Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.6% → 99.9%

Time: 17.9s

Alternatives: 12

Speedup: 1.6×

• 30.7% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
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 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 lambda1 around inf

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

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

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

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

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

   5. /-lowering-/.f64 91.7%

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

7. Simplified 91.7%

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

   1. div-inv N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. cos-sum N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   17. *-lowering-*.f64 93.6%

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

9. Applied egg-rr 93.6%

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

10. Taylor expanded in lambda1 around 0

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

    1. mul-1-neg N/A

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

    2. sub-neg N/A

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

    3. --lowering--.f64 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 93.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0115:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.0115)
   (* R (hypot (* (- lambda1 lambda2) (cos (/ phi1 2.0))) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (/ phi2 2.0))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.0115) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.0115) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi1 / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.0115:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi1 / 2.0))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi2 / 2.0))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.0115)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 / 2.0))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 / 2.0))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -0.0115)
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 / 2.0))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 / 2.0))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.0115], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.0115

   1. Initial program 62.5%

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

      1. *-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 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in phi1 around inf

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

   7. Simplified 92.4%

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

   if -0.0115 < phi1

   1. Initial program 65.0%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 0

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

   7. Simplified 95.1%

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

Alternative 3: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
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 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. Final simplification 97.1%

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

Alternative 4: 90.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

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

7. Simplified 89.7%

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

Alternative 5: 31.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4 \cdot 10^{-288}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_2 \leq 0.122:\\ \;\;\;\;\lambda_1 \cdot \left(\left(\frac{R \cdot \lambda_2}{\lambda_1} - R\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\phi_1}{\phi_2}\right) \cdot \left(R \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 4e-288)
   (* phi1 (- (* R (/ phi2 phi1)) R))
   (if (<= phi2 0.122)
     (*
      lambda1
      (* (- (/ (* R lambda2) lambda1) R) (cos (* 0.5 (+ phi2 phi1)))))
     (* (- 1.0 (/ phi1 phi2)) (* R phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4e-288) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi2 <= 0.122) {
		tmp = lambda1 * ((((R * lambda2) / lambda1) - R) * cos((0.5 * (phi2 + phi1))));
	} else {
		tmp = (1.0 - (phi1 / phi2)) * (R * phi2);
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 4d-288) then
        tmp = phi1 * ((r * (phi2 / phi1)) - r)
    else if (phi2 <= 0.122d0) then
        tmp = lambda1 * ((((r * lambda2) / lambda1) - r) * cos((0.5d0 * (phi2 + phi1))))
    else
        tmp = (1.0d0 - (phi1 / phi2)) * (r * phi2)
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4e-288) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi2 <= 0.122) {
		tmp = lambda1 * ((((R * lambda2) / lambda1) - R) * Math.cos((0.5 * (phi2 + phi1))));
	} else {
		tmp = (1.0 - (phi1 / phi2)) * (R * phi2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 4e-288:
		tmp = phi1 * ((R * (phi2 / phi1)) - R)
	elif phi2 <= 0.122:
		tmp = lambda1 * ((((R * lambda2) / lambda1) - R) * math.cos((0.5 * (phi2 + phi1))))
	else:
		tmp = (1.0 - (phi1 / phi2)) * (R * phi2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 4e-288)
		tmp = Float64(phi1 * Float64(Float64(R * Float64(phi2 / phi1)) - R));
	elseif (phi2 <= 0.122)
		tmp = Float64(lambda1 * Float64(Float64(Float64(Float64(R * lambda2) / lambda1) - R) * cos(Float64(0.5 * Float64(phi2 + phi1)))));
	else
		tmp = Float64(Float64(1.0 - Float64(phi1 / phi2)) * Float64(R * phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 4e-288)
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	elseif (phi2 <= 0.122)
		tmp = lambda1 * ((((R * lambda2) / lambda1) - R) * cos((0.5 * (phi2 + phi1))));
	else
		tmp = (1.0 - (phi1 / phi2)) * (R * phi2);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 4e-288], N[(phi1 * N[(N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 0.122], N[(lambda1 * N[(N[(N[(N[(R * lambda2), $MachinePrecision] / lambda1), $MachinePrecision] - R), $MachinePrecision] * N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision] * N[(R * phi2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 4.00000000000000023e-288

   1. Initial program 64.0%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. associate-/l* N/A

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

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

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

      10. /-lowering-/.f64 19.6%

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

   7. Simplified 19.6%

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

   if 4.00000000000000023e-288 < phi2 < 0.122

   1. Initial program 71.0%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

   7. Simplified 35.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

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

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 35.7%

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

   9. Applied egg-rr 35.7%

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

   if 0.122 < phi2

   1. Initial program 59.8%

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

      1. *-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 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in phi2 around inf

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

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

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

      2. associate-+r+ N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. associate-/l* N/A

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

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

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

   7. Simplified 59.9%

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

   9. Applied egg-rr 70.5%

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

   10. Taylor expanded in phi1 around inf

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

       1. /-lowering-/.f64 68.6%

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

   12. Simplified 68.6%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4 \cdot 10^{-288}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_2 \leq 0.122:\\ \;\;\;\;\lambda_1 \cdot \left(\left(\frac{R \cdot \lambda_2}{\lambda_1} - R\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\phi_1}{\phi_2}\right) \cdot \left(R \cdot \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 31.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -6.2 \cdot 10^{+170}:\\ \;\;\;\;R \cdot \left(\left(0 - \lambda_1\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 9.8 \cdot 10^{+148}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -6.2e+170)
   (* R (* (- 0.0 lambda1) (cos (* 0.5 (+ phi2 phi1)))))
   (if (<= lambda2 9.8e+148)
     (* phi2 (- R (/ (* R phi1) phi2)))
     (* R lambda2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -6.2e+170) {
		tmp = R * ((0.0 - lambda1) * cos((0.5 * (phi2 + phi1))));
	} else if (lambda2 <= 9.8e+148) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else {
		tmp = 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 <= (-6.2d+170)) then
        tmp = r * ((0.0d0 - lambda1) * cos((0.5d0 * (phi2 + phi1))))
    else if (lambda2 <= 9.8d+148) then
        tmp = phi2 * (r - ((r * phi1) / phi2))
    else
        tmp = 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 <= -6.2e+170) {
		tmp = R * ((0.0 - lambda1) * Math.cos((0.5 * (phi2 + phi1))));
	} else if (lambda2 <= 9.8e+148) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= -6.2e+170:
		tmp = R * ((0.0 - lambda1) * math.cos((0.5 * (phi2 + phi1))))
	elif lambda2 <= 9.8e+148:
		tmp = phi2 * (R - ((R * phi1) / phi2))
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -6.2e+170)
		tmp = Float64(R * Float64(Float64(0.0 - lambda1) * cos(Float64(0.5 * Float64(phi2 + phi1)))));
	elseif (lambda2 <= 9.8e+148)
		tmp = Float64(phi2 * Float64(R - Float64(Float64(R * phi1) / phi2)));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= -6.2e+170)
		tmp = R * ((0.0 - lambda1) * cos((0.5 * (phi2 + phi1))));
	elseif (lambda2 <= 9.8e+148)
		tmp = phi2 * (R - ((R * phi1) / phi2));
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -6.2e+170], N[(R * N[(N[(0.0 - lambda1), $MachinePrecision] * N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 9.8e+148], N[(phi2 * N[(R - N[(N[(R * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -6.2e170

   1. Initial program 46.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 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 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 13.7%

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

   7. Simplified 13.7%

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

   if -6.2e170 < lambda2 < 9.8e148

   1. Initial program 68.8%

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

      1. *-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 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in phi2 around inf

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. *-lowering-*.f64 37.5%

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

   7. Simplified 37.5%

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

   if 9.8e148 < lambda2

   1. Initial program 52.3%

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

      1. *-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 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in lambda2 around inf

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. +-lowering-+.f64 52.5%

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

   7. Simplified 52.5%

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

   8. Taylor expanded in phi2 around 0

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

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

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

      2. *-lowering-*.f64 48.2%

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

   10. Simplified 48.2%

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

   11. Taylor expanded in phi1 around 0

       \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 61.2%

          \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\lambda_2}\right) \]

   13. Simplified 61.2%

       \[\leadsto \color{blue}{R \cdot \lambda_2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -6.2 \cdot 10^{+170}:\\ \;\;\;\;R \cdot \left(\left(0 - \lambda_1\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 9.8 \cdot 10^{+148}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 28.7% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.4 \cdot 10^{-282}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-57}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(\left(\phi_1 \cdot \phi_1\right) \cdot 0.0026041666666666665 + -0.125\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.4e-282)
   (* phi1 (- (* R (/ phi2 phi1)) R))
   (if (<= phi2 3.8e-57)
     (*
      R
      (*
       lambda2
       (+
        (* (* phi1 phi1) (+ (* (* phi1 phi1) 0.0026041666666666665) -0.125))
        1.0)))
     (* R (* phi2 (- 1.0 (/ phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.4e-282) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi2 <= 3.8e-57) {
		tmp = R * (lambda2 * (((phi1 * phi1) * (((phi1 * phi1) * 0.0026041666666666665) + -0.125)) + 1.0));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 2.4d-282) then
        tmp = phi1 * ((r * (phi2 / phi1)) - r)
    else if (phi2 <= 3.8d-57) then
        tmp = r * (lambda2 * (((phi1 * phi1) * (((phi1 * phi1) * 0.0026041666666666665d0) + (-0.125d0))) + 1.0d0))
    else
        tmp = r * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.4e-282) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi2 <= 3.8e-57) {
		tmp = R * (lambda2 * (((phi1 * phi1) * (((phi1 * phi1) * 0.0026041666666666665) + -0.125)) + 1.0));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.4e-282:
		tmp = phi1 * ((R * (phi2 / phi1)) - R)
	elif phi2 <= 3.8e-57:
		tmp = R * (lambda2 * (((phi1 * phi1) * (((phi1 * phi1) * 0.0026041666666666665) + -0.125)) + 1.0))
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.4e-282)
		tmp = Float64(phi1 * Float64(Float64(R * Float64(phi2 / phi1)) - R));
	elseif (phi2 <= 3.8e-57)
		tmp = Float64(R * Float64(lambda2 * Float64(Float64(Float64(phi1 * phi1) * Float64(Float64(Float64(phi1 * phi1) * 0.0026041666666666665) + -0.125)) + 1.0)));
	else
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.4e-282)
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	elseif (phi2 <= 3.8e-57)
		tmp = R * (lambda2 * (((phi1 * phi1) * (((phi1 * phi1) * 0.0026041666666666665) + -0.125)) + 1.0));
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.4e-282], N[(phi1 * N[(N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 3.8e-57], N[(R * N[(lambda2 * N[(N[(N[(phi1 * phi1), $MachinePrecision] * N[(N[(N[(phi1 * phi1), $MachinePrecision] * 0.0026041666666666665), $MachinePrecision] + -0.125), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 2.39999999999999997e-282

   1. Initial program 62.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. associate-/l* N/A

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

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

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

      10. /-lowering-/.f64 19.2%

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

   7. Simplified 19.2%

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

   if 2.39999999999999997e-282 < phi2 < 3.7999999999999997e-57

   1. Initial program 73.0%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in lambda2 around inf

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. +-lowering-+.f64 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in phi2 around 0

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

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

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

      2. *-lowering-*.f64 20.8%

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

   10. Simplified 20.8%

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

   11. Taylor expanded in phi1 around 0

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\color{blue}{\left(1 + {\phi_1}^{2} \cdot \left(\frac{1}{384} \cdot {\phi_1}^{2} - \frac{1}{8}\right)\right)}, \lambda_2\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({\phi_1}^{2} \cdot \left(\frac{1}{384} \cdot {\phi_1}^{2} - \frac{1}{8}\right)\right)\right), \lambda_2\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({\phi_1}^{2}\right), \left(\frac{1}{384} \cdot {\phi_1}^{2} - \frac{1}{8}\right)\right)\right), \lambda_2\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\phi_1 \cdot \phi_1\right), \left(\frac{1}{384} \cdot {\phi_1}^{2} - \frac{1}{8}\right)\right)\right), \lambda_2\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \phi_1\right), \left(\frac{1}{384} \cdot {\phi_1}^{2} - \frac{1}{8}\right)\right)\right), \lambda_2\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \phi_1\right), \left(\frac{1}{384} \cdot {\phi_1}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)\right)\right), \lambda_2\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \phi_1\right), \left(\frac{1}{384} \cdot {\phi_1}^{2} + \frac{-1}{8}\right)\right)\right), \lambda_2\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \phi_1\right), \mathsf{+.f64}\left(\left(\frac{1}{384} \cdot {\phi_1}^{2}\right), \frac{-1}{8}\right)\right)\right), \lambda_2\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \phi_1\right), \mathsf{+.f64}\left(\left({\phi_1}^{2} \cdot \frac{1}{384}\right), \frac{-1}{8}\right)\right)\right), \lambda_2\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\phi_1}^{2}\right), \frac{1}{384}\right), \frac{-1}{8}\right)\right)\right), \lambda_2\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\phi_1 \cdot \phi_1\right), \frac{1}{384}\right), \frac{-1}{8}\right)\right)\right), \lambda_2\right)\right) \]

       11. *-lowering-*.f64 30.2%

           \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \phi_1\right), \frac{1}{384}\right), \frac{-1}{8}\right)\right)\right), \lambda_2\right)\right) \]

   13. Simplified 30.2%

       \[\leadsto R \cdot \left(\color{blue}{\left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot \left(\left(\phi_1 \cdot \phi_1\right) \cdot 0.0026041666666666665 + -0.125\right)\right)} \cdot \lambda_2\right) \]

   if 3.7999999999999997e-57 < phi2

   1. Initial program 62.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in phi2 around inf

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. /-lowering-/.f64 64.9%

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

   7. Simplified 64.9%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.4 \cdot 10^{-282}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-57}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(\left(\phi_1 \cdot \phi_1\right) \cdot 0.0026041666666666665 + -0.125\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 28.9% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.1 \cdot 10^{-282}:\\ \;\;\;\;\phi_1 \cdot \left(0 - R\right)\\ \mathbf{elif}\;\phi_2 \leq 1.75 \cdot 10^{-54}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.1e-282)
   (* phi1 (- 0.0 R))
   (if (<= phi2 1.75e-54) (* R lambda2) (* R (* phi2 (- 1.0 (/ phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.1e-282) {
		tmp = phi1 * (0.0 - R);
	} else if (phi2 <= 1.75e-54) {
		tmp = R * lambda2;
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 2.1d-282) then
        tmp = phi1 * (0.0d0 - r)
    else if (phi2 <= 1.75d-54) then
        tmp = r * lambda2
    else
        tmp = r * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.1e-282) {
		tmp = phi1 * (0.0 - R);
	} else if (phi2 <= 1.75e-54) {
		tmp = R * lambda2;
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.1e-282:
		tmp = phi1 * (0.0 - R)
	elif phi2 <= 1.75e-54:
		tmp = R * lambda2
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.1e-282)
		tmp = Float64(phi1 * Float64(0.0 - R));
	elseif (phi2 <= 1.75e-54)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.1e-282)
		tmp = phi1 * (0.0 - R);
	elseif (phi2 <= 1.75e-54)
		tmp = R * lambda2;
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.1e-282], N[(phi1 * N[(0.0 - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.75e-54], N[(R * lambda2), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 2.10000000000000012e-282

   1. Initial program 62.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in lambda1 around inf

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. /-lowering-/.f64 90.2%

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

   7. Simplified 90.2%

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

      1. div-inv N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. cos-sum N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      17. *-lowering-*.f64 92.2%

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

   9. Applied egg-rr 92.2%

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

   10. Taylor expanded in phi1 around -inf

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 21.3%

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

   12. Simplified 21.3%

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

   if 2.10000000000000012e-282 < phi2 < 1.74999999999999991e-54

   1. Initial program 73.0%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in lambda2 around inf

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. +-lowering-+.f64 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in phi2 around 0

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

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

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

      2. *-lowering-*.f64 20.8%

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

   10. Simplified 20.8%

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

   11. Taylor expanded in phi1 around 0

       \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 22.3%

          \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\lambda_2}\right) \]

   13. Simplified 22.3%

       \[\leadsto \color{blue}{R \cdot \lambda_2} \]

   if 1.74999999999999991e-54 < phi2

   1. Initial program 62.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in phi2 around inf

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. /-lowering-/.f64 64.9%

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

   7. Simplified 64.9%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.1 \cdot 10^{-282}:\\ \;\;\;\;\phi_1 \cdot \left(0 - R\right)\\ \mathbf{elif}\;\phi_2 \leq 1.75 \cdot 10^{-54}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.1% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.05 \cdot 10^{+149}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1.05e+149) (* phi2 (- R (/ (* R phi1) phi2))) (* R lambda2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.05e+149) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else {
		tmp = 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 <= 1.05d+149) then
        tmp = phi2 * (r - ((r * phi1) / phi2))
    else
        tmp = 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 <= 1.05e+149) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.05e+149:
		tmp = phi2 * (R - ((R * phi1) / phi2))
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.05e+149)
		tmp = Float64(phi2 * Float64(R - Float64(Float64(R * phi1) / phi2)));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1.05e+149)
		tmp = phi2 * (R - ((R * phi1) / phi2));
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.05e+149], N[(phi2 * N[(R - N[(N[(R * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.0500000000000001e149

   1. Initial program 65.8%

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

      1. *-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 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in phi2 around inf

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. *-lowering-*.f64 35.9%

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

   7. Simplified 35.9%

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

   if 1.0500000000000001e149 < lambda2

   1. Initial program 52.3%

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

      1. *-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 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in lambda2 around inf

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. +-lowering-+.f64 52.5%

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

   7. Simplified 52.5%

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

   8. Taylor expanded in phi2 around 0

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

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

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

      2. *-lowering-*.f64 48.2%

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

   10. Simplified 48.2%

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

   11. Taylor expanded in phi1 around 0

       \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 61.2%

          \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\lambda_2}\right) \]

   13. Simplified 61.2%

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

Alternative 10: 27.7% accurate, 25.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.05 \cdot 10^{-282}:\\ \;\;\;\;\phi_1 \cdot \left(0 - R\right)\\ \mathbf{elif}\;\phi_2 \leq 2.3 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.05e-282)
   (* phi1 (- 0.0 R))
   (if (<= phi2 2.3e-28) (* R lambda2) (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.05e-282) {
		tmp = phi1 * (0.0 - R);
	} else if (phi2 <= 2.3e-28) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 2.05d-282) then
        tmp = phi1 * (0.0d0 - r)
    else if (phi2 <= 2.3d-28) then
        tmp = r * lambda2
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.05e-282) {
		tmp = phi1 * (0.0 - R);
	} else if (phi2 <= 2.3e-28) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.05e-282:
		tmp = phi1 * (0.0 - R)
	elif phi2 <= 2.3e-28:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.05e-282)
		tmp = Float64(phi1 * Float64(0.0 - R));
	elseif (phi2 <= 2.3e-28)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.05e-282)
		tmp = phi1 * (0.0 - R);
	elseif (phi2 <= 2.3e-28)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.05e-282], N[(phi1 * N[(0.0 - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.3e-28], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 2.04999999999999989e-282

   1. Initial program 62.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in lambda1 around inf

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. /-lowering-/.f64 90.2%

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

   7. Simplified 90.2%

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

      1. div-inv N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. cos-sum N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      17. *-lowering-*.f64 92.2%

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

   9. Applied egg-rr 92.2%

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

   10. Taylor expanded in phi1 around -inf

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 21.3%

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

   12. Simplified 21.3%

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

   if 2.04999999999999989e-282 < phi2 < 2.29999999999999986e-28

   1. Initial program 75.4%

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

      1. *-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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in lambda2 around inf

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. +-lowering-+.f64 19.4%

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

   7. Simplified 19.4%

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

   8. Taylor expanded in phi2 around 0

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

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

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

      2. *-lowering-*.f64 19.4%

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

   10. Simplified 19.4%

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

   11. Taylor expanded in phi1 around 0

       \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 20.7%

          \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\lambda_2}\right) \]

   13. Simplified 20.7%

       \[\leadsto \color{blue}{R \cdot \lambda_2} \]

   if 2.29999999999999986e-28 < phi2

   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 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in phi2 around inf

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

      1. *-lowering-*.f64 60.8%

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

   7. Simplified 60.8%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.05 \cdot 10^{-282}:\\ \;\;\;\;\phi_1 \cdot \left(0 - R\right)\\ \mathbf{elif}\;\phi_2 \leq 2.3 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.6% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.05 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.05e-28) (* R lambda2) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.05e-28) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 1.05d-28) then
        tmp = r * lambda2
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.05e-28) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.05e-28:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.05e-28)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.05e-28)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.05e-28], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.05000000000000003e-28

   1. Initial program 65.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 lambda2 around inf

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. +-lowering-+.f64 19.5%

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

   7. Simplified 19.5%

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

   8. Taylor expanded in phi2 around 0

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

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

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

      2. *-lowering-*.f64 18.0%

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

   10. Simplified 18.0%

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

   11. Taylor expanded in phi1 around 0

       \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 16.3%

          \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\lambda_2}\right) \]

   13. Simplified 16.3%

       \[\leadsto \color{blue}{R \cdot \lambda_2} \]

   if 1.05000000000000003e-28 < phi2

   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 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in phi2 around inf

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

      1. *-lowering-*.f64 60.8%

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

   7. Simplified 60.8%

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

Alternative 12: 14.5% accurate, 109.7× speedup

Math

\[\begin{array}{l} \\ R \cdot \lambda_2 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * lambda2
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * lambda2), $MachinePrecision]

Derivation

1. Initial program 64.3%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
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 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 lambda2 around inf

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

   1. *-commutative N/A

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

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

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

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

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

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

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

   5. +-lowering-+.f64 17.7%

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

7. Simplified 17.7%

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

8. Taylor expanded in phi2 around 0

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

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

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

   2. *-lowering-*.f64 14.7%

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

10. Simplified 14.7%

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

11. Taylor expanded in phi1 around 0

    \[\leadsto \color{blue}{R \cdot \lambda_2} \]
12. Step-by-step derivation

    1. *-lowering-*.f64 13.8%

       \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\lambda_2}\right) \]

13. Simplified 13.8%

    \[\leadsto \color{blue}{R \cdot \lambda_2} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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))))))