Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.6% → 99.9%

Time: 18.9s

Alternatives: 12

Speedup: 1.6×

• 30.9% 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: 59.6% 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

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (-
     (* (cos (* 0.5 phi1)) (cos (* 0.5 phi2)))
     (* (sin (* 0.5 phi1)) (sin (* 0.5 phi2))))
    (- 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((0.5 * phi2))) - (sin((0.5 * phi1)) * sin((0.5 * phi2)))) * (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((0.5 * phi2))) - (Math.sin((0.5 * phi1)) * Math.sin((0.5 * phi2)))) * (lambda1 - lambda2)), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot((((math.cos((0.5 * phi1)) * math.cos((0.5 * phi2))) - (math.sin((0.5 * phi1)) * math.sin((0.5 * phi2)))) * (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(0.5 * phi2))) - Float64(sin(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot((((cos((0.5 * phi1)) * cos((0.5 * phi2))) - (sin((0.5 * phi1)) * sin((0.5 * phi2)))) * (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[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.7%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   9. --lowering--.f64 98.6%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

3. Simplified 98.6%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   2. flip-- N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \frac{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}{\lambda_1 + \lambda_2}\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \frac{1}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\frac{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right), \left(\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right), \left(\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right), \left(\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   9. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right), \left(\frac{1}{\frac{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}{\lambda_1 + \lambda_2}}\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   10. flip-- N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right), \left(\frac{1}{\lambda_1 - \lambda_2}\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right), \mathsf{/.f64}\left(1, \left(\lambda_1 - \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   12. --lowering--.f64 98.6%

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

6. Applied egg-rr 98.6%

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

   1. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_2 + \phi_1\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   5. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2 + \frac{1}{2} \cdot \phi_1\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\cos \left(\phi_2 \cdot \frac{1}{2} + \frac{1}{2} \cdot \phi_1\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. cos-sum N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\cos \left(\phi_2 \cdot \frac{1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   10. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\phi_2 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\phi_1 \cdot \frac{1}{2}\right)\right)\right), \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_1, \frac{1}{2}\right)\right)\right), \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_1, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   16. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_1, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\phi_2 \cdot \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_1, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   18. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_1, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_1, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   20. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_1, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\phi_1, \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

8. Applied egg-rr 99.9%

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

   1. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right) + \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right) + \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   4. fma-define N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{fma}\left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   5. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right), \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(0 - \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right), \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right), \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   8. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\left(\phi_2 \cdot \frac{1}{2}\right)\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right), \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right), \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   10. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right)\right), \mathsf{sin.f64}\left(\left(\phi_1 \cdot \frac{1}{2}\right)\right), \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right), \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right), \left(\cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right), \mathsf{*.f64}\left(\cos \left(\phi_2 \cdot \frac{1}{2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   14. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\phi_2 \cdot \frac{1}{2}\right)\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   16. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   18. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\phi_2, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

10. Applied egg-rr 99.9%

    \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\color{blue}{\mathsf{fma}\left(0 - \sin \left(\phi_2 \cdot 0.5\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}{\frac{1}{\lambda_1 - \lambda_2}}, \phi_1 - \phi_2\right) \]

11. Taylor expanded in phi2 around inf

    \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\left(-1 \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
12. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

13. Simplified 99.9%

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

Alternative 2: 92.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 5e-26)
   (* R (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) (- phi1 phi2)))
   (* R (hypot (* (cos (* 0.5 phi2)) (- lambda1 lambda2)) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5e-26) {
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	} else {
		tmp = R * hypot((cos((0.5 * phi2)) * (lambda1 - lambda2)), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5e-26) {
		tmp = R * Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((Math.cos((0.5 * phi2)) * (lambda1 - lambda2)), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 5e-26:
		tmp = R * math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2))
	else:
		tmp = R * math.hypot((math.cos((0.5 * phi2)) * (lambda1 - lambda2)), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5e-26)
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 5e-26)
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	else
		tmp = R * hypot((cos((0.5 * phi2)) * (lambda1 - lambda2)), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 5e-26], 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[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5.00000000000000019e-26

   1. Initial program 59.3%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 99.4%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 95.2%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 95.2%

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

   if 5.00000000000000019e-26 < phi2

   1. Initial program 60.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 96.8%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 96.8%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-lowering-*.f64 96.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 96.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * cos(((phi1 + phi2) / 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[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.7%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   9. --lowering--.f64 98.6%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

3. Simplified 98.6%

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

Alternative 4: 90.4% 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 59.7%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   9. --lowering--.f64 98.6%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

3. Simplified 98.6%

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

5. Taylor expanded in phi2 around 0

   \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   5. *-lowering-*.f64 93.7%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

7. Simplified 93.7%

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

8. Final simplification 93.7%

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

Alternative 5: 28.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.65 \cdot 10^{+193}:\\ \;\;\;\;\lambda_1 \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(0 - R\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.05 \cdot 10^{-151}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot \phi_1\right) \cdot \left(R \cdot \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.65e+193)
   (* lambda1 (* (cos (/ (+ phi1 phi2) 2.0)) (- 0.0 R)))
   (if (<= lambda1 1.05e-151)
     (* R (- phi2 phi1))
     (* (cos (* 0.5 phi1)) (* R lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.65e+193) {
		tmp = lambda1 * (cos(((phi1 + phi2) / 2.0)) * (0.0 - R));
	} else if (lambda1 <= 1.05e-151) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = cos((0.5 * phi1)) * (R * lambda2);
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-1.65d+193)) then
        tmp = lambda1 * (cos(((phi1 + phi2) / 2.0d0)) * (0.0d0 - r))
    else if (lambda1 <= 1.05d-151) then
        tmp = r * (phi2 - phi1)
    else
        tmp = cos((0.5d0 * phi1)) * (r * lambda2)
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.65e+193) {
		tmp = lambda1 * (Math.cos(((phi1 + phi2) / 2.0)) * (0.0 - R));
	} else if (lambda1 <= 1.05e-151) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = Math.cos((0.5 * phi1)) * (R * lambda2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.65e+193:
		tmp = lambda1 * (math.cos(((phi1 + phi2) / 2.0)) * (0.0 - R))
	elif lambda1 <= 1.05e-151:
		tmp = R * (phi2 - phi1)
	else:
		tmp = math.cos((0.5 * phi1)) * (R * lambda2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.65e+193)
		tmp = Float64(lambda1 * Float64(cos(Float64(Float64(phi1 + phi2) / 2.0)) * Float64(0.0 - R)));
	elseif (lambda1 <= 1.05e-151)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(cos(Float64(0.5 * phi1)) * Float64(R * lambda2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.65e+193)
		tmp = lambda1 * (cos(((phi1 + phi2) / 2.0)) * (0.0 - R));
	elseif (lambda1 <= 1.05e-151)
		tmp = R * (phi2 - phi1);
	else
		tmp = cos((0.5 * phi1)) * (R * lambda2);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.65e+193], N[(lambda1 * N[(N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(0.0 - R), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 1.05e-151], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(R * lambda2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.65e193

   1. Initial program 47.5%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 95.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   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. Add Preprocessing

   5. Taylor expanded in lambda1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{R \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto 0 - R \cdot \left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \color{blue}{\lambda_1}\right) \]

      4. associate-*l* N/A

         \[\leadsto 0 - \left(R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \cdot \color{blue}{\lambda_1} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\left(R \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \cdot \lambda_1\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot R\right) \cdot \lambda_1\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \color{blue}{\left(R \cdot \lambda_1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right), \color{blue}{\left(R \cdot \lambda_1\right)}\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right), \left(\color{blue}{R} \cdot \lambda_1\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\phi_1 + \phi_2\right)\right)\right), \left(R \cdot \lambda_1\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \left(R \cdot \lambda_1\right)\right)\right) \]

      12. *-lowering-*.f64 60.4%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \mathsf{*.f64}\left(R, \color{blue}{\lambda_1}\right)\right)\right) \]

   7. Simplified 60.4%

      \[\leadsto \color{blue}{0 - \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(R \cdot \lambda_1\right)} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(R \cdot \lambda_1\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot R\right) \cdot \lambda_1\right) \]

      3. distribute-rgt-neg-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot R\right), \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right) \cdot R\right), \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right) \cdot R\right), \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot R\right), \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(R \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\lambda_1}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\lambda_1}\right)\right)\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)\right), \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \cos \left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)\right), \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right), \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right), \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \mathsf{cos.f64}\left(\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)\right)\right), \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \mathsf{cos.f64}\left(\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)\right)\right), \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \]

      16. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \]

      19. neg-lowering-neg.f64 60.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\lambda_1\right)\right) \]

   9. Applied egg-rr 60.5%

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

   if -1.65e193 < lambda1 < 1.04999999999999995e-151

   1. Initial program 63.6%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 98.9%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 98.9%

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

   5. Taylor expanded in phi1 around -inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{neg}\left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(0 - \color{blue}{\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \color{blue}{\left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \color{blue}{\left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(1 + \left(\mathsf{neg}\left(\frac{\phi_2}{\phi_1}\right)\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(1 - \color{blue}{\frac{\phi_2}{\phi_1}}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\phi_2}{\phi_1}\right)}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 37.2%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right)\right)\right)\right) \]

   7. Simplified 37.2%

      \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 - \frac{\phi_2}{\phi_1}\right)\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(-1 \cdot \phi_1 - -1 \cdot \phi_2\right)}\right) \]
   9. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(\phi_1 + \color{blue}{\left(\mathsf{neg}\left(\phi_2\right)\right)}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(\phi_1 + -1 \cdot \color{blue}{\phi_2}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(-1 \cdot \phi_2 + \color{blue}{\phi_1}\right)\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(-1 \cdot \phi_2\right) + \color{blue}{-1 \cdot \phi_1}\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\left(\mathsf{neg}\left(-1 \cdot \phi_2\right)\right) + \color{blue}{-1} \cdot \phi_1\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) + -1 \cdot \phi_1\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 + \color{blue}{-1} \cdot \phi_1\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 + \left(\mathsf{neg}\left(\phi_1\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 - \color{blue}{\phi_1}\right)\right) \]

      11. --lowering--.f64 40.5%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right) \]

   10. Simplified 40.5%

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

   if 1.04999999999999995e-151 < lambda1

   1. Initial program 56.3%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 98.9%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 98.9%

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

   5. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right), \color{blue}{\left(R \cdot \lambda_2\right)}\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right), \left(\color{blue}{R} \cdot \lambda_2\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\phi_1 + \phi_2\right)\right)\right), \left(R \cdot \lambda_2\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \left(R \cdot \lambda_2\right)\right) \]

      9. *-lowering-*.f64 13.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \mathsf{*.f64}\left(R, \color{blue}{\lambda_2}\right)\right) \]

   7. Simplified 13.2%

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

   8. Taylor expanded in phi2 around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(R \cdot \lambda_2\right), \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \]

      5. *-lowering-*.f64 15.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right) \]

   10. Simplified 15.9%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.65 \cdot 10^{+193}:\\ \;\;\;\;\lambda_1 \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(0 - R\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.05 \cdot 10^{-151}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot \phi_1\right) \cdot \left(R \cdot \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 28.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.6 \cdot 10^{+191}:\\ \;\;\;\;R \cdot \left(\left(0 - \lambda_1\right) \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 3.8 \cdot 10^{-151}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot \phi_1\right) \cdot \left(R \cdot \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -4.6e+191)
   (* R (* (- 0.0 lambda1) (cos (* 0.5 (+ phi1 phi2)))))
   (if (<= lambda1 3.8e-151)
     (* R (- phi2 phi1))
     (* (cos (* 0.5 phi1)) (* R lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -4.6e+191) {
		tmp = R * ((0.0 - lambda1) * cos((0.5 * (phi1 + phi2))));
	} else if (lambda1 <= 3.8e-151) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = cos((0.5 * phi1)) * (R * lambda2);
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-4.6d+191)) then
        tmp = r * ((0.0d0 - lambda1) * cos((0.5d0 * (phi1 + phi2))))
    else if (lambda1 <= 3.8d-151) then
        tmp = r * (phi2 - phi1)
    else
        tmp = cos((0.5d0 * phi1)) * (r * lambda2)
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -4.6e+191) {
		tmp = R * ((0.0 - lambda1) * Math.cos((0.5 * (phi1 + phi2))));
	} else if (lambda1 <= 3.8e-151) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = Math.cos((0.5 * phi1)) * (R * lambda2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -4.6e+191:
		tmp = R * ((0.0 - lambda1) * math.cos((0.5 * (phi1 + phi2))))
	elif lambda1 <= 3.8e-151:
		tmp = R * (phi2 - phi1)
	else:
		tmp = math.cos((0.5 * phi1)) * (R * lambda2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -4.6e+191)
		tmp = Float64(R * Float64(Float64(0.0 - lambda1) * cos(Float64(0.5 * Float64(phi1 + phi2)))));
	elseif (lambda1 <= 3.8e-151)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(cos(Float64(0.5 * phi1)) * Float64(R * lambda2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -4.6e+191)
		tmp = R * ((0.0 - lambda1) * cos((0.5 * (phi1 + phi2))));
	elseif (lambda1 <= 3.8e-151)
		tmp = R * (phi2 - phi1);
	else
		tmp = cos((0.5 * phi1)) * (R * lambda2);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -4.6e+191], N[(R * N[(N[(0.0 - lambda1), $MachinePrecision] * N[Cos[N[(0.5 * N[(phi1 + phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 3.8e-151], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(R * lambda2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -4.5999999999999999e191

   1. Initial program 47.5%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 95.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   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. Add Preprocessing

   5. Taylor expanded in lambda1 around -inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{neg}\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \lambda_1\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(-1 \cdot \color{blue}{\lambda_1}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right), \color{blue}{\left(-1 \cdot \lambda_1\right)}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right), \left(\color{blue}{-1} \cdot \lambda_1\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\phi_1 + \phi_2\right)\right)\right), \left(-1 \cdot \lambda_1\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \left(-1 \cdot \lambda_1\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) \]

      10. neg-lowering-neg.f64 60.4%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \mathsf{neg.f64}\left(\lambda_1\right)\right)\right) \]

   7. Simplified 60.4%

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

   if -4.5999999999999999e191 < lambda1 < 3.7999999999999997e-151

   1. Initial program 63.6%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 98.9%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 98.9%

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

   5. Taylor expanded in phi1 around -inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{neg}\left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(0 - \color{blue}{\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \color{blue}{\left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \color{blue}{\left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(1 + \left(\mathsf{neg}\left(\frac{\phi_2}{\phi_1}\right)\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(1 - \color{blue}{\frac{\phi_2}{\phi_1}}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\phi_2}{\phi_1}\right)}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 37.2%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right)\right)\right)\right) \]

   7. Simplified 37.2%

      \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 - \frac{\phi_2}{\phi_1}\right)\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(-1 \cdot \phi_1 - -1 \cdot \phi_2\right)}\right) \]
   9. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(\phi_1 + \color{blue}{\left(\mathsf{neg}\left(\phi_2\right)\right)}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(\phi_1 + -1 \cdot \color{blue}{\phi_2}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(-1 \cdot \phi_2 + \color{blue}{\phi_1}\right)\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(-1 \cdot \phi_2\right) + \color{blue}{-1 \cdot \phi_1}\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\left(\mathsf{neg}\left(-1 \cdot \phi_2\right)\right) + \color{blue}{-1} \cdot \phi_1\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) + -1 \cdot \phi_1\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 + \color{blue}{-1} \cdot \phi_1\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 + \left(\mathsf{neg}\left(\phi_1\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 - \color{blue}{\phi_1}\right)\right) \]

      11. --lowering--.f64 40.5%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right) \]

   10. Simplified 40.5%

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

   if 3.7999999999999997e-151 < lambda1

   1. Initial program 56.3%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 98.9%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 98.9%

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

   5. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right), \color{blue}{\left(R \cdot \lambda_2\right)}\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right), \left(\color{blue}{R} \cdot \lambda_2\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\phi_1 + \phi_2\right)\right)\right), \left(R \cdot \lambda_2\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \left(R \cdot \lambda_2\right)\right) \]

      9. *-lowering-*.f64 13.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \mathsf{*.f64}\left(R, \color{blue}{\lambda_2}\right)\right) \]

   7. Simplified 13.2%

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

   8. Taylor expanded in phi2 around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(R \cdot \lambda_2\right), \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \]

      5. *-lowering-*.f64 15.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right) \]

   10. Simplified 15.9%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.6 \cdot 10^{+191}:\\ \;\;\;\;R \cdot \left(\left(0 - \lambda_1\right) \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 3.8 \cdot 10^{-151}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot \phi_1\right) \cdot \left(R \cdot \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 28.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.4 \cdot 10^{+190}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(0 - \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 3.8 \cdot 10^{-151}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot \phi_1\right) \cdot \left(R \cdot \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -3.4e+190)
   (* R (* (cos (* 0.5 phi2)) (- 0.0 lambda1)))
   (if (<= lambda1 3.8e-151)
     (* R (- phi2 phi1))
     (* (cos (* 0.5 phi1)) (* R lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3.4e+190) {
		tmp = R * (cos((0.5 * phi2)) * (0.0 - lambda1));
	} else if (lambda1 <= 3.8e-151) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = cos((0.5 * phi1)) * (R * lambda2);
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-3.4d+190)) then
        tmp = r * (cos((0.5d0 * phi2)) * (0.0d0 - lambda1))
    else if (lambda1 <= 3.8d-151) then
        tmp = r * (phi2 - phi1)
    else
        tmp = cos((0.5d0 * phi1)) * (r * lambda2)
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3.4e+190) {
		tmp = R * (Math.cos((0.5 * phi2)) * (0.0 - lambda1));
	} else if (lambda1 <= 3.8e-151) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = Math.cos((0.5 * phi1)) * (R * lambda2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -3.4e+190:
		tmp = R * (math.cos((0.5 * phi2)) * (0.0 - lambda1))
	elif lambda1 <= 3.8e-151:
		tmp = R * (phi2 - phi1)
	else:
		tmp = math.cos((0.5 * phi1)) * (R * lambda2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -3.4e+190)
		tmp = Float64(R * Float64(cos(Float64(0.5 * phi2)) * Float64(0.0 - lambda1)));
	elseif (lambda1 <= 3.8e-151)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(cos(Float64(0.5 * phi1)) * Float64(R * lambda2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -3.4e+190)
		tmp = R * (cos((0.5 * phi2)) * (0.0 - lambda1));
	elseif (lambda1 <= 3.8e-151)
		tmp = R * (phi2 - phi1);
	else
		tmp = cos((0.5 * phi1)) * (R * lambda2);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -3.4e+190], N[(R * N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(0.0 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 3.8e-151], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(R * lambda2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -3.3999999999999999e190

   1. Initial program 47.5%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 95.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   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. Add Preprocessing

   5. Taylor expanded in lambda1 around -inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{neg}\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \lambda_1\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(-1 \cdot \color{blue}{\lambda_1}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right), \color{blue}{\left(-1 \cdot \lambda_1\right)}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right), \left(\color{blue}{-1} \cdot \lambda_1\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\phi_1 + \phi_2\right)\right)\right), \left(-1 \cdot \lambda_1\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \left(-1 \cdot \lambda_1\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) \]

      10. neg-lowering-neg.f64 60.4%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \mathsf{neg.f64}\left(\lambda_1\right)\right)\right) \]

   7. Simplified 60.4%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_2\right)}, \mathsf{neg.f64}\left(\lambda_1\right)\right)\right) \]
   9. Step-by-step derivation

      1. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \mathsf{neg.f64}\left(\color{blue}{\lambda_1}\right)\right)\right) \]

      2. *-lowering-*.f64 59.2%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{neg.f64}\left(\lambda_1\right)\right)\right) \]

   10. Simplified 59.2%

       \[\leadsto R \cdot \left(\color{blue}{\cos \left(0.5 \cdot \phi_2\right)} \cdot \left(-\lambda_1\right)\right) \]

   if -3.3999999999999999e190 < lambda1 < 3.7999999999999997e-151

   1. Initial program 63.6%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 98.9%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 98.9%

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

   5. Taylor expanded in phi1 around -inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{neg}\left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(0 - \color{blue}{\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \color{blue}{\left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \color{blue}{\left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(1 + \left(\mathsf{neg}\left(\frac{\phi_2}{\phi_1}\right)\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(1 - \color{blue}{\frac{\phi_2}{\phi_1}}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\phi_2}{\phi_1}\right)}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 37.2%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right)\right)\right)\right) \]

   7. Simplified 37.2%

      \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 - \frac{\phi_2}{\phi_1}\right)\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(-1 \cdot \phi_1 - -1 \cdot \phi_2\right)}\right) \]
   9. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(\phi_1 + \color{blue}{\left(\mathsf{neg}\left(\phi_2\right)\right)}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(\phi_1 + -1 \cdot \color{blue}{\phi_2}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(-1 \cdot \phi_2 + \color{blue}{\phi_1}\right)\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(-1 \cdot \phi_2\right) + \color{blue}{-1 \cdot \phi_1}\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\left(\mathsf{neg}\left(-1 \cdot \phi_2\right)\right) + \color{blue}{-1} \cdot \phi_1\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) + -1 \cdot \phi_1\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 + \color{blue}{-1} \cdot \phi_1\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 + \left(\mathsf{neg}\left(\phi_1\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 - \color{blue}{\phi_1}\right)\right) \]

      11. --lowering--.f64 40.5%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right) \]

   10. Simplified 40.5%

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

   if 3.7999999999999997e-151 < lambda1

   1. Initial program 56.3%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 98.9%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 98.9%

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

   5. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right), \color{blue}{\left(R \cdot \lambda_2\right)}\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right), \left(\color{blue}{R} \cdot \lambda_2\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\phi_1 + \phi_2\right)\right)\right), \left(R \cdot \lambda_2\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \left(R \cdot \lambda_2\right)\right) \]

      9. *-lowering-*.f64 13.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \mathsf{*.f64}\left(R, \color{blue}{\lambda_2}\right)\right) \]

   7. Simplified 13.2%

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

   8. Taylor expanded in phi2 around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(R \cdot \lambda_2\right), \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \]

      5. *-lowering-*.f64 15.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right) \]

   10. Simplified 15.9%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.4 \cdot 10^{+190}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(0 - \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 3.8 \cdot 10^{-151}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot \phi_1\right) \cdot \left(R \cdot \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 28.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right)\\ \mathbf{if}\;\lambda_1 \leq -3.3 \cdot 10^{+190}:\\ \;\;\;\;R \cdot \left(t\_0 \cdot \left(0 - \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 3.8 \cdot 10^{-151}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(R \cdot \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 phi1))))
   (if (<= lambda1 -3.3e+190)
     (* R (* t_0 (- 0.0 lambda1)))
     (if (<= lambda1 3.8e-151) (* R (- phi2 phi1)) (* t_0 (* R lambda2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * phi1));
	double tmp;
	if (lambda1 <= -3.3e+190) {
		tmp = R * (t_0 * (0.0 - lambda1));
	} else if (lambda1 <= 3.8e-151) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = t_0 * (R * lambda2);
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((0.5d0 * phi1))
    if (lambda1 <= (-3.3d+190)) then
        tmp = r * (t_0 * (0.0d0 - lambda1))
    else if (lambda1 <= 3.8d-151) then
        tmp = r * (phi2 - phi1)
    else
        tmp = t_0 * (r * lambda2)
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((0.5 * phi1));
	double tmp;
	if (lambda1 <= -3.3e+190) {
		tmp = R * (t_0 * (0.0 - lambda1));
	} else if (lambda1 <= 3.8e-151) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = t_0 * (R * lambda2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((0.5 * phi1))
	tmp = 0
	if lambda1 <= -3.3e+190:
		tmp = R * (t_0 * (0.0 - lambda1))
	elif lambda1 <= 3.8e-151:
		tmp = R * (phi2 - phi1)
	else:
		tmp = t_0 * (R * lambda2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * phi1))
	tmp = 0.0
	if (lambda1 <= -3.3e+190)
		tmp = Float64(R * Float64(t_0 * Float64(0.0 - lambda1)));
	elseif (lambda1 <= 3.8e-151)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(t_0 * Float64(R * lambda2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((0.5 * phi1));
	tmp = 0.0;
	if (lambda1 <= -3.3e+190)
		tmp = R * (t_0 * (0.0 - lambda1));
	elseif (lambda1 <= 3.8e-151)
		tmp = R * (phi2 - phi1);
	else
		tmp = t_0 * (R * lambda2);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda1, -3.3e+190], N[(R * N[(t$95$0 * N[(0.0 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 3.8e-151], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(R * lambda2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -3.3e190

   1. Initial program 47.5%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 95.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   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. Add Preprocessing

   5. Taylor expanded in lambda1 around -inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{neg}\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \lambda_1\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(-1 \cdot \color{blue}{\lambda_1}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right), \color{blue}{\left(-1 \cdot \lambda_1\right)}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right), \left(\color{blue}{-1} \cdot \lambda_1\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\phi_1 + \phi_2\right)\right)\right), \left(-1 \cdot \lambda_1\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \left(-1 \cdot \lambda_1\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) \]

      10. neg-lowering-neg.f64 60.4%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \mathsf{neg.f64}\left(\lambda_1\right)\right)\right) \]

   7. Simplified 60.4%

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

   8. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}, \mathsf{neg.f64}\left(\lambda_1\right)\right)\right) \]
   9. Step-by-step derivation

      1. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right), \mathsf{neg.f64}\left(\color{blue}{\lambda_1}\right)\right)\right) \]

      2. *-lowering-*.f64 53.6%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right), \mathsf{neg.f64}\left(\lambda_1\right)\right)\right) \]

   10. Simplified 53.6%

       \[\leadsto R \cdot \left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right)} \cdot \left(-\lambda_1\right)\right) \]

   if -3.3e190 < lambda1 < 3.7999999999999997e-151

   1. Initial program 63.6%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 98.9%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 98.9%

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

   5. Taylor expanded in phi1 around -inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{neg}\left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(0 - \color{blue}{\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \color{blue}{\left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \color{blue}{\left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(1 + \left(\mathsf{neg}\left(\frac{\phi_2}{\phi_1}\right)\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(1 - \color{blue}{\frac{\phi_2}{\phi_1}}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\phi_2}{\phi_1}\right)}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 37.2%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right)\right)\right)\right) \]

   7. Simplified 37.2%

      \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 - \frac{\phi_2}{\phi_1}\right)\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(-1 \cdot \phi_1 - -1 \cdot \phi_2\right)}\right) \]
   9. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(\phi_1 + \color{blue}{\left(\mathsf{neg}\left(\phi_2\right)\right)}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(\phi_1 + -1 \cdot \color{blue}{\phi_2}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(-1 \cdot \phi_2 + \color{blue}{\phi_1}\right)\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(-1 \cdot \phi_2\right) + \color{blue}{-1 \cdot \phi_1}\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\left(\mathsf{neg}\left(-1 \cdot \phi_2\right)\right) + \color{blue}{-1} \cdot \phi_1\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) + -1 \cdot \phi_1\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 + \color{blue}{-1} \cdot \phi_1\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 + \left(\mathsf{neg}\left(\phi_1\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 - \color{blue}{\phi_1}\right)\right) \]

      11. --lowering--.f64 40.5%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right) \]

   10. Simplified 40.5%

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

   if 3.7999999999999997e-151 < lambda1

   1. Initial program 56.3%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 98.9%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 98.9%

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

   5. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right), \color{blue}{\left(R \cdot \lambda_2\right)}\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right), \left(\color{blue}{R} \cdot \lambda_2\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\phi_1 + \phi_2\right)\right)\right), \left(R \cdot \lambda_2\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \left(R \cdot \lambda_2\right)\right) \]

      9. *-lowering-*.f64 13.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \mathsf{*.f64}\left(R, \color{blue}{\lambda_2}\right)\right) \]

   7. Simplified 13.2%

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

   8. Taylor expanded in phi2 around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(R \cdot \lambda_2\right), \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \]

      5. *-lowering-*.f64 15.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right) \]

   10. Simplified 15.9%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.3 \cdot 10^{+190}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(0 - \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 3.8 \cdot 10^{-151}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot \phi_1\right) \cdot \left(R \cdot \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.2 \cdot 10^{+179}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot \phi_1\right) \cdot \left(R \cdot \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 5.2e+179)
   (* R (- phi2 phi1))
   (* (cos (* 0.5 phi1)) (* R lambda2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 5.2e+179) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = cos((0.5 * phi1)) * (R * 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 <= 5.2d+179) then
        tmp = r * (phi2 - phi1)
    else
        tmp = cos((0.5d0 * phi1)) * (r * 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 <= 5.2e+179) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = Math.cos((0.5 * phi1)) * (R * lambda2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 5.2e+179:
		tmp = R * (phi2 - phi1)
	else:
		tmp = math.cos((0.5 * phi1)) * (R * lambda2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 5.2e+179)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(cos(Float64(0.5 * phi1)) * Float64(R * lambda2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 5.2e+179)
		tmp = R * (phi2 - phi1);
	else
		tmp = cos((0.5 * phi1)) * (R * lambda2);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 5.2e+179], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(R * lambda2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 5.2000000000000004e179

   1. Initial program 61.6%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 98.9%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 98.9%

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

   5. Taylor expanded in phi1 around -inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{neg}\left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(0 - \color{blue}{\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \color{blue}{\left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \color{blue}{\left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(1 + \left(\mathsf{neg}\left(\frac{\phi_2}{\phi_1}\right)\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(1 - \color{blue}{\frac{\phi_2}{\phi_1}}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\phi_2}{\phi_1}\right)}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 30.6%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right)\right)\right)\right) \]

   7. Simplified 30.6%

      \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 - \frac{\phi_2}{\phi_1}\right)\right)} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(-1 \cdot \phi_1 - -1 \cdot \phi_2\right)}\right) \]
   9. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(\phi_1 + \color{blue}{\left(\mathsf{neg}\left(\phi_2\right)\right)}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(\phi_1 + -1 \cdot \color{blue}{\phi_2}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(-1 \cdot \phi_2 + \color{blue}{\phi_1}\right)\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(-1 \cdot \phi_2\right) + \color{blue}{-1 \cdot \phi_1}\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\left(\mathsf{neg}\left(-1 \cdot \phi_2\right)\right) + \color{blue}{-1} \cdot \phi_1\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) + -1 \cdot \phi_1\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 + \color{blue}{-1} \cdot \phi_1\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 + \left(\mathsf{neg}\left(\phi_1\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 - \color{blue}{\phi_1}\right)\right) \]

      11. --lowering--.f64 34.3%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right) \]

   10. Simplified 34.3%

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

   if 5.2000000000000004e179 < lambda2

   1. Initial program 41.2%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 96.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 96.0%

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

   5. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right), \color{blue}{\left(R \cdot \lambda_2\right)}\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right), \left(\color{blue}{R} \cdot \lambda_2\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\phi_1 + \phi_2\right)\right)\right), \left(R \cdot \lambda_2\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \left(R \cdot \lambda_2\right)\right) \]

      9. *-lowering-*.f64 56.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\phi_1, \phi_2\right)\right)\right), \mathsf{*.f64}\left(R, \color{blue}{\lambda_2}\right)\right) \]

   7. Simplified 56.7%

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

   8. Taylor expanded in phi2 around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(R \cdot \lambda_2\right), \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \]

      5. *-lowering-*.f64 67.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(R, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right) \]

   10. Simplified 67.2%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.2 \cdot 10^{+179}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot \phi_1\right) \cdot \left(R \cdot \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.0% accurate, 32.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;0 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -5.5e-9) (- 0.0 (* R phi1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5.5e-9) {
		tmp = 0.0 - (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 (phi1 <= (-5.5d-9)) then
        tmp = 0.0d0 - (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 (phi1 <= -5.5e-9) {
		tmp = 0.0 - (R * phi1);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -5.5e-9:
		tmp = 0.0 - (R * phi1)
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -5.5e-9)
		tmp = Float64(0.0 - Float64(R * phi1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -5.5e-9)
		tmp = 0.0 - (R * phi1);
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -5.5e-9], N[(0.0 - N[(R * phi1), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5.4999999999999996e-9

   1. Initial program 56.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 97.1%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 97.1%

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

   5. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{R \cdot \phi_1} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(R \cdot \phi_1\right)}\right) \]

      4. *-lowering-*.f64 58.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(R, \color{blue}{\phi_1}\right)\right) \]

   7. Simplified 58.7%

      \[\leadsto \color{blue}{0 - R \cdot \phi_1} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(R \cdot \phi_1\right)\right) \]

      3. *-lowering-*.f64 58.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(R, \phi_1\right)\right) \]

   9. Applied egg-rr 58.7%

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

   if -5.4999999999999996e-9 < phi1

   1. Initial program 60.6%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 99.1%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 99.1%

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

   5. Taylor expanded in phi2 around inf

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

      1. *-lowering-*.f64 24.5%

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\phi_2}\right) \]

   7. Simplified 24.5%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;0 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.1% 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 59.7%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   9. --lowering--.f64 98.6%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

3. Simplified 98.6%

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

5. Taylor expanded in phi1 around -inf

   \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{neg}\left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(0 - \color{blue}{\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \color{blue}{\left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \color{blue}{\left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right)\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(1 + \left(\mathsf{neg}\left(\frac{\phi_2}{\phi_1}\right)\right)\right)\right)\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(1 - \color{blue}{\frac{\phi_2}{\phi_1}}\right)\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\phi_2}{\phi_1}\right)}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 29.4%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right)\right)\right)\right) \]

7. Simplified 29.4%

   \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 - \frac{\phi_2}{\phi_1}\right)\right)} \]

8. Taylor expanded in phi1 around 0

   \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(-1 \cdot \phi_1 - -1 \cdot \phi_2\right)}\right) \]
9. Step-by-step derivation

   1. distribute-lft-out-- N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(\phi_1 + \color{blue}{\left(\mathsf{neg}\left(\phi_2\right)\right)}\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(\phi_1 + -1 \cdot \color{blue}{\phi_2}\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(-1 \cdot \phi_2 + \color{blue}{\phi_1}\right)\right)\right) \]

   5. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(-1 \cdot \left(-1 \cdot \phi_2\right) + \color{blue}{-1 \cdot \phi_1}\right)\right) \]

   6. neg-mul-1 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(\left(\mathsf{neg}\left(-1 \cdot \phi_2\right)\right) + \color{blue}{-1} \cdot \phi_1\right)\right) \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) + -1 \cdot \phi_1\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 + \color{blue}{-1} \cdot \phi_1\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 + \left(\mathsf{neg}\left(\phi_1\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(R, \left(\phi_2 - \color{blue}{\phi_1}\right)\right) \]

   11. --lowering--.f64 33.2%

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{\_.f64}\left(\phi_2, \color{blue}{\phi_1}\right)\right) \]

10. Simplified 33.2%

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

Alternative 12: 17.7% 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 59.7%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   9. --lowering--.f64 98.6%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

3. Simplified 98.6%

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

5. Taylor expanded in phi2 around inf

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

   1. *-lowering-*.f64 22.3%

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\phi_2}\right) \]

7. Simplified 22.3%

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024160 
(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))))))