999999991e186 < lambda1 < -3.99999999999999998e-70

   1. Initial program 61.1%

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

      1. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   if -3.99999999999999998e-70 < lambda1

   1. Initial program 64.7%

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

      1. hypot-def 95.5%

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

   3. Simplified 95.5%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. *-commutative 95.5%

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

      2. flip-- 75.0%

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

      3. associate-*r/ 75.0%

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

      4. associate-/l* 74.9%

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

      5. div-inv 74.9%

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

      6. metadata-eval 74.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}, \phi_1 - \phi_2\right) \]

      7. *-un-lft-identity 74.9%

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

      8. associate-/l* 74.9%

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

      9. flip-- 95.4%

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

   5. Applied egg-rr 95.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}{\frac{1}{\lambda_1 - \lambda_2}}}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 95.4%

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

      2. +-commutative 95.4%

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

      3. distribute-rgt-in 95.4%

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

      4. *-commutative 95.4%

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

      5. cos-sum 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in lambda1 around 0 87.7%

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

      1. associate-*r* 87.7%

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

      2. neg-mul-1 87.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(-\lambda_2\right)} \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right) \]

   10. Simplified 87.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.5 \cdot 10^{+186}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_1 \leq -4 \cdot 10^{-70}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 4: 88.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\\ t_1 := \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\\ \mathbf{if}\;\lambda_1 \leq -1 \cdot 10^{+186}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot t_1 - \lambda_1 \cdot t_0, \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_1 \leq -1 \cdot 10^{-68}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(t_0 - t_1\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (* 0.5 phi1)) (sin (* phi2 0.5))))
        (t_1 (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))))
   (if (<= lambda1 -1e+186)
     (* R (hypot (- (* lambda1 t_1) (* lambda1 t_0)) (- phi1 phi2)))
     (if (<= lambda1 -1e-68)
       (*
        R
        (hypot
         (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0)))
         (- phi1 phi2)))
       (* R (hypot (* lambda2 (- t_0 t_1)) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1)) * sin((phi2 * 0.5));
	double t_1 = cos((0.5 * phi1)) * cos((phi2 * 0.5));
	double tmp;
	if (lambda1 <= -1e+186) {
		tmp = R * hypot(((lambda1 * t_1) - (lambda1 * t_0)), (phi1 - phi2));
	} else if (lambda1 <= -1e-68) {
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5));
	double t_1 = Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5));
	double tmp;
	if (lambda1 <= -1e+186) {
		tmp = R * Math.hypot(((lambda1 * t_1) - (lambda1 * t_0)), (phi1 - phi2));
	} else if (lambda1 <= -1e-68) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))
	t_1 = math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))
	tmp = 0
	if lambda1 <= -1e+186:
		tmp = R * math.hypot(((lambda1 * t_1) - (lambda1 * t_0)), (phi1 - phi2))
	elif lambda1 <= -1e-68:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))
	else:
		tmp = R * math.hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5)))
	t_1 = Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5)))
	tmp = 0.0
	if (lambda1 <= -1e+186)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 * t_1) - Float64(lambda1 * t_0)), Float64(phi1 - phi2)));
	elseif (lambda1 <= -1e-68)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(lambda2 * Float64(t_0 - t_1)), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin((0.5 * phi1)) * sin((phi2 * 0.5));
	t_1 = cos((0.5 * phi1)) * cos((phi2 * 0.5));
	tmp = 0.0;
	if (lambda1 <= -1e+186)
		tmp = R * hypot(((lambda1 * t_1) - (lambda1 * t_0)), (phi1 - phi2));
	elseif (lambda1 <= -1e-68)
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	else
		tmp = R * hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -1e+186], N[(R * N[Sqrt[N[(N[(lambda1 * t$95$1), $MachinePrecision] - N[(lambda1 * t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, -1e-68], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda2 * N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -9.9999999999999998e185

   1. Initial program 52.3%

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

      1. hypot-def 88.2%

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

   3. Simplified 88.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. *-commutative 88.2%

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

      2. flip-- 44.7%

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

      3. associate-*r/ 44.7%

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

      4. associate-/l* 44.7%

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

      5. div-inv 44.7%

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

      6. metadata-eval 44.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}, \phi_1 - \phi_2\right) \]

      7. *-un-lft-identity 44.7%

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

      8. associate-/l* 44.7%

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

      9. flip-- 88.2%

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

   5. Applied egg-rr 88.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}{\frac{1}{\lambda_1 - \lambda_2}}}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 88.2%

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

      2. +-commutative 88.2%

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

      3. distribute-rgt-in 88.2%

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

      4. *-commutative 88.2%

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

      5. cos-sum 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. div-sub 99.8%

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

      2. div-inv 99.9%

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

      3. *-commutative 99.9%

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

      4. remove-double-div 99.8%

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

      5. div-inv 99.8%

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

      6. *-commutative 99.8%

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

      7. remove-double-div 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in lambda2 around 0 98.2%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right) - \lambda_1 \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}, \phi_1 - \phi_2\right) \]

   if -9.9999999999999998e185 < lambda1 < -1.00000000000000007e-68

   1. Initial program 61.1%

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

      1. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   if -1.00000000000000007e-68 < lambda1

   1. Initial program 64.7%

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

      1. hypot-def 95.5%

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

   3. Simplified 95.5%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. *-commutative 95.5%

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

      2. flip-- 75.0%

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

      3. associate-*r/ 75.0%

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

      4. associate-/l* 74.9%

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

      5. div-inv 74.9%

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

      6. metadata-eval 74.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}, \phi_1 - \phi_2\right) \]

      7. *-un-lft-identity 74.9%

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

      8. associate-/l* 74.9%

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

      9. flip-- 95.4%

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

   5. Applied egg-rr 95.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}{\frac{1}{\lambda_1 - \lambda_2}}}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 95.4%

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

      2. +-commutative 95.4%

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

      3. distribute-rgt-in 95.4%

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

      4. *-commutative 95.4%

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

      5. cos-sum 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in lambda1 around 0 87.7%

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

      1. associate-*r* 87.7%

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

      2. neg-mul-1 87.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(-\lambda_2\right)} \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right) \]

   10. Simplified 87.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1 \cdot 10^{+186}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right) - \lambda_1 \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_1 \leq -1 \cdot 10^{-68}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 5: 88.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := \sin \left(\phi_2 \cdot 0.5\right)\\ \mathbf{if}\;\lambda_1 \leq -2.8 \cdot 10^{-72}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot t_0 - \left(\lambda_1 \cdot t_1\right) \cdot t_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(t_1 \cdot t_2\right) - \lambda_2 \cdot t_0, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos (* 0.5 phi1)) (cos (* phi2 0.5))))
        (t_1 (sin (* 0.5 phi1)))
        (t_2 (sin (* phi2 0.5))))
   (if (<= lambda1 -2.8e-72)
     (*
      R
      (hypot
       (- (* (- lambda1 lambda2) t_0) (* (* lambda1 t_1) t_2))
       (- phi1 phi2)))
     (* R (hypot (- (* lambda2 (* t_1 t_2)) (* lambda2 t_0)) (- phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * phi1)) * cos((phi2 * 0.5));
	double t_1 = sin((0.5 * phi1));
	double t_2 = sin((phi2 * 0.5));
	double tmp;
	if (lambda1 <= -2.8e-72) {
		tmp = R * hypot((((lambda1 - lambda2) * t_0) - ((lambda1 * t_1) * t_2)), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda2 * (t_1 * t_2)) - (lambda2 * t_0)), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5));
	double t_1 = Math.sin((0.5 * phi1));
	double t_2 = Math.sin((phi2 * 0.5));
	double tmp;
	if (lambda1 <= -2.8e-72) {
		tmp = R * Math.hypot((((lambda1 - lambda2) * t_0) - ((lambda1 * t_1) * t_2)), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda2 * (t_1 * t_2)) - (lambda2 * t_0)), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))
	t_1 = math.sin((0.5 * phi1))
	t_2 = math.sin((phi2 * 0.5))
	tmp = 0
	if lambda1 <= -2.8e-72:
		tmp = R * math.hypot((((lambda1 - lambda2) * t_0) - ((lambda1 * t_1) * t_2)), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda2 * (t_1 * t_2)) - (lambda2 * t_0)), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5)))
	t_1 = sin(Float64(0.5 * phi1))
	t_2 = sin(Float64(phi2 * 0.5))
	tmp = 0.0
	if (lambda1 <= -2.8e-72)
		tmp = Float64(R * hypot(Float64(Float64(Float64(lambda1 - lambda2) * t_0) - Float64(Float64(lambda1 * t_1) * t_2)), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda2 * Float64(t_1 * t_2)) - Float64(lambda2 * t_0)), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((0.5 * phi1)) * cos((phi2 * 0.5));
	t_1 = sin((0.5 * phi1));
	t_2 = sin((phi2 * 0.5));
	tmp = 0.0;
	if (lambda1 <= -2.8e-72)
		tmp = R * hypot((((lambda1 - lambda2) * t_0) - ((lambda1 * t_1) * t_2)), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda2 * (t_1 * t_2)) - (lambda2 * t_0)), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda1, -2.8e-72], N[(R * N[Sqrt[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(N[(lambda1 * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda2 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(lambda2 * t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.7999999999999998e-72

   1. Initial program 58.2%

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

      1. hypot-def 96.1%

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

   3. Simplified 96.1%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. *-commutative 96.1%

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

      2. flip-- 67.8%

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

      3. associate-*r/ 67.8%

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

      4. associate-/l* 67.8%

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

      5. div-inv 67.8%

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

      6. metadata-eval 67.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}, \phi_1 - \phi_2\right) \]

      7. *-un-lft-identity 67.8%

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

      8. associate-/l* 67.8%

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

      9. flip-- 96.0%

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

   5. Applied egg-rr 96.0%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}{\frac{1}{\lambda_1 - \lambda_2}}}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 96.0%

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

      2. +-commutative 96.0%

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

      3. distribute-rgt-in 96.0%

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

      4. *-commutative 96.0%

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

      5. cos-sum 99.9%

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

      6. *-commutative 99.9%

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

      7. *-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. div-sub 99.9%

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

      2. div-inv 99.9%

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

      3. *-commutative 99.9%

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

      4. remove-double-div 99.9%

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

      5. div-inv 99.9%

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

      6. *-commutative 99.9%

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

      7. remove-double-div 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in lambda1 around inf 99.9%

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

       1. associate-*r* 99.9%

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

   12. Simplified 99.9%

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

   if -2.7999999999999998e-72 < lambda1

   1. Initial program 64.7%

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

      1. hypot-def 95.5%

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

   3. Simplified 95.5%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. *-commutative 95.5%

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

      2. flip-- 75.0%

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

      3. associate-*r/ 75.0%

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

      4. associate-/l* 74.9%

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

      5. div-inv 74.9%

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

      6. metadata-eval 74.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}, \phi_1 - \phi_2\right) \]

      7. *-un-lft-identity 74.9%

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

      8. associate-/l* 74.9%

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

      9. flip-- 95.4%

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

   5. Applied egg-rr 95.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}{\frac{1}{\lambda_1 - \lambda_2}}}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 95.4%

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

      2. +-commutative 95.4%

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

      3. distribute-rgt-in 95.4%

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

      4. *-commutative 95.4%

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

      5. cos-sum 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. div-sub 99.8%

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

      2. div-inv 99.8%

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

      3. *-commutative 99.8%

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

      4. remove-double-div 99.9%

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

      5. div-inv 99.9%

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

      6. *-commutative 99.9%

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

      7. remove-double-div 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in lambda1 around 0 87.7%

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

       1. distribute-lft-out-- 87.7%

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

   12. Simplified 87.7%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{-1 \cdot \left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right) - \lambda_2 \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)\right)}, \phi_1 - \phi_2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.8 \cdot 10^{-72}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right) - \left(\lambda_1 \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \sin \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right) - \lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 6: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.5 \cdot 10^{+186}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.5e+186)
   (*
    R
    (hypot
     (*
      lambda1
      (-
       (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
       (* (sin (* 0.5 phi1)) (sin (* phi2 0.5)))))
     (- phi1 phi2)))
   (*
    R
    (hypot
     (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0)))
     (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.5e+186) {
		tmp = R * hypot((lambda1 * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.5e+186) {
		tmp = R * Math.hypot((lambda1 * ((Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - (Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5))))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.5e+186:
		tmp = R * math.hypot((lambda1 * ((math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))) - (math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.5e+186)
		tmp = Float64(R * hypot(Float64(lambda1 * Float64(Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5))))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.5e+186)
		tmp = R * hypot((lambda1 * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.5e+186], N[(R * N[Sqrt[N[(lambda1 * N[(N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.49999999999999991e186

   1. Initial program 52.3%

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

      1. hypot-def 88.2%

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

   3. Simplified 88.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. *-commutative 88.2%

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

      2. flip-- 44.7%

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

      3. associate-*r/ 44.7%

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

      4. associate-/l* 44.7%

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

      5. div-inv 44.7%

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

      6. metadata-eval 44.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}, \phi_1 - \phi_2\right) \]

      7. *-un-lft-identity 44.7%

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

      8. associate-/l* 44.7%

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

      9. flip-- 88.2%

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

   5. Applied egg-rr 88.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}{\frac{1}{\lambda_1 - \lambda_2}}}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 88.2%

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

      2. +-commutative 88.2%

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

      3. distribute-rgt-in 88.2%

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

      4. *-commutative 88.2%

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

      5. cos-sum 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in lambda1 around inf 98.1%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}, \phi_1 - \phi_2\right) \]

   if -1.49999999999999991e186 < lambda1

   1. Initial program 63.8%

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

      1. hypot-def 96.6%

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

   3. Simplified 96.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.5 \cdot 10^{+186}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 7: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\frac{\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)}{\frac{1}{\lambda_1 - \lambda_2}}, \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (/
    (-
     (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
     (* (sin (* 0.5 phi1)) (sin (* phi2 0.5))))
    (/ 1.0 (- lambda1 lambda2)))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot((((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5)))) / (1.0 / (lambda1 - lambda2))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot((((Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - (Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5)))) / (1.0 / (lambda1 - lambda2))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot((((math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))) - (math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5)))) / (1.0 / (lambda1 - lambda2))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5)))) / Float64(1.0 / Float64(lambda1 - lambda2))), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot((((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5)))) / (1.0 / (lambda1 - lambda2))), (phi1 - phi2));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

   1. hypot-def 95.7%

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

3. Simplified 95.7%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Step-by-step derivation

   1. *-commutative 95.7%

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

   2. flip-- 72.6%

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

   3. associate-*r/ 72.6%

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

   4. associate-/l* 72.6%

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

   5. div-inv 72.6%

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

   6. metadata-eval 72.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}, \phi_1 - \phi_2\right) \]

   7. *-un-lft-identity 72.6%

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

   8. associate-/l* 72.6%

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

   9. flip-- 95.6%

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

5. Applied egg-rr 95.6%

   \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}{\frac{1}{\lambda_1 - \lambda_2}}}, \phi_1 - \phi_2\right) \]
6. Step-by-step derivation

   1. *-commutative 95.6%

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

   2. +-commutative 95.6%

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

   3. distribute-rgt-in 95.6%

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

   4. *-commutative 95.6%

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

   5. cos-sum 99.8%

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

   6. *-commutative 99.8%

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

   7. *-commutative 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)}{\frac{1}{\lambda_1 - \lambda_2}}, \phi_1 - \phi_2\right) \]

Alternative 8: 93.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 4e-23)
   (* R (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (* phi2 0.5))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4e-23) {
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4e-23) {
		tmp = R * Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 4e-23:
		tmp = R * math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi2 * 0.5))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 4e-23)
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 4e-23)
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 4e-23], N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.99999999999999984e-23

   1. Initial program 64.1%

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

      1. hypot-def 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in phi2 around 0 93.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

   if 3.99999999999999984e-23 < phi2

   1. Initial program 57.4%

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

      1. hypot-def 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in phi1 around 0 90.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 9: 95.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0))) (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

   1. hypot-def 95.7%

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

3. Simplified 95.7%

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

4. Final simplification 95.7%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right) \]

Alternative 10: 90.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

   1. hypot-def 95.7%

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

3. Simplified 95.7%

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

4. Taylor expanded in phi2 around 0 91.5%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

5. Final simplification 91.5%

   \[\leadsto R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \]

Alternative 11: 34.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4.2 \cdot 10^{+106} \lor \neg \left(\lambda_2 \leq 2 \cdot 10^{+136}\right) \land \lambda_2 \leq 1.02 \cdot 10^{+167}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda2 4.2e+106)
         (and (not (<= lambda2 2e+136)) (<= lambda2 1.02e+167)))
   (* R (- phi2 phi1))
   (* R (* (cos (* 0.5 phi1)) lambda2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= 4.2e+106) || (!(lambda2 <= 2e+136) && (lambda2 <= 1.02e+167))) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (cos((0.5 * phi1)) * lambda2);
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((lambda2 <= 4.2d+106) .or. (.not. (lambda2 <= 2d+136)) .and. (lambda2 <= 1.02d+167)) then
        tmp = r * (phi2 - phi1)
    else
        tmp = r * (cos((0.5d0 * phi1)) * lambda2)
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= 4.2e+106) || (!(lambda2 <= 2e+136) && (lambda2 <= 1.02e+167))) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (Math.cos((0.5 * phi1)) * lambda2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda2 <= 4.2e+106) or (not (lambda2 <= 2e+136) and (lambda2 <= 1.02e+167)):
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * (math.cos((0.5 * phi1)) * lambda2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda2 <= 4.2e+106) || (!(lambda2 <= 2e+136) && (lambda2 <= 1.02e+167)))
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * Float64(cos(Float64(0.5 * phi1)) * lambda2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda2 <= 4.2e+106) || (~((lambda2 <= 2e+136)) && (lambda2 <= 1.02e+167)))
		tmp = R * (phi2 - phi1);
	else
		tmp = R * (cos((0.5 * phi1)) * lambda2);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda2, 4.2e+106], And[N[Not[LessEqual[lambda2, 2e+136]], $MachinePrecision], LessEqual[lambda2, 1.02e+167]]], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 4.2000000000000001e106 or 2.00000000000000012e136 < lambda2 < 1.02e167

   1. Initial program 64.6%

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

      1. hypot-def 96.2%

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

   3. Simplified 96.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 91.0%

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

      2. *-commutative 91.0%

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

      3. div-inv 91.0%

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

      4. metadata-eval 91.0%

         \[\leadsto R \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)\right) \]

   5. Applied egg-rr 91.0%

      \[\leadsto R \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)\right)} \]

   6. Taylor expanded in phi1 around -inf 28.8%

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

      1. mul-1-neg 28.8%

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

      2. sub-neg 28.8%

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

   8. Simplified 28.8%

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

   if 4.2000000000000001e106 < lambda2 < 2.00000000000000012e136 or 1.02e167 < lambda2

   1. Initial program 49.9%

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

      1. hypot-def 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in lambda2 around inf 65.2%

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 65.2%

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

      2. +-commutative 65.2%

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

   6. Simplified 65.2%

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

   7. Taylor expanded in phi2 around 0 67.4%

      \[\leadsto \color{blue}{\left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4.2 \cdot 10^{+106} \lor \neg \left(\lambda_2 \leq 2 \cdot 10^{+136}\right) \land \lambda_2 \leq 1.02 \cdot 10^{+167}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_2\right)\\ \end{array} \]

Alternative 12: 85.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

   1. hypot-def 95.7%

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

3. Simplified 95.7%

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

4. Taylor expanded in phi2 around 0 82.8%

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

   1. associate-*r* 82.8%

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

   2. *-commutative 82.8%

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

   3. *-commutative 82.8%

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

6. Simplified 82.8%

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

7. Taylor expanded in phi1 around 0 87.2%

   \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]

8. Final simplification 87.2%

   \[\leadsto R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right) \]

Alternative 13: 29.5% accurate, 36.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.032:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.032) (* R (- phi1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.032) {
		tmp = R * -phi1;
	} else {
		tmp = R * 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 <= 0.032d0) then
        tmp = r * -phi1
    else
        tmp = r * 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 <= 0.032) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.032:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.032)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 0.032)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.032], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.032000000000000001

   1. Initial program 63.4%

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

      1. hypot-def 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in phi1 around -inf 15.4%

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

      1. mul-1-neg 15.4%

         \[\leadsto \color{blue}{-R \cdot \phi_1} \]

      2. *-commutative 15.4%

         \[\leadsto -\color{blue}{\phi_1 \cdot R} \]

      3. distribute-rgt-neg-in 15.4%

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

   6. Simplified 15.4%

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

   if 0.032000000000000001 < phi2

   1. Initial program 59.4%

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

      1. hypot-def 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in phi2 around inf 64.3%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   5. Step-by-step derivation

      1. *-commutative 64.3%

         \[\leadsto \color{blue}{\phi_2 \cdot R} \]

   6. Simplified 64.3%

      \[\leadsto \color{blue}{\phi_2 \cdot R} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.032:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 14: 30.6% accurate, 65.8× speedup

Math

\[\begin{array}{l} \\ R \cdot \left(\phi_2 - \phi_1\right) \end{array} \]

FPCore

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

C

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

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 * (phi2 - phi1)
end function

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * (phi2 - phi1)

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

   1. hypot-def 95.7%

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

3. Simplified 95.7%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 90.5%

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

   2. *-commutative 90.5%

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

   3. div-inv 90.5%

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

   4. metadata-eval 90.5%

      \[\leadsto R \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)\right) \]

5. Applied egg-rr 90.5%

   \[\leadsto R \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)\right)} \]

6. Taylor expanded in phi1 around -inf 26.3%

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

   1. mul-1-neg 26.3%

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

   2. sub-neg 26.3%

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

8. Simplified 26.3%

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

9. Final simplification 26.3%

   \[\leadsto R \cdot \left(\phi_2 - \phi_1\right) \]

Alternative 15: 17.8% accurate, 109.7× speedup

Math

\[\begin{array}{l} \\ R \cdot \phi_2 \end{array} \]

FPCore

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

C

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

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 * phi2
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * phi2;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * phi2

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * phi2), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

   1. hypot-def 95.7%

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

3. Simplified 95.7%

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

4. Taylor expanded in phi2 around inf 17.6%

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
5. Step-by-step derivation

   1. *-commutative 17.6%

      \[\leadsto \color{blue}{\phi_2 \cdot R} \]

6. Simplified 17.6%

   \[\leadsto \color{blue}{\phi_2 \cdot R} \]

7. Final simplification 17.6%

   \[\leadsto R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023336 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))