Equirectangular approximation to distance on a great circle

Percentage Accurate: 58.2% → 91.1%

Time: 14.9s

Alternatives: 13

Speedup: 20.5×

• 30.2% 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: 58.2% 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: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2050000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) + \left(\phi_2 \cdot -0.5\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2050000.0)
   (*
    R
    (hypot
     (- phi1 phi2)
     (+
      (* (- lambda1 lambda2) (cos (* phi1 0.5)))
      (* (* phi2 -0.5) (* (- lambda1 lambda2) (sin (* phi1 0.5)))))))
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* phi2 0.5)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2050000.0) {
		tmp = R * hypot((phi1 - phi2), (((lambda1 - lambda2) * cos((phi1 * 0.5))) + ((phi2 * -0.5) * ((lambda1 - lambda2) * sin((phi1 * 0.5))))));
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2050000.0) {
		tmp = R * Math.hypot((phi1 - phi2), (((lambda1 - lambda2) * Math.cos((phi1 * 0.5))) + ((phi2 * -0.5) * ((lambda1 - lambda2) * Math.sin((phi1 * 0.5))))));
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((phi2 * 0.5))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2050000.0:
		tmp = R * math.hypot((phi1 - phi2), (((lambda1 - lambda2) * math.cos((phi1 * 0.5))) + ((phi2 * -0.5) * ((lambda1 - lambda2) * math.sin((phi1 * 0.5))))))
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((phi2 * 0.5))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2050000.0)
		tmp = Float64(R * hypot(Float64(phi1 - phi2), Float64(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5))) + Float64(Float64(phi2 * -0.5) * Float64(Float64(lambda1 - lambda2) * sin(Float64(phi1 * 0.5)))))));
	else
		tmp = Float64(R * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2050000.0)
		tmp = R * hypot((phi1 - phi2), (((lambda1 - lambda2) * cos((phi1 * 0.5))) + ((phi2 * -0.5) * ((lambda1 - lambda2) * sin((phi1 * 0.5))))));
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2050000.0], N[(R * N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(phi2 * -0.5), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.05e6

   1. Initial program 64.6%

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. exp-lft-sqr N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. +-lowering-+.f64 50.5%

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

   4. Applied egg-rr 50.5%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 89.6%

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

   7. Simplified 89.6%

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

   if 2.05e6 < phi2

   1. Initial program 56.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

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

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

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

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

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

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

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

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

      12. --lowering--.f64 80.3%

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

   5. Simplified 80.3%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2050000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) + \left(\phi_2 \cdot -0.5\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1700000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(-0.125 \cdot \left(\phi_2 \cdot \phi_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1700000.0)
   (*
    R
    (hypot
     (- phi1 phi2)
     (+
      (* (- lambda1 lambda2) (cos (* phi1 0.5)))
      (* (- lambda1 lambda2) (* -0.125 (* phi2 phi2))))))
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* phi2 0.5)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1700000.0) {
		tmp = R * hypot((phi1 - phi2), (((lambda1 - lambda2) * cos((phi1 * 0.5))) + ((lambda1 - lambda2) * (-0.125 * (phi2 * phi2)))));
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1700000.0) {
		tmp = R * Math.hypot((phi1 - phi2), (((lambda1 - lambda2) * Math.cos((phi1 * 0.5))) + ((lambda1 - lambda2) * (-0.125 * (phi2 * phi2)))));
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((phi2 * 0.5))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1700000.0:
		tmp = R * math.hypot((phi1 - phi2), (((lambda1 - lambda2) * math.cos((phi1 * 0.5))) + ((lambda1 - lambda2) * (-0.125 * (phi2 * phi2)))))
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((phi2 * 0.5))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1700000.0)
		tmp = Float64(R * hypot(Float64(phi1 - phi2), Float64(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5))) + Float64(Float64(lambda1 - lambda2) * Float64(-0.125 * Float64(phi2 * phi2))))));
	else
		tmp = Float64(R * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1700000.0)
		tmp = R * hypot((phi1 - phi2), (((lambda1 - lambda2) * cos((phi1 * 0.5))) + ((lambda1 - lambda2) * (-0.125 * (phi2 * phi2)))));
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1700000.0], N[(R * N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(-0.125 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.7e6

   1. Initial program 64.6%

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. exp-lft-sqr N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. +-lowering-+.f64 50.5%

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

   4. Applied egg-rr 50.5%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \color{blue}{\left(\phi_2 \cdot \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \frac{-1}{8} \cdot \left(\phi_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right) + \color{blue}{\phi_2 \cdot \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \frac{-1}{8} \cdot \left(\phi_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{+.f64}\left(\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right), \color{blue}{\left(\phi_2 \cdot \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \frac{-1}{8} \cdot \left(\phi_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{+.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \left(\color{blue}{\phi_2} \cdot \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \frac{-1}{8} \cdot \left(\phi_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \left(\color{blue}{\phi_2} \cdot \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \frac{-1}{8} \cdot \left(\phi_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \left(\phi_2 \cdot \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \frac{-1}{8} \cdot \left(\phi_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_1\right)\right)\right), \left(\phi_2 \cdot \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \frac{-1}{8} \cdot \left(\phi_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \left(\phi_2 \cdot \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \frac{-1}{8} \cdot \left(\phi_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{*.f64}\left(\phi_2, \color{blue}{\left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \frac{-1}{8} \cdot \left(\phi_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{*.f64}\left(\phi_2, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \color{blue}{\left(\frac{-1}{8} \cdot \left(\phi_2 \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)\right)\right)\right)\right) \]

   7. Simplified 83.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) + \phi_2 \cdot \left(-0.5 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) + \left(-0.125 \cdot \phi_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\right)}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \color{blue}{\left(\frac{-1}{8} \cdot \left({\phi_2}^{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \left(\left(\frac{-1}{8} \cdot {\phi_2}^{2}\right) \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {\phi_2}^{2}\right), \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({\phi_2}^{2}\right)\right), \left(\color{blue}{\lambda_1} - \lambda_2\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\phi_2 \cdot \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\phi_2, \phi_2\right)\right), \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right) \]

      6. --lowering--.f64 83.1%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\phi_1, \phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_1\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\phi_2, \phi_2\right)\right), \mathsf{\_.f64}\left(\lambda_1, \color{blue}{\lambda_2}\right)\right)\right)\right)\right) \]

   10. Simplified 83.1%

       \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) + \color{blue}{\left(-0.125 \cdot \left(\phi_2 \cdot \phi_2\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right) \]

   if 1.7e6 < phi2

   1. Initial program 56.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

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

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

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

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

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

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

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

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

      12. --lowering--.f64 80.3%

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

   5. Simplified 80.3%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1700000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(-0.125 \cdot \left(\phi_2 \cdot \phi_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.2e-7)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))))
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* phi2 0.5)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.2e-7) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.2e-7) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((phi1 * 0.5))));
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((phi2 * 0.5))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.2e-7:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((phi1 * 0.5))))
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((phi2 * 0.5))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.2e-7)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))));
	else
		tmp = Float64(R * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3.2e-7)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3.2e-7], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.2000000000000001e-7

   1. Initial program 54.9%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. *-lowering-*.f64 73.6%

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

   5. Simplified 73.6%

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

   if -3.2000000000000001e-7 < phi1

   1. Initial program 65.4%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

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

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

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

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

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

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

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

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

      12. --lowering--.f64 79.8%

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

   5. Simplified 79.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2050000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2050000.0)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))))
   (* R (hypot phi2 (- lambda1 lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2050000.0) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2050000.0) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((phi1 * 0.5))));
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2050000.0:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((phi1 * 0.5))))
	else:
		tmp = R * math.hypot(phi2, (lambda1 - lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2050000.0)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2050000.0)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	else
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2050000.0], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.05e6

   1. Initial program 64.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. *-lowering-*.f64 77.1%

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

   5. Simplified 77.1%

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

   if 2.05e6 < phi2

   1. Initial program 56.9%

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. exp-lft-sqr N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. +-lowering-+.f64 40.8%

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

   4. Applied egg-rr 40.8%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_2}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\sqrt{\phi_2 \cdot \phi_2 + \left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)\right) \]

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

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

      5. --lowering--.f64 71.5%

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

   10. Simplified 71.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2050000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.8 \cdot 10^{-45}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 6.8e-45)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (hypot phi2 (- lambda1 lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 6.8e-45) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 6.8e-45) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 6.8e-45:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * math.hypot(phi2, (lambda1 - lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 6.8e-45)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 6.8e-45)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 6.8e-45], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 6.80000000000000008e-45

   1. Initial program 63.8%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. *-lowering-*.f64 77.4%

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

   5. Simplified 77.4%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\phi_1, \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 71.9%

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

   8. Simplified 71.9%

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

   if 6.80000000000000008e-45 < phi2

   1. Initial program 60.0%

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. exp-lft-sqr N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. +-lowering-+.f64 38.2%

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

   4. Applied egg-rr 38.2%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 73.6%

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

   7. Simplified 73.6%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_2}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\sqrt{\phi_2 \cdot \phi_2 + \left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)\right) \]

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

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

      5. --lowering--.f64 69.9%

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

   10. Simplified 69.9%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.8 \cdot 10^{-45}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.5e-7)
   (* R (- phi2 phi1))
   (* R (hypot phi2 (- lambda1 lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.5e-7) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.5e-7) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.5e-7:
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * math.hypot(phi2, (lambda1 - lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.5e-7)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3.5e-7)
		tmp = R * (phi2 - phi1);
	else
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3.5e-7], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.49999999999999984e-7

   1. Initial program 54.9%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
   4. 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. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 60.9%

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

   5. Simplified 60.9%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]
   7. Step-by-step derivation

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

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

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

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

      3. --lowering--.f64 60.9%

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

   8. Simplified 60.9%

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

   if -3.49999999999999984e-7 < phi1

   1. Initial program 65.4%

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. exp-lft-sqr N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. +-lowering-+.f64 48.8%

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

   4. Applied egg-rr 48.8%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 85.4%

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

   7. Simplified 85.4%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_2}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\sqrt{\phi_2 \cdot \phi_2 + \left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}\right)\right) \]

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

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

      5. --lowering--.f64 75.0%

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

   10. Simplified 75.0%

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

Alternative 7: 59.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+257}:\\ \;\;\;\;R \cdot \left(\lambda_1 + \left(\frac{0.5 \cdot \left(\phi_1 \cdot \left(\phi_1 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot -0.25\right) + 1\right)\right)\right)}{\lambda_1 - \lambda_2} - \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -4 \cdot 10^{+135}:\\ \;\;\;\;\phi_1 \cdot \left(\phi_2 \cdot \left(\frac{R}{\phi_1} - \frac{R}{\phi_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -1e+257)
   (*
    R
    (+
     lambda1
     (-
      (/
       (*
        0.5
        (*
         phi1
         (*
          phi1
          (+ (* (- lambda1 lambda2) (* (- lambda1 lambda2) -0.25)) 1.0))))
       (- lambda1 lambda2))
      lambda2)))
   (if (<= (- lambda1 lambda2) -4e+135)
     (* phi1 (* phi2 (- (/ R phi1) (/ R phi2))))
     (* R (- phi2 phi1)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -1e+257) {
		tmp = R * (lambda1 + (((0.5 * (phi1 * (phi1 * (((lambda1 - lambda2) * ((lambda1 - lambda2) * -0.25)) + 1.0)))) / (lambda1 - lambda2)) - lambda2));
	} else if ((lambda1 - lambda2) <= -4e+135) {
		tmp = phi1 * (phi2 * ((R / phi1) - (R / phi2)));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((lambda1 - lambda2) <= (-1d+257)) then
        tmp = r * (lambda1 + (((0.5d0 * (phi1 * (phi1 * (((lambda1 - lambda2) * ((lambda1 - lambda2) * (-0.25d0))) + 1.0d0)))) / (lambda1 - lambda2)) - lambda2))
    else if ((lambda1 - lambda2) <= (-4d+135)) then
        tmp = phi1 * (phi2 * ((r / phi1) - (r / phi2)))
    else
        tmp = r * (phi2 - phi1)
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -1e+257) {
		tmp = R * (lambda1 + (((0.5 * (phi1 * (phi1 * (((lambda1 - lambda2) * ((lambda1 - lambda2) * -0.25)) + 1.0)))) / (lambda1 - lambda2)) - lambda2));
	} else if ((lambda1 - lambda2) <= -4e+135) {
		tmp = phi1 * (phi2 * ((R / phi1) - (R / phi2)));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda1 - lambda2) <= -1e+257:
		tmp = R * (lambda1 + (((0.5 * (phi1 * (phi1 * (((lambda1 - lambda2) * ((lambda1 - lambda2) * -0.25)) + 1.0)))) / (lambda1 - lambda2)) - lambda2))
	elif (lambda1 - lambda2) <= -4e+135:
		tmp = phi1 * (phi2 * ((R / phi1) - (R / phi2)))
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -1e+257)
		tmp = Float64(R * Float64(lambda1 + Float64(Float64(Float64(0.5 * Float64(phi1 * Float64(phi1 * Float64(Float64(Float64(lambda1 - lambda2) * Float64(Float64(lambda1 - lambda2) * -0.25)) + 1.0)))) / Float64(lambda1 - lambda2)) - lambda2)));
	elseif (Float64(lambda1 - lambda2) <= -4e+135)
		tmp = Float64(phi1 * Float64(phi2 * Float64(Float64(R / phi1) - Float64(R / phi2))));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -1e+257)
		tmp = R * (lambda1 + (((0.5 * (phi1 * (phi1 * (((lambda1 - lambda2) * ((lambda1 - lambda2) * -0.25)) + 1.0)))) / (lambda1 - lambda2)) - lambda2));
	elseif ((lambda1 - lambda2) <= -4e+135)
		tmp = phi1 * (phi2 * ((R / phi1) - (R / phi2)));
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -1e+257], N[(R * N[(lambda1 + N[(N[(N[(0.5 * N[(phi1 * N[(phi1 * N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -4e+135], N[(phi1 * N[(phi2 * N[(N[(R / phi1), $MachinePrecision] - N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 lambda1 lambda2) < -1.00000000000000003e257

   1. Initial program 54.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. *-lowering-*.f64 79.0%

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

   5. Simplified 79.0%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\left(\lambda_1 + \frac{1}{2} \cdot \frac{{\phi_1}^{2} \cdot \left(1 + \frac{-1}{4} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}\right)}{\lambda_1 - \lambda_2}\right) - \lambda_2\right)}, R\right) \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\lambda_1 + \left(\frac{1}{2} \cdot \frac{{\phi_1}^{2} \cdot \left(1 + \frac{-1}{4} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}\right)}{\lambda_1 - \lambda_2} - \lambda_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\lambda_1, \left(\frac{1}{2} \cdot \frac{{\phi_1}^{2} \cdot \left(1 + \frac{-1}{4} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}\right)}{\lambda_1 - \lambda_2} - \lambda_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\lambda_1, \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{{\phi_1}^{2} \cdot \left(1 + \frac{-1}{4} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2}\right)}{\lambda_1 - \lambda_2}\right), \lambda_2\right)\right), R\right) \]

   8. Simplified 44.2%

      \[\leadsto \color{blue}{\left(\lambda_1 + \left(\frac{0.5 \cdot \left(\left(\phi_1 \cdot \phi_1\right) \cdot \left(1 + -0.25 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{\lambda_1 - \lambda_2} - \lambda_2\right)\right)} \cdot R \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\lambda_1, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\phi_1 \cdot \left(\phi_1 \cdot \left(1 + \frac{-1}{4} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \lambda_2\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\lambda_1, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\phi_1 \cdot \left(1 + \frac{-1}{4} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \phi_1\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \lambda_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\lambda_1, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(\phi_1 \cdot \left(1 + \frac{-1}{4} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), \phi_1\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \lambda_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\lambda_1, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \left(1 + \frac{-1}{4} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), \phi_1\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \lambda_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\lambda_1, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(1, \left(\frac{-1}{4} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), \phi_1\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \lambda_2\right)\right), R\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\lambda_1, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{4} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), \phi_1\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \lambda_2\right)\right), R\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\lambda_1, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(1, \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\frac{-1}{4} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), \phi_1\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \lambda_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\lambda_1, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \left(\frac{-1}{4} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), \phi_1\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \lambda_2\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\lambda_1, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \left(\frac{-1}{4} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), \phi_1\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \lambda_2\right)\right), R\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\lambda_1, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{-1}{4}\right)\right)\right)\right), \phi_1\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \lambda_2\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\lambda_1, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \frac{-1}{4}\right)\right)\right)\right), \phi_1\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \lambda_2\right)\right), R\right) \]

      12. --lowering--.f64 54.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\lambda_1, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \frac{-1}{4}\right)\right)\right)\right), \phi_1\right)\right), \mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \lambda_2\right)\right), R\right) \]

   10. Applied egg-rr 54.7%

       \[\leadsto \left(\lambda_1 + \left(\frac{0.5 \cdot \color{blue}{\left(\left(\phi_1 \cdot \left(1 + \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot -0.25\right)\right)\right) \cdot \phi_1\right)}}{\lambda_1 - \lambda_2} - \lambda_2\right)\right) \cdot R \]

   if -1.00000000000000003e257 < (-.f64 lambda1 lambda2) < -3.99999999999999985e135

   1. Initial program 40.4%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
   4. 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. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 25.6%

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

   5. Simplified 25.6%

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

   6. Taylor expanded in phi2 around inf

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

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

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

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

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

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

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

      4. /-lowering-/.f64 28.1%

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

   8. Simplified 28.1%

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

   if -3.99999999999999985e135 < (-.f64 lambda1 lambda2)

   1. Initial program 68.4%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
   4. 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. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 29.5%

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

   5. Simplified 29.5%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]
   7. Step-by-step derivation

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

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

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

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

      3. --lowering--.f64 27.5%

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

   8. Simplified 27.5%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+257}:\\ \;\;\;\;R \cdot \left(\lambda_1 + \left(\frac{0.5 \cdot \left(\phi_1 \cdot \left(\phi_1 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot -0.25\right) + 1\right)\right)\right)}{\lambda_1 - \lambda_2} - \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -4 \cdot 10^{+135}:\\ \;\;\;\;\phi_1 \cdot \left(\phi_2 \cdot \left(\frac{R}{\phi_1} - \frac{R}{\phi_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.5% accurate, 17.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.5 \cdot 10^{+89}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 2.15 \cdot 10^{+20}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -6.5e+89)
   (* R (- phi2 phi1))
   (if (<= phi1 2.15e+20)
     (* phi2 (- R (/ (* R phi1) phi2)))
     (* phi1 (* R (+ (/ phi2 phi1) -1.0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -6.5e+89) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= 2.15e+20) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else {
		tmp = phi1 * (R * ((phi2 / phi1) + -1.0));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-6.5d+89)) then
        tmp = r * (phi2 - phi1)
    else if (phi1 <= 2.15d+20) then
        tmp = phi2 * (r - ((r * phi1) / phi2))
    else
        tmp = phi1 * (r * ((phi2 / phi1) + (-1.0d0)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -6.5e+89) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= 2.15e+20) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else {
		tmp = phi1 * (R * ((phi2 / phi1) + -1.0));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -6.5e+89:
		tmp = R * (phi2 - phi1)
	elif phi1 <= 2.15e+20:
		tmp = phi2 * (R - ((R * phi1) / phi2))
	else:
		tmp = phi1 * (R * ((phi2 / phi1) + -1.0))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -6.5e+89)
		tmp = Float64(R * Float64(phi2 - phi1));
	elseif (phi1 <= 2.15e+20)
		tmp = Float64(phi2 * Float64(R - Float64(Float64(R * phi1) / phi2)));
	else
		tmp = Float64(phi1 * Float64(R * Float64(Float64(phi2 / phi1) + -1.0)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -6.5e+89)
		tmp = R * (phi2 - phi1);
	elseif (phi1 <= 2.15e+20)
		tmp = phi2 * (R - ((R * phi1) / phi2));
	else
		tmp = phi1 * (R * ((phi2 / phi1) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -6.5e+89], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 2.15e+20], N[(phi2 * N[(R - N[(N[(R * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(phi1 * N[(R * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -6.4999999999999996e89

   1. Initial program 46.5%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
   4. 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. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 70.2%

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

   5. Simplified 70.2%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]
   7. Step-by-step derivation

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

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

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

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

      3. --lowering--.f64 70.2%

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

   8. Simplified 70.2%

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

   if -6.4999999999999996e89 < phi1 < 2.15e20

   1. Initial program 70.5%

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

   3. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{\phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)} \]
   4. 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. *-commutative N/A

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

      7. *-lowering-*.f64 24.9%

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

   5. Simplified 24.9%

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

   if 2.15e20 < phi1

   1. Initial program 54.5%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
   4. 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. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 14.4%

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

   5. Simplified 14.4%

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

   6. Taylor expanded in phi1 around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

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

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

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

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

      15. /-lowering-/.f64 5.9%

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

   8. Simplified 5.9%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.5 \cdot 10^{+89}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 2.15 \cdot 10^{+20}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.6% accurate, 20.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;R \leq 1.7 \cdot 10^{+151}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(\phi_2 \cdot \left(\frac{R}{\phi_1} - \frac{R}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= R 1.7e+151)
   (* R (- phi2 phi1))
   (* phi1 (* phi2 (- (/ R phi1) (/ R phi2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (R <= 1.7e+151) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = phi1 * (phi2 * ((R / phi1) - (R / phi2)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (r <= 1.7d+151) then
        tmp = r * (phi2 - phi1)
    else
        tmp = phi1 * (phi2 * ((r / phi1) - (r / phi2)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (R <= 1.7e+151) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = phi1 * (phi2 * ((R / phi1) - (R / phi2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if R <= 1.7e+151:
		tmp = R * (phi2 - phi1)
	else:
		tmp = phi1 * (phi2 * ((R / phi1) - (R / phi2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (R <= 1.7e+151)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(phi1 * Float64(phi2 * Float64(Float64(R / phi1) - Float64(R / phi2))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (R <= 1.7e+151)
		tmp = R * (phi2 - phi1);
	else
		tmp = phi1 * (phi2 * ((R / phi1) - (R / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[R, 1.7e+151], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(phi1 * N[(phi2 * N[(N[(R / phi1), $MachinePrecision] - N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if R < 1.7e151

   1. Initial program 57.7%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
   4. 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. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 25.8%

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

   5. Simplified 25.8%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]
   7. Step-by-step derivation

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

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

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

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

      3. --lowering--.f64 25.4%

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

   8. Simplified 25.4%

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

   if 1.7e151 < R

   1. Initial program 100.0%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
   4. 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. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 41.4%

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

   5. Simplified 41.4%

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

   6. Taylor expanded in phi2 around inf

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

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

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

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

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

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

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

      4. /-lowering-/.f64 37.7%

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

   8. Simplified 37.7%

      \[\leadsto 0 - \phi_1 \cdot \color{blue}{\left(\phi_2 \cdot \left(\frac{R}{\phi_2} - \frac{R}{\phi_1}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;R \leq 1.7 \cdot 10^{+151}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(\phi_2 \cdot \left(\frac{R}{\phi_1} - \frac{R}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.9% accurate, 23.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 10^{+132}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(\phi_2 \cdot \frac{R}{\phi_1} - R\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1e+132)
   (* R (- phi2 phi1))
   (* phi1 (- (* phi2 (/ R phi1)) R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1e+132) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = phi1 * ((phi2 * (R / phi1)) - R);
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 <= 1d+132) then
        tmp = r * (phi2 - phi1)
    else
        tmp = phi1 * ((phi2 * (r / phi1)) - r)
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1e+132) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = phi1 * ((phi2 * (R / phi1)) - R);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1e+132:
		tmp = R * (phi2 - phi1)
	else:
		tmp = phi1 * ((phi2 * (R / phi1)) - R)
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1e+132)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(phi1 * Float64(Float64(phi2 * Float64(R / phi1)) - R));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1e+132)
		tmp = R * (phi2 - phi1);
	else
		tmp = phi1 * ((phi2 * (R / phi1)) - R);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1e+132], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(phi1 * N[(N[(phi2 * N[(R / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 9.99999999999999991e131

   1. Initial program 63.8%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
   4. 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. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 28.7%

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

   5. Simplified 28.7%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]
   7. Step-by-step derivation

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

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

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

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

      3. --lowering--.f64 26.5%

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

   8. Simplified 26.5%

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

   if 9.99999999999999991e131 < lambda2

   1. Initial program 55.6%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
   4. 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. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 20.5%

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

   5. Simplified 20.5%

      \[\leadsto \color{blue}{0 - \phi_1 \cdot \left(R - \frac{\phi_2 \cdot R}{\phi_1}\right)} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 25.9%

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

   7. Applied egg-rr 25.9%

      \[\leadsto 0 - \phi_1 \cdot \left(R - \color{blue}{\frac{R}{\phi_1} \cdot \phi_2}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 10^{+132}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(\phi_2 \cdot \frac{R}{\phi_1} - R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.3% accurate, 65.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \left(\phi_2 - \phi_1\right) \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* R (- phi2 phi1)))

C

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

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * (phi2 - phi1)
end function

Java

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

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return R * (phi2 - phi1)

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * Float64(phi2 - phi1))
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * (phi2 - phi1);
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in phi1 around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
4. 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. /-lowering-/.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 27.6%

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

5. Simplified 27.6%

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

6. Taylor expanded in phi2 around 0

   \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]
7. Step-by-step derivation

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

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

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

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

   3. --lowering--.f64 26.1%

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

8. Simplified 26.1%

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

Alternative 12: 31.3% accurate, 109.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \phi_2 \cdot R \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* phi2 R))

C

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

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 = phi2 * r
end function

Java

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

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return phi2 * R

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(phi2 * R)
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = phi2 * R;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(phi2 * R), $MachinePrecision]

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in phi2 around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 18.7%

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

5. Simplified 18.7%

   \[\leadsto \color{blue}{\phi_2 \cdot R} \]
6. Add Preprocessing

Alternative 13: 13.8% accurate, 109.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \lambda_1 \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* R lambda1))

C

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

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 * lambda1
end function

Java

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

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return R * lambda1

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * lambda1)
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * lambda1;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * lambda1), $MachinePrecision]

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in phi2 around 0

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

   1. *-commutative N/A

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

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

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. unswap-sqr N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

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

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

   13. *-lowering-*.f64 71.5%

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

5. Simplified 71.5%

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

6. Taylor expanded in lambda1 around inf

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

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

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

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

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

   3. *-lowering-*.f64 18.3%

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

8. Simplified 18.3%

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

9. Taylor expanded in phi1 around 0

   \[\leadsto \color{blue}{R \cdot \lambda_1} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \lambda_1 \cdot \color{blue}{R} \]

    2. *-lowering-*.f64 14.5%

       \[\leadsto \mathsf{*.f64}\left(\lambda_1, \color{blue}{R}\right) \]

11. Simplified 14.5%

    \[\leadsto \color{blue}{\lambda_1 \cdot R} \]

12. Final simplification 14.5%

    \[\leadsto R \cdot \lambda_1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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))))))