Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.2% → 99.4%

Time: 19.3s

Alternatives: 17

Speedup: 3.0×

• 31.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (pow
   (sqrt
    (hypot
     (*
      (-
       (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
       (* (sin (* phi2 0.5)) (sin (* 0.5 phi1))))
      (- lambda1 lambda2))
     (- phi1 phi2)))
   2.0)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * pow(sqrt(hypot((((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1)))) * (lambda1 - lambda2)), (phi1 - phi2))), 2.0);
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.pow(Math.sqrt(Math.hypot((((Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - (Math.sin((phi2 * 0.5)) * Math.sin((0.5 * phi1)))) * (lambda1 - lambda2)), (phi1 - phi2))), 2.0);
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.pow(math.sqrt(math.hypot((((math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))) - (math.sin((phi2 * 0.5)) * math.sin((0.5 * phi1)))) * (lambda1 - lambda2)), (phi1 - phi2))), 2.0)

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * (sqrt(hypot(Float64(Float64(Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(sin(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2))) ^ 2.0))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * (sqrt(hypot((((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1)))) * (lambda1 - lambda2)), (phi1 - phi2))) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 57.0%

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

   1. hypot-define 97.6%

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

3. Simplified 97.6%

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

   1. add-sqr-sqrt 97.1%

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

   2. pow2 97.1%

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

   3. *-commutative 97.1%

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

   4. div-inv 97.1%

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

   5. metadata-eval 97.1%

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

6. Applied egg-rr 97.1%

   \[\leadsto R \cdot \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2}} \]
7. Step-by-step derivation

   1. *-commutative 97.1%

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

   2. +-commutative 97.1%

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

   3. distribute-rgt-in 97.1%

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

   4. cos-sum 99.4%

      \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

8. Applied egg-rr 99.4%

   \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

9. Final simplification 99.4%

   \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2} \]
10. Add Preprocessing

Alternative 2: 93.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.1 \cdot 10^{-40}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3.1e-40)
   (* R (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (* phi2 0.5))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.1e-40) {
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.1e-40) {
		tmp = R * Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3.1e-40:
		tmp = R * math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi2 * 0.5))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3.1e-40)
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 3.1e-40)
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3.1e-40], N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.10000000000000011e-40

   1. Initial program 56.6%

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

      1. hypot-define 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in phi2 around 0 93.4%

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

   if 3.10000000000000011e-40 < phi2

   1. Initial program 57.8%

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

      1. hypot-define 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in phi1 around 0 95.2%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.1 \cdot 10^{-40}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.25 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(-\cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.25e-23)
   (* R (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) (- phi1 phi2)))
   (* R (hypot (* lambda2 (- (cos (* phi2 0.5)))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.25e-23) {
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	} else {
		tmp = R * hypot((lambda2 * -cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.25e-23) {
		tmp = R * Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((lambda2 * -Math.cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.25e-23:
		tmp = R * math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2))
	else:
		tmp = R * math.hypot((lambda2 * -math.cos((phi2 * 0.5))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.25e-23)
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(lambda2 * Float64(-cos(Float64(phi2 * 0.5)))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.25e-23)
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	else
		tmp = R * hypot((lambda2 * -cos((phi2 * 0.5))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.25e-23], N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda2 * (-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 < 1.2500000000000001e-23

   1. Initial program 56.3%

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

      1. hypot-define 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in phi2 around 0 93.6%

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

   if 1.2500000000000001e-23 < phi2

   1. Initial program 58.9%

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

      1. hypot-define 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in phi1 around 0 96.1%

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

   6. Taylor expanded in lambda1 around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. distribute-rgt-neg-in 86.8%

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

   8. Simplified 86.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.25 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(-\cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.25 \cdot 10^{+38}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(-\cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.25e+38)
   (* R (hypot (- lambda1 lambda2) (- phi1 phi2)))
   (* R (hypot (* lambda2 (- (cos (* phi2 0.5)))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.25e+38) {
		tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
	} else {
		tmp = R * hypot((lambda2 * -cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.25e+38) {
		tmp = R * Math.hypot((lambda1 - lambda2), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((lambda2 * -Math.cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.25e+38:
		tmp = R * math.hypot((lambda1 - lambda2), (phi1 - phi2))
	else:
		tmp = R * math.hypot((lambda2 * -math.cos((phi2 * 0.5))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.25e+38)
		tmp = Float64(R * hypot(Float64(lambda1 - lambda2), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(lambda2 * Float64(-cos(Float64(phi2 * 0.5)))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.25e+38)
		tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
	else
		tmp = R * hypot((lambda2 * -cos((phi2 * 0.5))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.25e+38], N[(R * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda2 * (-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 lambda1 < -1.24999999999999992e38

   1. Initial program 53.7%

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

      1. hypot-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in phi1 around 0 94.7%

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

   6. Taylor expanded in phi2 around 0 88.8%

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

   if -1.24999999999999992e38 < lambda1

   1. Initial program 57.6%

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

      1. hypot-define 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in phi1 around 0 92.1%

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

   6. Taylor expanded in lambda1 around 0 79.7%

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

      1. mul-1-neg 79.7%

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

      2. distribute-rgt-neg-in 79.7%

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

   8. Simplified 79.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.25 \cdot 10^{+38}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(-\cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

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

   1. hypot-define 97.6%

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

3. Simplified 97.6%

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

5. Final simplification 97.6%

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

Alternative 6: 71.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.35 \cdot 10^{+58}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.35e+58)
   (* R (hypot (- lambda1 lambda2) phi1))
   (* R (* phi2 (- 1.0 (/ phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.35e+58) {
		tmp = R * hypot((lambda1 - lambda2), phi1);
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.35e+58) {
		tmp = R * Math.hypot((lambda1 - lambda2), phi1);
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.35e+58:
		tmp = R * math.hypot((lambda1 - lambda2), phi1)
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.35e+58)
		tmp = Float64(R * hypot(Float64(lambda1 - lambda2), phi1));
	else
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.35e+58)
		tmp = R * hypot((lambda1 - lambda2), phi1);
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.35e+58], N[(R * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.34999999999999986e58

   1. Initial program 57.4%

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

      1. hypot-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 phi1 around 0 91.5%

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

   6. Taylor expanded in phi2 around 0 46.0%

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

      1. +-commutative 46.0%

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

      2. unpow2 46.0%

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

      3. unpow2 46.0%

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

      4. hypot-define 69.1%

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

   8. Simplified 69.1%

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

   if 2.34999999999999986e58 < phi2

   1. Initial program 55.0%

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

      1. hypot-define 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in phi2 around inf 74.3%

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

      1. mul-1-neg 74.3%

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

      2. unsub-neg 74.3%

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

   7. Simplified 74.3%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 85.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

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

   1. hypot-define 97.6%

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

3. Simplified 97.6%

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

5. Taylor expanded in phi1 around 0 92.6%

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

6. Taylor expanded in phi2 around 0 86.6%

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

Alternative 8: 33.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ \mathbf{if}\;\lambda_1 \leq -3.2 \cdot 10^{+198}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq -1.35 \cdot 10^{+158}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -9.2 \cdot 10^{+148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq 1.5 \cdot 10^{-299} \lor \neg \left(\lambda_1 \leq 1.9 \cdot 10^{-16}\right):\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* R (- lambda1))))
   (if (<= lambda1 -3.2e+198)
     t_0
     (if (<= lambda1 -1.35e+158)
       (* R (* phi1 (+ (/ phi2 phi1) -1.0)))
       (if (<= lambda1 -9.2e+148)
         t_0
         (if (or (<= lambda1 1.5e-299) (not (<= lambda1 1.9e-16)))
           (- (* R phi2) (* R phi1))
           (* phi2 (- R (* phi1 (/ R phi2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * -lambda1;
	double tmp;
	if (lambda1 <= -3.2e+198) {
		tmp = t_0;
	} else if (lambda1 <= -1.35e+158) {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	} else if (lambda1 <= -9.2e+148) {
		tmp = t_0;
	} else if ((lambda1 <= 1.5e-299) || !(lambda1 <= 1.9e-16)) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * -lambda1
    if (lambda1 <= (-3.2d+198)) then
        tmp = t_0
    else if (lambda1 <= (-1.35d+158)) then
        tmp = r * (phi1 * ((phi2 / phi1) + (-1.0d0)))
    else if (lambda1 <= (-9.2d+148)) then
        tmp = t_0
    else if ((lambda1 <= 1.5d-299) .or. (.not. (lambda1 <= 1.9d-16))) then
        tmp = (r * phi2) - (r * phi1)
    else
        tmp = phi2 * (r - (phi1 * (r / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * -lambda1;
	double tmp;
	if (lambda1 <= -3.2e+198) {
		tmp = t_0;
	} else if (lambda1 <= -1.35e+158) {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	} else if (lambda1 <= -9.2e+148) {
		tmp = t_0;
	} else if ((lambda1 <= 1.5e-299) || !(lambda1 <= 1.9e-16)) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = R * -lambda1
	tmp = 0
	if lambda1 <= -3.2e+198:
		tmp = t_0
	elif lambda1 <= -1.35e+158:
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0))
	elif lambda1 <= -9.2e+148:
		tmp = t_0
	elif (lambda1 <= 1.5e-299) or not (lambda1 <= 1.9e-16):
		tmp = (R * phi2) - (R * phi1)
	else:
		tmp = phi2 * (R - (phi1 * (R / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * Float64(-lambda1))
	tmp = 0.0
	if (lambda1 <= -3.2e+198)
		tmp = t_0;
	elseif (lambda1 <= -1.35e+158)
		tmp = Float64(R * Float64(phi1 * Float64(Float64(phi2 / phi1) + -1.0)));
	elseif (lambda1 <= -9.2e+148)
		tmp = t_0;
	elseif ((lambda1 <= 1.5e-299) || !(lambda1 <= 1.9e-16))
		tmp = Float64(Float64(R * phi2) - Float64(R * phi1));
	else
		tmp = Float64(phi2 * Float64(R - Float64(phi1 * Float64(R / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = R * -lambda1;
	tmp = 0.0;
	if (lambda1 <= -3.2e+198)
		tmp = t_0;
	elseif (lambda1 <= -1.35e+158)
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	elseif (lambda1 <= -9.2e+148)
		tmp = t_0;
	elseif ((lambda1 <= 1.5e-299) || ~((lambda1 <= 1.9e-16)))
		tmp = (R * phi2) - (R * phi1);
	else
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * (-lambda1)), $MachinePrecision]}, If[LessEqual[lambda1, -3.2e+198], t$95$0, If[LessEqual[lambda1, -1.35e+158], N[(R * N[(phi1 * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, -9.2e+148], t$95$0, If[Or[LessEqual[lambda1, 1.5e-299], N[Not[LessEqual[lambda1, 1.9e-16]], $MachinePrecision]], N[(N[(R * phi2), $MachinePrecision] - N[(R * phi1), $MachinePrecision]), $MachinePrecision], N[(phi2 * N[(R - N[(phi1 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if lambda1 < -3.1999999999999998e198 or -1.34999999999999989e158 < lambda1 < -9.2000000000000002e148

   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 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in lambda1 around -inf 53.0%

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

      1. mul-1-neg 53.0%

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

      2. associate-*r* 53.0%

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

      3. distribute-lft-neg-in 53.0%

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

      4. +-commutative 53.0%

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

   7. Simplified 53.0%

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

   8. Taylor expanded in phi2 around 0 59.6%

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

      1. associate-*r* 59.6%

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

      2. mul-1-neg 59.6%

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

      3. *-commutative 59.6%

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

   10. Simplified 59.6%

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

   11. Taylor expanded in phi1 around 0 76.8%

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

   if -3.1999999999999998e198 < lambda1 < -1.34999999999999989e158

   1. Initial program 31.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 phi1 around -inf 10.3%

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

      1. mul-1-neg 10.3%

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

      2. distribute-rgt-neg-in 10.3%

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

      3. mul-1-neg 10.3%

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

      4. unsub-neg 10.3%

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

   7. Simplified 10.3%

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

   if -9.2000000000000002e148 < lambda1 < 1.49999999999999992e-299 or 1.90000000000000006e-16 < lambda1

   1. Initial program 55.3%

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

      1. hypot-define 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in phi1 around -inf 27.9%

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

      1. mul-1-neg 27.9%

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

      2. distribute-rgt-neg-in 27.9%

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

      3. mul-1-neg 27.9%

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

      4. unsub-neg 27.9%

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

      5. *-commutative 27.9%

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

      6. associate-/l* 30.1%

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

   7. Simplified 30.1%

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

   8. Taylor expanded in phi1 around 0 29.6%

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

      1. +-commutative 29.6%

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

      2. mul-1-neg 29.6%

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

      3. unsub-neg 29.6%

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

      4. *-commutative 29.6%

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

      5. *-commutative 29.6%

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

   10. Simplified 29.6%

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

   if 1.49999999999999992e-299 < lambda1 < 1.90000000000000006e-16

   1. Initial program 63.0%

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

      1. hypot-define 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in phi2 around inf 43.7%

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

      1. mul-1-neg 43.7%

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

      2. unsub-neg 43.7%

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

      3. *-commutative 43.7%

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

      4. associate-/l* 43.6%

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

   7. Simplified 43.6%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.2 \cdot 10^{+198}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -1.35 \cdot 10^{+158}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -9.2 \cdot 10^{+148}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.5 \cdot 10^{-299} \lor \neg \left(\lambda_1 \leq 1.9 \cdot 10^{-16}\right):\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ \mathbf{if}\;\lambda_1 \leq -3.2 \cdot 10^{+198}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq -1.45 \cdot 10^{+159}:\\ \;\;\;\;\left(R \cdot \phi_1\right) \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\\ \mathbf{elif}\;\lambda_1 \leq -4.5 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq 2.9 \cdot 10^{-297} \lor \neg \left(\lambda_1 \leq 4.8 \cdot 10^{-18}\right):\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* R (- lambda1))))
   (if (<= lambda1 -3.2e+198)
     t_0
     (if (<= lambda1 -1.45e+159)
       (* (* R phi1) (+ (/ phi2 phi1) -1.0))
       (if (<= lambda1 -4.5e+149)
         t_0
         (if (or (<= lambda1 2.9e-297) (not (<= lambda1 4.8e-18)))
           (- (* R phi2) (* R phi1))
           (* phi2 (- R (* phi1 (/ R phi2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * -lambda1;
	double tmp;
	if (lambda1 <= -3.2e+198) {
		tmp = t_0;
	} else if (lambda1 <= -1.45e+159) {
		tmp = (R * phi1) * ((phi2 / phi1) + -1.0);
	} else if (lambda1 <= -4.5e+149) {
		tmp = t_0;
	} else if ((lambda1 <= 2.9e-297) || !(lambda1 <= 4.8e-18)) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * -lambda1
    if (lambda1 <= (-3.2d+198)) then
        tmp = t_0
    else if (lambda1 <= (-1.45d+159)) then
        tmp = (r * phi1) * ((phi2 / phi1) + (-1.0d0))
    else if (lambda1 <= (-4.5d+149)) then
        tmp = t_0
    else if ((lambda1 <= 2.9d-297) .or. (.not. (lambda1 <= 4.8d-18))) then
        tmp = (r * phi2) - (r * phi1)
    else
        tmp = phi2 * (r - (phi1 * (r / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * -lambda1;
	double tmp;
	if (lambda1 <= -3.2e+198) {
		tmp = t_0;
	} else if (lambda1 <= -1.45e+159) {
		tmp = (R * phi1) * ((phi2 / phi1) + -1.0);
	} else if (lambda1 <= -4.5e+149) {
		tmp = t_0;
	} else if ((lambda1 <= 2.9e-297) || !(lambda1 <= 4.8e-18)) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = R * -lambda1
	tmp = 0
	if lambda1 <= -3.2e+198:
		tmp = t_0
	elif lambda1 <= -1.45e+159:
		tmp = (R * phi1) * ((phi2 / phi1) + -1.0)
	elif lambda1 <= -4.5e+149:
		tmp = t_0
	elif (lambda1 <= 2.9e-297) or not (lambda1 <= 4.8e-18):
		tmp = (R * phi2) - (R * phi1)
	else:
		tmp = phi2 * (R - (phi1 * (R / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * Float64(-lambda1))
	tmp = 0.0
	if (lambda1 <= -3.2e+198)
		tmp = t_0;
	elseif (lambda1 <= -1.45e+159)
		tmp = Float64(Float64(R * phi1) * Float64(Float64(phi2 / phi1) + -1.0));
	elseif (lambda1 <= -4.5e+149)
		tmp = t_0;
	elseif ((lambda1 <= 2.9e-297) || !(lambda1 <= 4.8e-18))
		tmp = Float64(Float64(R * phi2) - Float64(R * phi1));
	else
		tmp = Float64(phi2 * Float64(R - Float64(phi1 * Float64(R / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = R * -lambda1;
	tmp = 0.0;
	if (lambda1 <= -3.2e+198)
		tmp = t_0;
	elseif (lambda1 <= -1.45e+159)
		tmp = (R * phi1) * ((phi2 / phi1) + -1.0);
	elseif (lambda1 <= -4.5e+149)
		tmp = t_0;
	elseif ((lambda1 <= 2.9e-297) || ~((lambda1 <= 4.8e-18)))
		tmp = (R * phi2) - (R * phi1);
	else
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * (-lambda1)), $MachinePrecision]}, If[LessEqual[lambda1, -3.2e+198], t$95$0, If[LessEqual[lambda1, -1.45e+159], N[(N[(R * phi1), $MachinePrecision] * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, -4.5e+149], t$95$0, If[Or[LessEqual[lambda1, 2.9e-297], N[Not[LessEqual[lambda1, 4.8e-18]], $MachinePrecision]], N[(N[(R * phi2), $MachinePrecision] - N[(R * phi1), $MachinePrecision]), $MachinePrecision], N[(phi2 * N[(R - N[(phi1 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if lambda1 < -3.1999999999999998e198 or -1.45000000000000007e159 < lambda1 < -4.49999999999999982e149

   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 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in lambda1 around -inf 53.0%

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

      1. mul-1-neg 53.0%

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

      2. associate-*r* 53.0%

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

      3. distribute-lft-neg-in 53.0%

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

      4. +-commutative 53.0%

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

   7. Simplified 53.0%

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

   8. Taylor expanded in phi2 around 0 59.6%

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

      1. associate-*r* 59.6%

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

      2. mul-1-neg 59.6%

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

      3. *-commutative 59.6%

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

   10. Simplified 59.6%

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

   11. Taylor expanded in phi1 around 0 76.8%

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

   if -3.1999999999999998e198 < lambda1 < -1.45000000000000007e159

   1. Initial program 31.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 phi1 around -inf 19.3%

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

      1. mul-1-neg 19.3%

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

      2. distribute-rgt-neg-in 19.3%

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

      3. mul-1-neg 19.3%

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

      4. unsub-neg 19.3%

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

      5. *-commutative 19.3%

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

      6. associate-/l* 10.2%

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

   7. Simplified 10.2%

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

   8. Taylor expanded in R around 0 10.3%

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

      1. associate-*r* 10.3%

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

      2. *-commutative 10.3%

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

      3. sub-neg 10.3%

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

      4. metadata-eval 10.3%

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

   10. Simplified 10.3%

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

   if -4.49999999999999982e149 < lambda1 < 2.89999999999999989e-297 or 4.79999999999999988e-18 < lambda1

   1. Initial program 55.3%

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

      1. hypot-define 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in phi1 around -inf 27.9%

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

      1. mul-1-neg 27.9%

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

      2. distribute-rgt-neg-in 27.9%

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

      3. mul-1-neg 27.9%

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

      4. unsub-neg 27.9%

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

      5. *-commutative 27.9%

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

      6. associate-/l* 30.1%

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

   7. Simplified 30.1%

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

   8. Taylor expanded in phi1 around 0 29.6%

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

      1. +-commutative 29.6%

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

      2. mul-1-neg 29.6%

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

      3. unsub-neg 29.6%

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

      4. *-commutative 29.6%

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

      5. *-commutative 29.6%

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

   10. Simplified 29.6%

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

   if 2.89999999999999989e-297 < lambda1 < 4.79999999999999988e-18

   1. Initial program 63.0%

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

      1. hypot-define 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in phi2 around inf 43.7%

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

      1. mul-1-neg 43.7%

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

      2. unsub-neg 43.7%

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

      3. *-commutative 43.7%

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

      4. associate-/l* 43.6%

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

   7. Simplified 43.6%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.2 \cdot 10^{+198}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -1.45 \cdot 10^{+159}:\\ \;\;\;\;\left(R \cdot \phi_1\right) \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\\ \mathbf{elif}\;\lambda_1 \leq -4.5 \cdot 10^{+149}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 2.9 \cdot 10^{-297} \lor \neg \left(\lambda_1 \leq 4.8 \cdot 10^{-18}\right):\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ \mathbf{if}\;\lambda_1 \leq -7.5 \cdot 10^{+198}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq -9.5 \cdot 10^{+156}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -5.4 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq 3.8 \cdot 10^{-297} \lor \neg \left(\lambda_1 \leq 8 \cdot 10^{-18}\right):\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* R (- lambda1))))
   (if (<= lambda1 -7.5e+198)
     t_0
     (if (<= lambda1 -9.5e+156)
       (* phi1 (* R (+ (/ phi2 phi1) -1.0)))
       (if (<= lambda1 -5.4e+146)
         t_0
         (if (or (<= lambda1 3.8e-297) (not (<= lambda1 8e-18)))
           (- (* R phi2) (* R phi1))
           (* phi2 (- R (* phi1 (/ R phi2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * -lambda1;
	double tmp;
	if (lambda1 <= -7.5e+198) {
		tmp = t_0;
	} else if (lambda1 <= -9.5e+156) {
		tmp = phi1 * (R * ((phi2 / phi1) + -1.0));
	} else if (lambda1 <= -5.4e+146) {
		tmp = t_0;
	} else if ((lambda1 <= 3.8e-297) || !(lambda1 <= 8e-18)) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * -lambda1
    if (lambda1 <= (-7.5d+198)) then
        tmp = t_0
    else if (lambda1 <= (-9.5d+156)) then
        tmp = phi1 * (r * ((phi2 / phi1) + (-1.0d0)))
    else if (lambda1 <= (-5.4d+146)) then
        tmp = t_0
    else if ((lambda1 <= 3.8d-297) .or. (.not. (lambda1 <= 8d-18))) then
        tmp = (r * phi2) - (r * phi1)
    else
        tmp = phi2 * (r - (phi1 * (r / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * -lambda1;
	double tmp;
	if (lambda1 <= -7.5e+198) {
		tmp = t_0;
	} else if (lambda1 <= -9.5e+156) {
		tmp = phi1 * (R * ((phi2 / phi1) + -1.0));
	} else if (lambda1 <= -5.4e+146) {
		tmp = t_0;
	} else if ((lambda1 <= 3.8e-297) || !(lambda1 <= 8e-18)) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = R * -lambda1
	tmp = 0
	if lambda1 <= -7.5e+198:
		tmp = t_0
	elif lambda1 <= -9.5e+156:
		tmp = phi1 * (R * ((phi2 / phi1) + -1.0))
	elif lambda1 <= -5.4e+146:
		tmp = t_0
	elif (lambda1 <= 3.8e-297) or not (lambda1 <= 8e-18):
		tmp = (R * phi2) - (R * phi1)
	else:
		tmp = phi2 * (R - (phi1 * (R / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * Float64(-lambda1))
	tmp = 0.0
	if (lambda1 <= -7.5e+198)
		tmp = t_0;
	elseif (lambda1 <= -9.5e+156)
		tmp = Float64(phi1 * Float64(R * Float64(Float64(phi2 / phi1) + -1.0)));
	elseif (lambda1 <= -5.4e+146)
		tmp = t_0;
	elseif ((lambda1 <= 3.8e-297) || !(lambda1 <= 8e-18))
		tmp = Float64(Float64(R * phi2) - Float64(R * phi1));
	else
		tmp = Float64(phi2 * Float64(R - Float64(phi1 * Float64(R / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = R * -lambda1;
	tmp = 0.0;
	if (lambda1 <= -7.5e+198)
		tmp = t_0;
	elseif (lambda1 <= -9.5e+156)
		tmp = phi1 * (R * ((phi2 / phi1) + -1.0));
	elseif (lambda1 <= -5.4e+146)
		tmp = t_0;
	elseif ((lambda1 <= 3.8e-297) || ~((lambda1 <= 8e-18)))
		tmp = (R * phi2) - (R * phi1);
	else
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * (-lambda1)), $MachinePrecision]}, If[LessEqual[lambda1, -7.5e+198], t$95$0, If[LessEqual[lambda1, -9.5e+156], N[(phi1 * N[(R * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, -5.4e+146], t$95$0, If[Or[LessEqual[lambda1, 3.8e-297], N[Not[LessEqual[lambda1, 8e-18]], $MachinePrecision]], N[(N[(R * phi2), $MachinePrecision] - N[(R * phi1), $MachinePrecision]), $MachinePrecision], N[(phi2 * N[(R - N[(phi1 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if lambda1 < -7.5000000000000002e198 or -9.5000000000000002e156 < lambda1 < -5.39999999999999977e146

   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 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in lambda1 around -inf 53.0%

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

      1. mul-1-neg 53.0%

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

      2. associate-*r* 53.0%

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

      3. distribute-lft-neg-in 53.0%

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

      4. +-commutative 53.0%

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

   7. Simplified 53.0%

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

   8. Taylor expanded in phi2 around 0 59.6%

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

      1. associate-*r* 59.6%

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

      2. mul-1-neg 59.6%

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

      3. *-commutative 59.6%

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

   10. Simplified 59.6%

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

   11. Taylor expanded in phi1 around 0 76.8%

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

   if -7.5000000000000002e198 < lambda1 < -9.5000000000000002e156

   1. Initial program 31.2%

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

      1. add-sqr-sqrt 99.6%

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

      2. pow2 99.6%

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

      3. *-commutative 99.6%

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

      4. div-inv 99.6%

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

      5. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in phi1 around -inf 19.3%

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

      1. mul-1-neg 19.3%

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

      2. mul-1-neg 19.3%

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

      3. *-commutative 19.3%

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

      4. associate-*r/ 10.2%

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

      5. sub-neg 10.2%

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

      6. distribute-rgt-neg-in 10.2%

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

      7. sub-neg 10.2%

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

      8. associate-*r/ 19.3%

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

      9. *-commutative 19.3%

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

      10. mul-1-neg 19.3%

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

      11. distribute-neg-in 19.3%

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

      12. mul-1-neg 19.3%

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

      13. mul-1-neg 19.3%

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

      14. *-commutative 19.3%

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

      15. associate-*r/ 10.2%

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

      16. remove-double-neg 10.2%

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

      17. associate-*r/ 19.3%

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

      18. *-commutative 19.3%

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

   9. Simplified 10.2%

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

   if -5.39999999999999977e146 < lambda1 < 3.80000000000000005e-297 or 8.0000000000000006e-18 < lambda1

   1. Initial program 55.3%

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

      1. hypot-define 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in phi1 around -inf 27.9%

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

      1. mul-1-neg 27.9%

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

      2. distribute-rgt-neg-in 27.9%

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

      3. mul-1-neg 27.9%

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

      4. unsub-neg 27.9%

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

      5. *-commutative 27.9%

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

      6. associate-/l* 30.1%

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

   7. Simplified 30.1%

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

   8. Taylor expanded in phi1 around 0 29.6%

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

      1. +-commutative 29.6%

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

      2. mul-1-neg 29.6%

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

      3. unsub-neg 29.6%

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

      4. *-commutative 29.6%

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

      5. *-commutative 29.6%

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

   10. Simplified 29.6%

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

   if 3.80000000000000005e-297 < lambda1 < 8.0000000000000006e-18

   1. Initial program 63.0%

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

      1. hypot-define 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in phi2 around inf 43.7%

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

      1. mul-1-neg 43.7%

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

      2. unsub-neg 43.7%

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

      3. *-commutative 43.7%

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

      4. associate-/l* 43.6%

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

   7. Simplified 43.6%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -7.5 \cdot 10^{+198}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -9.5 \cdot 10^{+156}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -5.4 \cdot 10^{+146}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 3.8 \cdot 10^{-297} \lor \neg \left(\lambda_1 \leq 8 \cdot 10^{-18}\right):\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 33.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ t_1 := \phi_1 \cdot \left(R \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{if}\;\lambda_1 \leq -3.5 \cdot 10^{+200}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq -1.12 \cdot 10^{+160}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\lambda_1 \leq -1.86 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq -2.15 \cdot 10^{-306} \lor \neg \left(\lambda_1 \leq 2 \cdot 10^{-181}\right):\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* R (- lambda1))) (t_1 (* phi1 (* R (+ (/ phi2 phi1) -1.0)))))
   (if (<= lambda1 -3.5e+200)
     t_0
     (if (<= lambda1 -1.12e+160)
       t_1
       (if (<= lambda1 -1.86e+149)
         t_0
         (if (or (<= lambda1 -2.15e-306) (not (<= lambda1 2e-181)))
           (- (* R phi2) (* R phi1))
           t_1))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * -lambda1;
	double t_1 = phi1 * (R * ((phi2 / phi1) + -1.0));
	double tmp;
	if (lambda1 <= -3.5e+200) {
		tmp = t_0;
	} else if (lambda1 <= -1.12e+160) {
		tmp = t_1;
	} else if (lambda1 <= -1.86e+149) {
		tmp = t_0;
	} else if ((lambda1 <= -2.15e-306) || !(lambda1 <= 2e-181)) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = r * -lambda1
    t_1 = phi1 * (r * ((phi2 / phi1) + (-1.0d0)))
    if (lambda1 <= (-3.5d+200)) then
        tmp = t_0
    else if (lambda1 <= (-1.12d+160)) then
        tmp = t_1
    else if (lambda1 <= (-1.86d+149)) then
        tmp = t_0
    else if ((lambda1 <= (-2.15d-306)) .or. (.not. (lambda1 <= 2d-181))) then
        tmp = (r * phi2) - (r * phi1)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * -lambda1;
	double t_1 = phi1 * (R * ((phi2 / phi1) + -1.0));
	double tmp;
	if (lambda1 <= -3.5e+200) {
		tmp = t_0;
	} else if (lambda1 <= -1.12e+160) {
		tmp = t_1;
	} else if (lambda1 <= -1.86e+149) {
		tmp = t_0;
	} else if ((lambda1 <= -2.15e-306) || !(lambda1 <= 2e-181)) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = R * -lambda1
	t_1 = phi1 * (R * ((phi2 / phi1) + -1.0))
	tmp = 0
	if lambda1 <= -3.5e+200:
		tmp = t_0
	elif lambda1 <= -1.12e+160:
		tmp = t_1
	elif lambda1 <= -1.86e+149:
		tmp = t_0
	elif (lambda1 <= -2.15e-306) or not (lambda1 <= 2e-181):
		tmp = (R * phi2) - (R * phi1)
	else:
		tmp = t_1
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * Float64(-lambda1))
	t_1 = Float64(phi1 * Float64(R * Float64(Float64(phi2 / phi1) + -1.0)))
	tmp = 0.0
	if (lambda1 <= -3.5e+200)
		tmp = t_0;
	elseif (lambda1 <= -1.12e+160)
		tmp = t_1;
	elseif (lambda1 <= -1.86e+149)
		tmp = t_0;
	elseif ((lambda1 <= -2.15e-306) || !(lambda1 <= 2e-181))
		tmp = Float64(Float64(R * phi2) - Float64(R * phi1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = R * -lambda1;
	t_1 = phi1 * (R * ((phi2 / phi1) + -1.0));
	tmp = 0.0;
	if (lambda1 <= -3.5e+200)
		tmp = t_0;
	elseif (lambda1 <= -1.12e+160)
		tmp = t_1;
	elseif (lambda1 <= -1.86e+149)
		tmp = t_0;
	elseif ((lambda1 <= -2.15e-306) || ~((lambda1 <= 2e-181)))
		tmp = (R * phi2) - (R * phi1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * (-lambda1)), $MachinePrecision]}, Block[{t$95$1 = N[(phi1 * N[(R * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -3.5e+200], t$95$0, If[LessEqual[lambda1, -1.12e+160], t$95$1, If[LessEqual[lambda1, -1.86e+149], t$95$0, If[Or[LessEqual[lambda1, -2.15e-306], N[Not[LessEqual[lambda1, 2e-181]], $MachinePrecision]], N[(N[(R * phi2), $MachinePrecision] - N[(R * phi1), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -3.50000000000000006e200 or -1.12e160 < lambda1 < -1.85999999999999997e149

   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 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in lambda1 around -inf 53.0%

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

      1. mul-1-neg 53.0%

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

      2. associate-*r* 53.0%

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

      3. distribute-lft-neg-in 53.0%

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

      4. +-commutative 53.0%

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

   7. Simplified 53.0%

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

   8. Taylor expanded in phi2 around 0 59.6%

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

      1. associate-*r* 59.6%

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

      2. mul-1-neg 59.6%

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

      3. *-commutative 59.6%

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

   10. Simplified 59.6%

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

   11. Taylor expanded in phi1 around 0 76.8%

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

   if -3.50000000000000006e200 < lambda1 < -1.12e160 or -2.15e-306 < lambda1 < 2.00000000000000009e-181

   1. Initial program 56.9%

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

      1. hypot-define 97.7%

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

   3. Simplified 97.7%

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

      1. add-sqr-sqrt 97.3%

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

      2. pow2 97.3%

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

      3. *-commutative 97.3%

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

      4. div-inv 97.3%

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

      5. metadata-eval 97.3%

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

   6. Applied egg-rr 97.3%

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

   7. Taylor expanded in phi1 around -inf 30.1%

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

      1. mul-1-neg 30.1%

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

      2. mul-1-neg 30.1%

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

      3. *-commutative 30.1%

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

      4. associate-*r/ 24.3%

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

      5. sub-neg 24.3%

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

      6. distribute-rgt-neg-in 24.3%

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

      7. sub-neg 24.3%

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

      8. associate-*r/ 30.1%

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

      9. *-commutative 30.1%

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

      10. mul-1-neg 30.1%

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

      11. distribute-neg-in 30.1%

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

      12. mul-1-neg 30.1%

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

      13. mul-1-neg 30.1%

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

      14. *-commutative 30.1%

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

      15. associate-*r/ 24.3%

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

      16. remove-double-neg 24.3%

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

      17. associate-*r/ 30.1%

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

      18. *-commutative 30.1%

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

   9. Simplified 24.2%

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

   if -1.85999999999999997e149 < lambda1 < -2.15e-306 or 2.00000000000000009e-181 < lambda1

   1. Initial program 55.8%

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

      1. hypot-define 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in phi1 around -inf 30.8%

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

      1. mul-1-neg 30.8%

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

      2. distribute-rgt-neg-in 30.8%

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

      3. mul-1-neg 30.8%

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

      4. unsub-neg 30.8%

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

      5. *-commutative 30.8%

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

      6. associate-/l* 32.1%

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

   7. Simplified 32.1%

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

   8. Taylor expanded in phi1 around 0 32.3%

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

      1. +-commutative 32.3%

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

      2. mul-1-neg 32.3%

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

      3. unsub-neg 32.3%

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

      4. *-commutative 32.3%

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

      5. *-commutative 32.3%

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

   10. Simplified 32.3%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.5 \cdot 10^{+200}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -1.12 \cdot 10^{+160}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -1.86 \cdot 10^{+149}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -2.15 \cdot 10^{-306} \lor \neg \left(\lambda_1 \leq 2 \cdot 10^{-181}\right):\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.8% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ \mathbf{if}\;\lambda_1 \leq -3.2 \cdot 10^{+198}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq -5.8 \cdot 10^{+156}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -9 \cdot 10^{+144}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\lambda_1 \leq 1.95 \cdot 10^{-307}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(\phi_2 \cdot \frac{R}{\phi_1} - R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* R (- lambda1))))
   (if (<= lambda1 -3.2e+198)
     t_0
     (if (<= lambda1 -5.8e+156)
       (* R (* phi1 (+ (/ phi2 phi1) -1.0)))
       (if (<= lambda1 -9e+144)
         t_0
         (if (<= lambda1 1.95e-307)
           (- (* R phi2) (* R phi1))
           (* phi1 (- (* phi2 (/ R phi1)) R))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * -lambda1;
	double tmp;
	if (lambda1 <= -3.2e+198) {
		tmp = t_0;
	} else if (lambda1 <= -5.8e+156) {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	} else if (lambda1 <= -9e+144) {
		tmp = t_0;
	} else if (lambda1 <= 1.95e-307) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = phi1 * ((phi2 * (R / phi1)) - R);
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * -lambda1
    if (lambda1 <= (-3.2d+198)) then
        tmp = t_0
    else if (lambda1 <= (-5.8d+156)) then
        tmp = r * (phi1 * ((phi2 / phi1) + (-1.0d0)))
    else if (lambda1 <= (-9d+144)) then
        tmp = t_0
    else if (lambda1 <= 1.95d-307) then
        tmp = (r * phi2) - (r * phi1)
    else
        tmp = phi1 * ((phi2 * (r / phi1)) - r)
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * -lambda1;
	double tmp;
	if (lambda1 <= -3.2e+198) {
		tmp = t_0;
	} else if (lambda1 <= -5.8e+156) {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	} else if (lambda1 <= -9e+144) {
		tmp = t_0;
	} else if (lambda1 <= 1.95e-307) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = phi1 * ((phi2 * (R / phi1)) - R);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = R * -lambda1
	tmp = 0
	if lambda1 <= -3.2e+198:
		tmp = t_0
	elif lambda1 <= -5.8e+156:
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0))
	elif lambda1 <= -9e+144:
		tmp = t_0
	elif lambda1 <= 1.95e-307:
		tmp = (R * phi2) - (R * phi1)
	else:
		tmp = phi1 * ((phi2 * (R / phi1)) - R)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * Float64(-lambda1))
	tmp = 0.0
	if (lambda1 <= -3.2e+198)
		tmp = t_0;
	elseif (lambda1 <= -5.8e+156)
		tmp = Float64(R * Float64(phi1 * Float64(Float64(phi2 / phi1) + -1.0)));
	elseif (lambda1 <= -9e+144)
		tmp = t_0;
	elseif (lambda1 <= 1.95e-307)
		tmp = Float64(Float64(R * phi2) - Float64(R * phi1));
	else
		tmp = Float64(phi1 * Float64(Float64(phi2 * Float64(R / phi1)) - R));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = R * -lambda1;
	tmp = 0.0;
	if (lambda1 <= -3.2e+198)
		tmp = t_0;
	elseif (lambda1 <= -5.8e+156)
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	elseif (lambda1 <= -9e+144)
		tmp = t_0;
	elseif (lambda1 <= 1.95e-307)
		tmp = (R * phi2) - (R * phi1);
	else
		tmp = phi1 * ((phi2 * (R / phi1)) - R);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * (-lambda1)), $MachinePrecision]}, If[LessEqual[lambda1, -3.2e+198], t$95$0, If[LessEqual[lambda1, -5.8e+156], N[(R * N[(phi1 * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, -9e+144], t$95$0, If[LessEqual[lambda1, 1.95e-307], N[(N[(R * phi2), $MachinePrecision] - N[(R * phi1), $MachinePrecision]), $MachinePrecision], N[(phi1 * N[(N[(phi2 * N[(R / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if lambda1 < -3.1999999999999998e198 or -5.80000000000000021e156 < lambda1 < -8.99999999999999935e144

   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 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in lambda1 around -inf 53.0%

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

      1. mul-1-neg 53.0%

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

      2. associate-*r* 53.0%

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

      3. distribute-lft-neg-in 53.0%

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

      4. +-commutative 53.0%

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

   7. Simplified 53.0%

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

   8. Taylor expanded in phi2 around 0 59.6%

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

      1. associate-*r* 59.6%

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

      2. mul-1-neg 59.6%

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

      3. *-commutative 59.6%

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

   10. Simplified 59.6%

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

   11. Taylor expanded in phi1 around 0 76.8%

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

   if -3.1999999999999998e198 < lambda1 < -5.80000000000000021e156

   1. Initial program 31.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 phi1 around -inf 10.3%

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

      1. mul-1-neg 10.3%

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

      2. distribute-rgt-neg-in 10.3%

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

      3. mul-1-neg 10.3%

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

      4. unsub-neg 10.3%

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

   7. Simplified 10.3%

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

   if -8.99999999999999935e144 < lambda1 < 1.95e-307

   1. Initial program 58.8%

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

      1. hypot-define 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in phi1 around -inf 33.0%

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

      1. mul-1-neg 33.0%

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

      2. distribute-rgt-neg-in 33.0%

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

      3. mul-1-neg 33.0%

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

      4. unsub-neg 33.0%

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

      5. *-commutative 33.0%

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

      6. associate-/l* 34.9%

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

   7. Simplified 34.9%

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

   8. Taylor expanded in phi1 around 0 36.2%

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

      1. +-commutative 36.2%

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

      2. mul-1-neg 36.2%

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

      3. unsub-neg 36.2%

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

      4. *-commutative 36.2%

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

      5. *-commutative 36.2%

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

   10. Simplified 36.2%

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

   if 1.95e-307 < lambda1

   1. Initial program 56.0%

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

      1. hypot-define 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in phi1 around -inf 30.1%

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

      1. mul-1-neg 30.1%

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

      2. distribute-rgt-neg-in 30.1%

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

      3. mul-1-neg 30.1%

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

      4. unsub-neg 30.1%

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

      5. *-commutative 30.1%

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

      6. associate-/l* 30.0%

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

   7. Simplified 30.0%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.2 \cdot 10^{+198}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -5.8 \cdot 10^{+156}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -9 \cdot 10^{+144}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.95 \cdot 10^{-307}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(\phi_2 \cdot \frac{R}{\phi_1} - R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 29.2% accurate, 23.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -215:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -5 \cdot 10^{-61}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -215.0)
   (* R (- phi1))
   (if (<= phi1 -5e-61) (* R (- lambda1)) (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -215.0) {
		tmp = R * -phi1;
	} else if (phi1 <= -5e-61) {
		tmp = R * -lambda1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-215.0d0)) then
        tmp = r * -phi1
    else if (phi1 <= (-5d-61)) then
        tmp = r * -lambda1
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -215.0) {
		tmp = R * -phi1;
	} else if (phi1 <= -5e-61) {
		tmp = R * -lambda1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -215.0:
		tmp = R * -phi1
	elif phi1 <= -5e-61:
		tmp = R * -lambda1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -215.0)
		tmp = Float64(R * Float64(-phi1));
	elseif (phi1 <= -5e-61)
		tmp = Float64(R * Float64(-lambda1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -215.0)
		tmp = R * -phi1;
	elseif (phi1 <= -5e-61)
		tmp = R * -lambda1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -215.0], N[(R * (-phi1)), $MachinePrecision], If[LessEqual[phi1, -5e-61], N[(R * (-lambda1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -215

   1. Initial program 47.4%

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

      1. hypot-define 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in phi1 around -inf 65.5%

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

      1. mul-1-neg 65.5%

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

      2. *-commutative 65.5%

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

      3. distribute-rgt-neg-in 65.5%

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

   7. Simplified 65.5%

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

   if -215 < phi1 < -4.9999999999999999e-61

   1. Initial program 66.9%

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

      1. hypot-define 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 lambda1 around -inf 23.7%

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

      1. mul-1-neg 23.7%

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

      2. associate-*r* 23.7%

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

      3. distribute-lft-neg-in 23.7%

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

      4. +-commutative 23.7%

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

   7. Simplified 23.7%

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

   8. Taylor expanded in phi2 around 0 24.6%

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

      1. associate-*r* 24.6%

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

      2. mul-1-neg 24.6%

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

      3. *-commutative 24.6%

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

   10. Simplified 24.6%

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

   11. Taylor expanded in phi1 around 0 24.6%

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

   if -4.9999999999999999e-61 < phi1

   1. Initial program 59.4%

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

      1. hypot-define 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in phi2 around inf 19.7%

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

      1. *-commutative 19.7%

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

   7. Simplified 19.7%

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

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -215:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -5 \cdot 10^{-61}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 33.5% accurate, 27.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.25 \cdot 10^{+195}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -2.25e+195) (* R (- lambda1)) (- (* R phi2) (* R phi1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.25e+195) {
		tmp = R * -lambda1;
	} else {
		tmp = (R * phi2) - (R * phi1);
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-2.25d+195)) then
        tmp = r * -lambda1
    else
        tmp = (r * phi2) - (r * phi1)
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.25e+195) {
		tmp = R * -lambda1;
	} else {
		tmp = (R * phi2) - (R * phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -2.25e+195:
		tmp = R * -lambda1
	else:
		tmp = (R * phi2) - (R * phi1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -2.25e+195)
		tmp = Float64(R * Float64(-lambda1));
	else
		tmp = Float64(Float64(R * phi2) - Float64(R * phi1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -2.25e+195)
		tmp = R * -lambda1;
	else
		tmp = (R * phi2) - (R * phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -2.25e+195], N[(R * (-lambda1)), $MachinePrecision], N[(N[(R * phi2), $MachinePrecision] - N[(R * phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.25000000000000005e195

   1. Initial program 65.2%

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

      1. hypot-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in lambda1 around -inf 42.8%

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

      1. mul-1-neg 42.8%

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

      2. associate-*r* 42.8%

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

      3. distribute-lft-neg-in 42.8%

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

      4. +-commutative 42.8%

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

   7. Simplified 42.8%

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

   8. Taylor expanded in phi2 around 0 54.4%

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

      1. associate-*r* 54.4%

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

      2. mul-1-neg 54.4%

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

      3. *-commutative 54.4%

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

   10. Simplified 54.4%

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

   11. Taylor expanded in phi1 around 0 78.7%

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

   if -2.25000000000000005e195 < lambda1

   1. Initial program 56.3%

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

      1. 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 30.1%

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

      1. mul-1-neg 30.1%

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

      2. distribute-rgt-neg-in 30.1%

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

      3. mul-1-neg 30.1%

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

      4. unsub-neg 30.1%

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

      5. *-commutative 30.1%

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

      6. associate-/l* 31.2%

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

   7. Simplified 31.2%

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

   8. Taylor expanded in phi1 around 0 30.5%

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

      1. +-commutative 30.5%

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

      2. mul-1-neg 30.5%

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

      3. unsub-neg 30.5%

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

      4. *-commutative 30.5%

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

      5. *-commutative 30.5%

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

   10. Simplified 30.5%

       \[\leadsto \color{blue}{\phi_2 \cdot R - \phi_1 \cdot R} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.25 \cdot 10^{+195}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.4% accurate, 36.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1820:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1820.0) (* R (- phi1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1820.0) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-1820.0d0)) then
        tmp = r * -phi1
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1820.0) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1820.0:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1820.0)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1820.0)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1820.0], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1820

   1. Initial program 47.4%

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

      1. hypot-define 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in phi1 around -inf 65.5%

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

      1. mul-1-neg 65.5%

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

      2. *-commutative 65.5%

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

      3. distribute-rgt-neg-in 65.5%

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

   7. Simplified 65.5%

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

   if -1820 < phi1

   1. Initial program 60.3%

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

      1. hypot-define 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in phi2 around inf 19.2%

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

      1. *-commutative 19.2%

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

   7. Simplified 19.2%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1820:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 18.2% accurate, 109.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

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

   1. hypot-define 97.6%

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

3. Simplified 97.6%

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

5. Taylor expanded in phi2 around inf 17.9%

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

   1. *-commutative 17.9%

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

7. Simplified 17.9%

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

8. Final simplification 17.9%

   \[\leadsto R \cdot \phi_2 \]
9. Add Preprocessing

Alternative 17: 14.0% accurate, 109.7× speedup

Math

\[\begin{array}{l} \\ R \cdot \lambda_1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * lambda1), $MachinePrecision]

Derivation

1. Initial program 57.0%

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

   1. hypot-define 97.6%

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

3. Simplified 97.6%

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

5. Taylor expanded in lambda1 around -inf 12.2%

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

   1. mul-1-neg 12.2%

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

   2. associate-*r* 12.2%

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

   3. distribute-lft-neg-in 12.2%

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

   4. +-commutative 12.2%

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

7. Simplified 12.2%

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

   1. add-sqr-sqrt 7.0%

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

   2. pow2 7.0%

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

   3. *-commutative 7.0%

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

   4. +-commutative 7.0%

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

   5. *-commutative 7.0%

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

   6. add-sqr-sqrt 4.7%

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

   7. sqrt-unprod 12.3%

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

   8. sqr-neg 12.3%

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

   9. sqrt-unprod 7.9%

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

   10. add-sqr-sqrt 11.5%

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

   11. *-commutative 11.5%

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

9. Applied egg-rr 11.5%

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

10. Taylor expanded in phi1 around 0 16.7%

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

    1. associate-*r* 16.7%

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

    2. *-commutative 16.7%

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

12. Simplified 16.7%

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

13. Taylor expanded in phi2 around 0 14.0%

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

    1. *-commutative 14.0%

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

15. Simplified 14.0%

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

16. Final simplification 14.0%

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

Reproduce

herbie shell --seed 2024091 
(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))))))