Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.1% → 94.5%

Time: 25.7s

Alternatives: 14

Speedup: 3.0×

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

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.04 \cdot 10^{+33}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

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 1.04e+33)
   (* R (hypot (* (- lambda1 lambda2) (cos (* phi1 0.5))) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (* phi2 0.5))) (- phi1 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 (phi2 <= 1.04e+33) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	}
	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 <= 1.04e+33) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 * 0.5))), (phi1 - 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 phi2 <= 1.04e+33:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi1 * 0.5))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi2 * 0.5))), (phi1 - 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 (phi2 <= 1.04e+33)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

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 <= 1.04e+33)
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - 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[phi2, 1.04e+33], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.0400000000000001e33

   1. Initial program 66.2%

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

      1. hypot-define 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in phi2 around 0 96.3%

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

   if 1.0400000000000001e33 < phi2

   1. Initial program 54.3%

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

      1. hypot-define 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in phi1 around 0 89.4%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.04 \cdot 10^{+33}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

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
  (hypot (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (- phi1 phi2))))

C

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

Java

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 * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0))), (phi1 - phi2));
}

Python

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

Julia

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

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))), (phi1 - phi2));
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[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.6%

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

   1. hypot-define 96.3%

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

3. Simplified 96.3%

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

Alternative 3: 90.5% accurate, 1.6× speedup

Math

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

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 (hypot (* (- lambda1 lambda2) (cos (* phi1 0.5))) (- phi1 phi2))))

C

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

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 * Math.hypot(((lambda1 - lambda2) * Math.cos((phi1 * 0.5))), (phi1 - phi2));
}

Python

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

Julia

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

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
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[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.6%

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

   1. hypot-define 96.3%

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

3. Simplified 96.3%

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

5. Taylor expanded in phi2 around 0 93.5%

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

6. Final simplification 93.5%

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

Alternative 4: 60.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.2 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -2.7 \cdot 10^{-304}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(-1 + \frac{\lambda_2}{\lambda_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1\right)\\ \end{array} \end{array} \]

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 (<= phi1 -2.2e-28)
   (* R (- phi2 phi1))
   (if (<= phi1 -2.7e-304)
     (* R (* lambda1 (+ -1.0 (/ lambda2 lambda1))))
     (* R (hypot phi2 lambda1)))))

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 <= -2.2e-28) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= -2.7e-304) {
		tmp = R * (lambda1 * (-1.0 + (lambda2 / lambda1)));
	} else {
		tmp = R * hypot(phi2, lambda1);
	}
	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 <= -2.2e-28) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= -2.7e-304) {
		tmp = R * (lambda1 * (-1.0 + (lambda2 / lambda1)));
	} else {
		tmp = R * Math.hypot(phi2, lambda1);
	}
	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 <= -2.2e-28:
		tmp = R * (phi2 - phi1)
	elif phi1 <= -2.7e-304:
		tmp = R * (lambda1 * (-1.0 + (lambda2 / lambda1)))
	else:
		tmp = R * math.hypot(phi2, lambda1)
	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 <= -2.2e-28)
		tmp = Float64(R * Float64(phi2 - phi1));
	elseif (phi1 <= -2.7e-304)
		tmp = Float64(R * Float64(lambda1 * Float64(-1.0 + Float64(lambda2 / lambda1))));
	else
		tmp = Float64(R * hypot(phi2, lambda1));
	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 <= -2.2e-28)
		tmp = R * (phi2 - phi1);
	elseif (phi1 <= -2.7e-304)
		tmp = R * (lambda1 * (-1.0 + (lambda2 / lambda1)));
	else
		tmp = R * hypot(phi2, lambda1);
	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, -2.2e-28], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -2.7e-304], N[(R * N[(lambda1 * N[(-1.0 + N[(lambda2 / lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + lambda1 ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.19999999999999996e-28

   1. Initial program 52.5%

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

      1. hypot-define 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in phi1 around -inf 64.1%

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

      1. associate-*r* 64.1%

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

      2. mul-1-neg 64.1%

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

      3. associate-*r/ 64.1%

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

      4. mul-1-neg 64.1%

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

      5. *-commutative 64.1%

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

   7. Simplified 64.1%

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

   8. Taylor expanded in phi1 around 0 64.1%

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

      1. +-commutative 64.1%

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

      2. mul-1-neg 64.1%

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

      3. unsub-neg 64.1%

         \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   10. Simplified 64.1%

       \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   11. Taylor expanded in R around 0 70.1%

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

   if -2.19999999999999996e-28 < phi1 < -2.7000000000000001e-304

   1. Initial program 72.5%

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

      1. hypot-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in phi2 around 0 94.7%

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

      1. expm1-log1p-u 94.7%

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

      2. expm1-undefine 94.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 \cdot \phi_1\right)} - 1\right)}, \phi_1 - \phi_2\right) \]

      3. *-commutative 94.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{\phi_1 \cdot 0.5}\right)} - 1\right), \phi_1 - \phi_2\right) \]

   7. Applied egg-rr 94.7%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\phi_1 \cdot 0.5\right)} - 1\right)}, \phi_1 - \phi_2\right) \]
   8. Step-by-step derivation

      1. expm1-define 94.7%

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

   9. Simplified 94.7%

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

   10. Taylor expanded in phi1 around 0 68.2%

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

       1. +-commutative 68.2%

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

       2. unpow2 68.2%

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

       3. unpow2 68.2%

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

       4. hypot-define 91.6%

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

   12. Simplified 91.6%

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

   13. Taylor expanded in lambda1 around -inf 37.6%

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

       1. associate-*r* 37.6%

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

       2. mul-1-neg 37.6%

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

       3. associate-*r/ 37.6%

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

       4. neg-mul-1 37.6%

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

   15. Simplified 37.6%

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

   if -2.7000000000000001e-304 < phi1

   1. Initial program 67.2%

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

      1. hypot-define 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in phi2 around 0 91.5%

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

      1. expm1-log1p-u 84.7%

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

      2. expm1-undefine 84.6%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 \cdot \phi_1\right)} - 1\right)}, \phi_1 - \phi_2\right) \]

      3. *-commutative 84.6%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{\phi_1 \cdot 0.5}\right)} - 1\right), \phi_1 - \phi_2\right) \]

   7. Applied egg-rr 84.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\phi_1 \cdot 0.5\right)} - 1\right)}, \phi_1 - \phi_2\right) \]
   8. Step-by-step derivation

      1. expm1-define 84.7%

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

   9. Simplified 84.7%

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

   10. Taylor expanded in phi1 around 0 50.2%

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

       1. +-commutative 50.2%

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

       2. unpow2 50.2%

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

       3. unpow2 50.2%

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

       4. hypot-define 66.2%

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

   12. Simplified 66.2%

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

   13. Taylor expanded in lambda2 around 0 32.8%

       \[\leadsto \color{blue}{R \cdot \sqrt{{\lambda_1}^{2} + {\phi_2}^{2}}} \]
   14. Step-by-step derivation

       1. +-commutative 32.8%

          \[\leadsto R \cdot \sqrt{\color{blue}{{\phi_2}^{2} + {\lambda_1}^{2}}} \]

       2. unpow2 32.8%

          \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\lambda_1}^{2}} \]

       3. unpow2 32.8%

          \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\lambda_1 \cdot \lambda_1}} \]

       4. hypot-define 43.6%

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

   15. Simplified 43.6%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.2 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -2.7 \cdot 10^{-304}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(-1 + \frac{\lambda_2}{\lambda_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{+21}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

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 -2.8e+21)
   (* 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 <= -2.8e+21) {
		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 <= -2.8e+21) {
		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 <= -2.8e+21:
		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 <= -2.8e+21)
		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 <= -2.8e+21)
		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, -2.8e+21], 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 < -2.8e21

   1. Initial program 52.6%

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

      1. hypot-define 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in phi1 around -inf 63.8%

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

      1. associate-*r* 63.8%

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

      2. mul-1-neg 63.8%

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

      3. associate-*r/ 63.8%

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

      4. mul-1-neg 63.8%

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

      5. *-commutative 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in phi1 around 0 63.8%

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

      1. +-commutative 63.8%

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

      2. mul-1-neg 63.8%

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

      3. unsub-neg 63.8%

         \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   10. Simplified 63.8%

       \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   11. Taylor expanded in R around 0 70.3%

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

   if -2.8e21 < phi1

   1. Initial program 68.5%

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

      1. hypot-define 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in phi2 around 0 92.4%

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

   6. Taylor expanded in phi1 around 0 55.7%

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

      1. unpow2 55.7%

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

      2. unpow2 55.7%

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

      3. hypot-define 74.8%

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

   8. Simplified 74.8%

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

Alternative 6: 80.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.45 \cdot 10^{+55}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

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 (<= phi2 1.45e+55)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* 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 (phi2 <= 1.45e+55) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	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 <= 1.45e+55) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} 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 phi2 <= 1.45e+55:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	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 (phi2 <= 1.45e+55)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	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 (phi2 <= 1.45e+55)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	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[phi2, 1.45e+55], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.4499999999999999e55

   1. Initial program 66.6%

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

      1. hypot-define 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in phi1 around 0 86.7%

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

   6. Taylor expanded in phi2 around 0 58.3%

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

      1. unpow2 58.3%

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

      2. unpow2 58.3%

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

      3. hypot-define 75.3%

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

   8. Simplified 75.3%

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

   if 1.4499999999999999e55 < phi2

   1. Initial program 52.7%

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

      1. hypot-define 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in phi1 around -inf 74.4%

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

      1. associate-*r* 74.4%

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

      2. mul-1-neg 74.4%

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

      3. associate-*r/ 74.4%

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

      4. mul-1-neg 74.4%

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

      5. *-commutative 74.4%

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

   7. Simplified 74.4%

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

   8. Taylor expanded in phi1 around 0 67.6%

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

      1. +-commutative 67.6%

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

      2. mul-1-neg 67.6%

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

      3. unsub-neg 67.6%

         \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   10. Simplified 67.6%

       \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   11. Taylor expanded in R around 0 69.4%

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

Alternative 7: 85.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right) \end{array} \]

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 (hypot (- lambda1 lambda2) (- phi1 phi2))))

C

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

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 * Math.hypot((lambda1 - lambda2), (phi1 - phi2));
}

Python

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

Julia

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

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
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[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.6%

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

   1. hypot-define 96.3%

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

3. Simplified 96.3%

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

5. Taylor expanded in phi2 around 0 93.5%

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

6. Taylor expanded in phi1 around 0 89.1%

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

Alternative 8: 49.8% accurate, 13.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -4.3 \cdot 10^{+24}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.4 \cdot 10^{-159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq -1.85 \cdot 10^{-202}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_1 \leq -4.6 \cdot 10^{-269}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* R (- lambda1))))
   (if (<= phi1 -4.3e+24)
     (* R (- phi1))
     (if (<= phi1 -1.4e-159)
       t_0
       (if (<= phi1 -1.85e-202)
         (* R lambda2)
         (if (<= phi1 -4.6e-269) t_0 (* 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 t_0 = R * -lambda1;
	double tmp;
	if (phi1 <= -4.3e+24) {
		tmp = R * -phi1;
	} else if (phi1 <= -1.4e-159) {
		tmp = t_0;
	} else if (phi1 <= -1.85e-202) {
		tmp = R * lambda2;
	} else if (phi1 <= -4.6e-269) {
		tmp = t_0;
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = r * -lambda1
    if (phi1 <= (-4.3d+24)) then
        tmp = r * -phi1
    else if (phi1 <= (-1.4d-159)) then
        tmp = t_0
    else if (phi1 <= (-1.85d-202)) then
        tmp = r * lambda2
    else if (phi1 <= (-4.6d-269)) then
        tmp = t_0
    else
        tmp = 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 t_0 = R * -lambda1;
	double tmp;
	if (phi1 <= -4.3e+24) {
		tmp = R * -phi1;
	} else if (phi1 <= -1.4e-159) {
		tmp = t_0;
	} else if (phi1 <= -1.85e-202) {
		tmp = R * lambda2;
	} else if (phi1 <= -4.6e-269) {
		tmp = t_0;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = R * -lambda1
	tmp = 0
	if phi1 <= -4.3e+24:
		tmp = R * -phi1
	elif phi1 <= -1.4e-159:
		tmp = t_0
	elif phi1 <= -1.85e-202:
		tmp = R * lambda2
	elif phi1 <= -4.6e-269:
		tmp = t_0
	else:
		tmp = R * phi2
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * Float64(-lambda1))
	tmp = 0.0
	if (phi1 <= -4.3e+24)
		tmp = Float64(R * Float64(-phi1));
	elseif (phi1 <= -1.4e-159)
		tmp = t_0;
	elseif (phi1 <= -1.85e-202)
		tmp = Float64(R * lambda2);
	elseif (phi1 <= -4.6e-269)
		tmp = t_0;
	else
		tmp = 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)
	t_0 = R * -lambda1;
	tmp = 0.0;
	if (phi1 <= -4.3e+24)
		tmp = R * -phi1;
	elseif (phi1 <= -1.4e-159)
		tmp = t_0;
	elseif (phi1 <= -1.85e-202)
		tmp = R * lambda2;
	elseif (phi1 <= -4.6e-269)
		tmp = t_0;
	else
		tmp = 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_] := Block[{t$95$0 = N[(R * (-lambda1)), $MachinePrecision]}, If[LessEqual[phi1, -4.3e+24], N[(R * (-phi1)), $MachinePrecision], If[LessEqual[phi1, -1.4e-159], t$95$0, If[LessEqual[phi1, -1.85e-202], N[(R * lambda2), $MachinePrecision], If[LessEqual[phi1, -4.6e-269], t$95$0, N[(R * phi2), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -4.29999999999999987e24

   1. Initial program 52.6%

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

      1. hypot-define 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in phi1 around -inf 67.6%

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

      1. mul-1-neg 67.6%

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

   7. Simplified 67.6%

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

   if -4.29999999999999987e24 < phi1 < -1.4000000000000001e-159 or -1.84999999999999995e-202 < phi1 < -4.5999999999999999e-269

   1. Initial program 61.7%

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

      1. hypot-define 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in phi2 around 0 92.9%

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

      1. expm1-log1p-u 92.5%

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

      2. expm1-undefine 92.5%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 \cdot \phi_1\right)} - 1\right)}, \phi_1 - \phi_2\right) \]

      3. *-commutative 92.5%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{\phi_1 \cdot 0.5}\right)} - 1\right), \phi_1 - \phi_2\right) \]

   7. Applied egg-rr 92.5%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\phi_1 \cdot 0.5\right)} - 1\right)}, \phi_1 - \phi_2\right) \]
   8. Step-by-step derivation

      1. expm1-define 92.5%

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

   9. Simplified 92.5%

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

   10. Taylor expanded in phi1 around 0 55.6%

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

       1. +-commutative 55.6%

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

       2. unpow2 55.6%

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

       3. unpow2 55.6%

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

       4. hypot-define 86.8%

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

   12. Simplified 86.8%

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

   13. Taylor expanded in lambda1 around -inf 27.2%

       \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \lambda_1\right)} \]
   14. Step-by-step derivation

       1. associate-*r* 27.2%

          \[\leadsto \color{blue}{\left(-1 \cdot R\right) \cdot \lambda_1} \]

       2. neg-mul-1 27.2%

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

   15. Simplified 27.2%

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

   if -1.4000000000000001e-159 < phi1 < -1.84999999999999995e-202

   1. Initial program 92.7%

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

      1. hypot-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in phi2 around 0 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-undefine 100.0%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 \cdot \phi_1\right)} - 1\right)}, \phi_1 - \phi_2\right) \]

      3. *-commutative 100.0%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{\phi_1 \cdot 0.5}\right)} - 1\right), \phi_1 - \phi_2\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\phi_1 \cdot 0.5\right)} - 1\right)}, \phi_1 - \phi_2\right) \]
   8. Step-by-step derivation

      1. expm1-define 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in phi1 around 0 92.7%

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

       1. +-commutative 92.7%

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

       2. unpow2 92.7%

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

       3. unpow2 92.7%

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

       4. hypot-define 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in lambda2 around inf 25.5%

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

   if -4.5999999999999999e-269 < phi1

   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. Step-by-step derivation

      1. hypot-define 95.5%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in phi2 around inf 11.0%

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

      1. *-commutative 11.0%

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

   7. Simplified 11.0%

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

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.3 \cdot 10^{+24}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.4 \cdot 10^{-159}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.85 \cdot 10^{-202}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_1 \leq -4.6 \cdot 10^{-269}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.4% accurate, 17.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.25 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 5 \cdot 10^{-282}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(-1 + \frac{\lambda_2}{\lambda_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \end{array} \]

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 -1.25e-23)
   (* R (- phi2 phi1))
   (if (<= phi1 5e-282)
     (* R (* lambda1 (+ -1.0 (/ lambda2 lambda1))))
     (* 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 (phi1 <= -1.25e-23) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= 5e-282) {
		tmp = R * (lambda1 * (-1.0 + (lambda2 / lambda1)));
	} else {
		tmp = 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 (phi1 <= (-1.25d-23)) then
        tmp = r * (phi2 - phi1)
    else if (phi1 <= 5d-282) then
        tmp = r * (lambda1 * ((-1.0d0) + (lambda2 / lambda1)))
    else
        tmp = 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 (phi1 <= -1.25e-23) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= 5e-282) {
		tmp = R * (lambda1 * (-1.0 + (lambda2 / lambda1)));
	} else {
		tmp = 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 phi1 <= -1.25e-23:
		tmp = R * (phi2 - phi1)
	elif phi1 <= 5e-282:
		tmp = R * (lambda1 * (-1.0 + (lambda2 / lambda1)))
	else:
		tmp = 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 (phi1 <= -1.25e-23)
		tmp = Float64(R * Float64(phi2 - phi1));
	elseif (phi1 <= 5e-282)
		tmp = Float64(R * Float64(lambda1 * Float64(-1.0 + Float64(lambda2 / lambda1))));
	else
		tmp = Float64(phi2 * Float64(R - Float64(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 (phi1 <= -1.25e-23)
		tmp = R * (phi2 - phi1);
	elseif (phi1 <= 5e-282)
		tmp = R * (lambda1 * (-1.0 + (lambda2 / lambda1)));
	else
		tmp = 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[phi1, -1.25e-23], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 5e-282], N[(R * N[(lambda1 * N[(-1.0 + N[(lambda2 / lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(phi2 * N[(R - N[(phi1 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.2500000000000001e-23

   1. Initial program 52.5%

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

      1. hypot-define 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in phi1 around -inf 64.1%

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

      1. associate-*r* 64.1%

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

      2. mul-1-neg 64.1%

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

      3. associate-*r/ 64.1%

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

      4. mul-1-neg 64.1%

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

      5. *-commutative 64.1%

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

   7. Simplified 64.1%

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

   8. Taylor expanded in phi1 around 0 64.1%

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

      1. +-commutative 64.1%

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

      2. mul-1-neg 64.1%

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

      3. unsub-neg 64.1%

         \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   10. Simplified 64.1%

       \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   11. Taylor expanded in R around 0 70.1%

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

   if -1.2500000000000001e-23 < phi1 < 5.0000000000000001e-282

   1. Initial program 72.2%

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

      1. hypot-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in phi2 around 0 95.2%

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

      1. expm1-log1p-u 95.2%

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

      2. expm1-undefine 95.2%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 \cdot \phi_1\right)} - 1\right)}, \phi_1 - \phi_2\right) \]

      3. *-commutative 95.2%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{\phi_1 \cdot 0.5}\right)} - 1\right), \phi_1 - \phi_2\right) \]

   7. Applied egg-rr 95.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\phi_1 \cdot 0.5\right)} - 1\right)}, \phi_1 - \phi_2\right) \]
   8. Step-by-step derivation

      1. expm1-define 95.2%

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

   9. Simplified 95.2%

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

   10. Taylor expanded in phi1 around 0 68.2%

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

       1. +-commutative 68.2%

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

       2. unpow2 68.2%

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

       3. unpow2 68.2%

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

       4. hypot-define 92.4%

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

   12. Simplified 92.4%

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

   13. Taylor expanded in lambda1 around -inf 37.4%

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

       1. associate-*r* 37.4%

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

       2. mul-1-neg 37.4%

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

       3. associate-*r/ 37.4%

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

       4. neg-mul-1 37.4%

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

   15. Simplified 37.4%

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

   if 5.0000000000000001e-282 < phi1

   1. Initial program 67.1%

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

      1. hypot-define 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in phi2 around inf 8.3%

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

      1. mul-1-neg 8.3%

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

      2. unsub-neg 8.3%

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

      3. *-commutative 8.3%

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

      4. associate-/l* 9.2%

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

   7. Simplified 9.2%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.25 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 5 \cdot 10^{-282}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(-1 + \frac{\lambda_2}{\lambda_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.9% accurate, 17.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.2 \cdot 10^{-24}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 4.3 \cdot 10^{-288}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \end{array} \]

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 -4.2e-24)
   (* R (- phi2 phi1))
   (if (<= phi1 4.3e-288)
     (* lambda2 (- R (/ (* R lambda1) lambda2)))
     (* 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 (phi1 <= -4.2e-24) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= 4.3e-288) {
		tmp = lambda2 * (R - ((R * lambda1) / lambda2));
	} else {
		tmp = 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 (phi1 <= (-4.2d-24)) then
        tmp = r * (phi2 - phi1)
    else if (phi1 <= 4.3d-288) then
        tmp = lambda2 * (r - ((r * lambda1) / lambda2))
    else
        tmp = 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 (phi1 <= -4.2e-24) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= 4.3e-288) {
		tmp = lambda2 * (R - ((R * lambda1) / lambda2));
	} else {
		tmp = 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 phi1 <= -4.2e-24:
		tmp = R * (phi2 - phi1)
	elif phi1 <= 4.3e-288:
		tmp = lambda2 * (R - ((R * lambda1) / lambda2))
	else:
		tmp = 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 (phi1 <= -4.2e-24)
		tmp = Float64(R * Float64(phi2 - phi1));
	elseif (phi1 <= 4.3e-288)
		tmp = Float64(lambda2 * Float64(R - Float64(Float64(R * lambda1) / lambda2)));
	else
		tmp = Float64(phi2 * Float64(R - Float64(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 (phi1 <= -4.2e-24)
		tmp = R * (phi2 - phi1);
	elseif (phi1 <= 4.3e-288)
		tmp = lambda2 * (R - ((R * lambda1) / lambda2));
	else
		tmp = 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[phi1, -4.2e-24], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4.3e-288], N[(lambda2 * N[(R - N[(N[(R * lambda1), $MachinePrecision] / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(phi2 * N[(R - N[(phi1 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -4.1999999999999999e-24

   1. Initial program 52.5%

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

      1. hypot-define 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in phi1 around -inf 64.1%

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

      1. associate-*r* 64.1%

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

      2. mul-1-neg 64.1%

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

      3. associate-*r/ 64.1%

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

      4. mul-1-neg 64.1%

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

      5. *-commutative 64.1%

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

   7. Simplified 64.1%

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

   8. Taylor expanded in phi1 around 0 64.1%

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

      1. +-commutative 64.1%

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

      2. mul-1-neg 64.1%

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

      3. unsub-neg 64.1%

         \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   10. Simplified 64.1%

       \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   11. Taylor expanded in R around 0 70.1%

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

   if -4.1999999999999999e-24 < phi1 < 4.29999999999999976e-288

   1. Initial program 71.7%

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

      1. hypot-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in phi2 around 0 95.1%

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

      1. expm1-log1p-u 95.1%

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

      2. expm1-undefine 95.1%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 \cdot \phi_1\right)} - 1\right)}, \phi_1 - \phi_2\right) \]

      3. *-commutative 95.1%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{\phi_1 \cdot 0.5}\right)} - 1\right), \phi_1 - \phi_2\right) \]

   7. Applied egg-rr 95.1%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\phi_1 \cdot 0.5\right)} - 1\right)}, \phi_1 - \phi_2\right) \]
   8. Step-by-step derivation

      1. expm1-define 95.1%

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

   9. Simplified 95.1%

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

   10. Taylor expanded in phi1 around 0 67.7%

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

       1. +-commutative 67.7%

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

       2. unpow2 67.7%

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

       3. unpow2 67.7%

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

       4. hypot-define 92.3%

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

   12. Simplified 92.3%

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

   13. Taylor expanded in lambda2 around inf 36.3%

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

       1. associate-*r/ 36.3%

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

       2. associate-*r* 36.3%

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

       3. neg-mul-1 36.3%

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

   15. Simplified 36.3%

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

   if 4.29999999999999976e-288 < phi1

   1. Initial program 67.4%

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

      1. hypot-define 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in phi2 around inf 8.2%

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

      1. mul-1-neg 8.2%

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

      2. unsub-neg 8.2%

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

      3. *-commutative 8.2%

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

      4. associate-/l* 9.2%

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

   7. Simplified 9.2%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.2 \cdot 10^{-24}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 4.3 \cdot 10^{-288}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.9% accurate, 21.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.7 \cdot 10^{+145}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

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 -3.7e+145)
   (* R (- lambda1))
   (if (<= lambda1 2.3e-6) (* R (- phi2 phi1)) (* R 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 (lambda1 <= -3.7e+145) {
		tmp = R * -lambda1;
	} else if (lambda1 <= 2.3e-6) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * lambda2;
	}
	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 <= (-3.7d+145)) then
        tmp = r * -lambda1
    else if (lambda1 <= 2.3d-6) then
        tmp = r * (phi2 - phi1)
    else
        tmp = r * lambda2
    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 <= -3.7e+145) {
		tmp = R * -lambda1;
	} else if (lambda1 <= 2.3e-6) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * 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 lambda1 <= -3.7e+145:
		tmp = R * -lambda1
	elif lambda1 <= 2.3e-6:
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * 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 (lambda1 <= -3.7e+145)
		tmp = Float64(R * Float64(-lambda1));
	elseif (lambda1 <= 2.3e-6)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * 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 (lambda1 <= -3.7e+145)
		tmp = R * -lambda1;
	elseif (lambda1 <= 2.3e-6)
		tmp = R * (phi2 - phi1);
	else
		tmp = R * 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[lambda1, -3.7e+145], N[(R * (-lambda1)), $MachinePrecision], If[LessEqual[lambda1, 2.3e-6], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -3.69999999999999993e145

   1. Initial program 50.6%

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

      1. hypot-define 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in phi2 around 0 80.1%

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

      1. expm1-log1p-u 50.4%

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

      2. expm1-undefine 50.4%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 \cdot \phi_1\right)} - 1\right)}, \phi_1 - \phi_2\right) \]

      3. *-commutative 50.4%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{\phi_1 \cdot 0.5}\right)} - 1\right), \phi_1 - \phi_2\right) \]

   7. Applied egg-rr 50.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\phi_1 \cdot 0.5\right)} - 1\right)}, \phi_1 - \phi_2\right) \]
   8. Step-by-step derivation

      1. expm1-define 50.4%

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

   9. Simplified 50.4%

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

   10. Taylor expanded in phi1 around 0 47.9%

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

       1. +-commutative 47.9%

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

       2. unpow2 47.9%

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

       3. unpow2 47.9%

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

       4. hypot-define 67.5%

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

   12. Simplified 67.5%

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

   13. Taylor expanded in lambda1 around -inf 60.1%

       \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \lambda_1\right)} \]
   14. Step-by-step derivation

       1. associate-*r* 60.1%

          \[\leadsto \color{blue}{\left(-1 \cdot R\right) \cdot \lambda_1} \]

       2. neg-mul-1 60.1%

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

   15. Simplified 60.1%

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

   if -3.69999999999999993e145 < lambda1 < 2.3e-6

   1. Initial program 69.3%

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

      1. hypot-define 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in phi1 around -inf 32.7%

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

      1. associate-*r* 32.7%

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

      2. mul-1-neg 32.7%

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

      3. associate-*r/ 32.7%

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

      4. mul-1-neg 32.7%

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

      5. *-commutative 32.7%

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

   7. Simplified 32.7%

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

   8. Taylor expanded in phi1 around 0 32.1%

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

      1. +-commutative 32.1%

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

      2. mul-1-neg 32.1%

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

      3. unsub-neg 32.1%

         \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   10. Simplified 32.1%

       \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

   11. Taylor expanded in R around 0 33.3%

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

   if 2.3e-6 < lambda1

   1. Initial program 53.9%

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

      1. hypot-define 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in phi2 around 0 93.9%

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

      1. expm1-log1p-u 57.7%

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

      2. expm1-undefine 57.6%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 \cdot \phi_1\right)} - 1\right)}, \phi_1 - \phi_2\right) \]

      3. *-commutative 57.6%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{\phi_1 \cdot 0.5}\right)} - 1\right), \phi_1 - \phi_2\right) \]

   7. Applied egg-rr 57.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\phi_1 \cdot 0.5\right)} - 1\right)}, \phi_1 - \phi_2\right) \]
   8. Step-by-step derivation

      1. expm1-define 57.7%

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

   9. Simplified 57.7%

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

   10. Taylor expanded in phi1 around 0 51.5%

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

       1. +-commutative 51.5%

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

       2. unpow2 51.5%

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

       3. unpow2 51.5%

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

       4. hypot-define 75.9%

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

   12. Simplified 75.9%

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

   13. Taylor expanded in lambda2 around inf 14.4%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.7 \cdot 10^{+145}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.2% accurate, 25.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.6 \cdot 10^{-59}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.15 \cdot 10^{-304}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

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 -6.6e-59)
   (* R (- phi1))
   (if (<= phi1 -1.15e-304) (* R lambda2) (* 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 (phi1 <= -6.6e-59) {
		tmp = R * -phi1;
	} else if (phi1 <= -1.15e-304) {
		tmp = R * lambda2;
	} else {
		tmp = 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 (phi1 <= (-6.6d-59)) then
        tmp = r * -phi1
    else if (phi1 <= (-1.15d-304)) then
        tmp = r * lambda2
    else
        tmp = 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 (phi1 <= -6.6e-59) {
		tmp = R * -phi1;
	} else if (phi1 <= -1.15e-304) {
		tmp = R * lambda2;
	} else {
		tmp = 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 phi1 <= -6.6e-59:
		tmp = R * -phi1
	elif phi1 <= -1.15e-304:
		tmp = R * lambda2
	else:
		tmp = 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 (phi1 <= -6.6e-59)
		tmp = Float64(R * Float64(-phi1));
	elseif (phi1 <= -1.15e-304)
		tmp = Float64(R * lambda2);
	else
		tmp = 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 (phi1 <= -6.6e-59)
		tmp = R * -phi1;
	elseif (phi1 <= -1.15e-304)
		tmp = R * lambda2;
	else
		tmp = 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[phi1, -6.6e-59], N[(R * (-phi1)), $MachinePrecision], If[LessEqual[phi1, -1.15e-304], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -6.59999999999999964e-59

   1. Initial program 53.6%

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

      1. hypot-define 95.5%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in phi1 around -inf 61.3%

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

      1. mul-1-neg 61.3%

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

   7. Simplified 61.3%

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

   if -6.59999999999999964e-59 < phi1 < -1.15e-304

   1. Initial program 73.0%

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

      1. hypot-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in phi2 around 0 94.1%

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

      1. expm1-log1p-u 94.1%

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

      2. expm1-undefine 94.1%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 \cdot \phi_1\right)} - 1\right)}, \phi_1 - \phi_2\right) \]

      3. *-commutative 94.1%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{\phi_1 \cdot 0.5}\right)} - 1\right), \phi_1 - \phi_2\right) \]

   7. Applied egg-rr 94.1%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\phi_1 \cdot 0.5\right)} - 1\right)}, \phi_1 - \phi_2\right) \]
   8. Step-by-step derivation

      1. expm1-define 94.1%

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

   9. Simplified 94.1%

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

   10. Taylor expanded in phi1 around 0 69.9%

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

       1. +-commutative 69.9%

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

       2. unpow2 69.9%

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

       3. unpow2 69.9%

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

       4. hypot-define 92.5%

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

   12. Simplified 92.5%

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

   13. Taylor expanded in lambda2 around inf 28.6%

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

   if -1.15e-304 < phi1

   1. Initial program 67.2%

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

      1. hypot-define 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in phi2 around inf 11.4%

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

      1. *-commutative 11.4%

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

   7. Simplified 11.4%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.6 \cdot 10^{-59}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.15 \cdot 10^{-304}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.8% accurate, 41.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.8 \cdot 10^{+55}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

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 2.8e+55) (* R lambda2) (* 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 (phi2 <= 2.8e+55) {
		tmp = R * lambda2;
	} else {
		tmp = 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 (phi2 <= 2.8d+55) then
        tmp = r * lambda2
    else
        tmp = 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 (phi2 <= 2.8e+55) {
		tmp = R * lambda2;
	} else {
		tmp = 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 phi2 <= 2.8e+55:
		tmp = R * lambda2
	else:
		tmp = 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 (phi2 <= 2.8e+55)
		tmp = Float64(R * lambda2);
	else
		tmp = 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 (phi2 <= 2.8e+55)
		tmp = R * lambda2;
	else
		tmp = 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[phi2, 2.8e+55], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.8000000000000001e55

   1. Initial program 66.6%

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

      1. hypot-define 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in phi2 around 0 95.9%

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

      1. expm1-log1p-u 66.1%

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

      2. expm1-undefine 66.1%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 \cdot \phi_1\right)} - 1\right)}, \phi_1 - \phi_2\right) \]

      3. *-commutative 66.1%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{\phi_1 \cdot 0.5}\right)} - 1\right), \phi_1 - \phi_2\right) \]

   7. Applied egg-rr 66.1%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\phi_1 \cdot 0.5\right)} - 1\right)}, \phi_1 - \phi_2\right) \]
   8. Step-by-step derivation

      1. expm1-define 66.1%

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

   9. Simplified 66.1%

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

   10. Taylor expanded in phi1 around 0 51.5%

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

       1. +-commutative 51.5%

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

       2. unpow2 51.5%

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

       3. unpow2 51.5%

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

       4. hypot-define 67.7%

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

   12. Simplified 67.7%

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

   13. Taylor expanded in lambda2 around inf 17.5%

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

   if 2.8000000000000001e55 < phi2

   1. Initial program 52.7%

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

      1. hypot-define 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in phi2 around inf 56.7%

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

      1. *-commutative 56.7%

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

   7. Simplified 56.7%

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

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.8 \cdot 10^{+55}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 13.3% accurate, 109.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \lambda_2 \end{array} \]

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 lambda2))

C

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

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 * lambda2
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 * lambda2;
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.6%

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

   1. hypot-define 96.3%

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

3. Simplified 96.3%

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

5. Taylor expanded in phi2 around 0 93.5%

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

   1. expm1-log1p-u 61.2%

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

   2. expm1-undefine 61.1%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 \cdot \phi_1\right)} - 1\right)}, \phi_1 - \phi_2\right) \]

   3. *-commutative 61.1%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{\phi_1 \cdot 0.5}\right)} - 1\right), \phi_1 - \phi_2\right) \]

7. Applied egg-rr 61.1%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\phi_1 \cdot 0.5\right)} - 1\right)}, \phi_1 - \phi_2\right) \]
8. Step-by-step derivation

   1. expm1-define 61.2%

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

9. Simplified 61.2%

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

10. Taylor expanded in phi1 around 0 50.3%

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

    1. +-commutative 50.3%

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

    2. unpow2 50.3%

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

    3. unpow2 50.3%

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

    4. hypot-define 67.4%

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

12. Simplified 67.4%

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

13. Taylor expanded in lambda2 around inf 17.1%

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

Reproduce

herbie shell --seed 2024110 
(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))))))