Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.4% → 99.9%

Time: 15.7s

Alternatives: 11

Speedup: 1.6×

• 31.6% 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: 60.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

   1. hypot-define 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. Step-by-step derivation

   1. add-cbrt-cube 97.0%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}, \phi_1 - \phi_2\right) \]

   2. pow3 97.0%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\color{blue}{{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}^{3}}}, \phi_1 - \phi_2\right) \]

   3. div-inv 97.0%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}^{3}}, \phi_1 - \phi_2\right) \]

   4. metadata-eval 97.0%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}^{3}}, \phi_1 - \phi_2\right) \]

6. Applied egg-rr 97.0%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sqrt[3]{{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}^{3}}}, \phi_1 - \phi_2\right) \]
7. Step-by-step derivation

   1. rem-cbrt-cube 97.1%

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

   2. *-commutative 97.1%

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

   3. distribute-rgt-in 97.1%

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

   4. cos-sum 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 93.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 880:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 880.0)
   (* R (hypot (* (- lambda1 lambda2) (cos (* phi1 0.5))) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi2))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 880.0) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi2))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 880.0) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi2))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 880.0:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi1 * 0.5))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi2))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 880.0)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi2))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 880.0)
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi2))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 880.0], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 880

   1. Initial program 62.3%

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

      1. hypot-define 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in phi2 around 0 93.9%

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

      1. *-commutative 93.9%

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

   7. Simplified 93.9%

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

   if 880 < phi2

   1. Initial program 52.4%

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

      1. hypot-define 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in phi1 around 0 94.3%

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

      1. *-commutative 94.3%

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

   7. Simplified 94.3%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 880:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

   1. hypot-define 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. Add Preprocessing

Alternative 4: 90.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

   1. hypot-define 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 phi2 around 0 90.6%

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

   1. *-commutative 90.6%

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

7. Simplified 90.6%

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

8. Final simplification 90.6%

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

Alternative 5: 29.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.3 \cdot 10^{-85}:\\ \;\;\;\;\lambda_1 \cdot \left(\left(-R\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.95 \cdot 10^{+187}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -2.3e-85)
   (* lambda1 (* (- R) (cos (* phi1 0.5))))
   (if (<= lambda2 1.95e+187)
     (* phi1 (- (/ (* R phi2) phi1) R))
     (* R lambda2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -2.3e-85) {
		tmp = lambda1 * (-R * cos((phi1 * 0.5)));
	} else if (lambda2 <= 1.95e+187) {
		tmp = phi1 * (((R * phi2) / phi1) - R);
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= (-2.3d-85)) then
        tmp = lambda1 * (-r * cos((phi1 * 0.5d0)))
    else if (lambda2 <= 1.95d+187) then
        tmp = phi1 * (((r * phi2) / phi1) - r)
    else
        tmp = r * lambda2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -2.3e-85) {
		tmp = lambda1 * (-R * Math.cos((phi1 * 0.5)));
	} else if (lambda2 <= 1.95e+187) {
		tmp = phi1 * (((R * phi2) / phi1) - R);
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= -2.3e-85:
		tmp = lambda1 * (-R * math.cos((phi1 * 0.5)))
	elif lambda2 <= 1.95e+187:
		tmp = phi1 * (((R * phi2) / phi1) - R)
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -2.3e-85)
		tmp = Float64(lambda1 * Float64(Float64(-R) * cos(Float64(phi1 * 0.5))));
	elseif (lambda2 <= 1.95e+187)
		tmp = Float64(phi1 * Float64(Float64(Float64(R * phi2) / phi1) - R));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= -2.3e-85)
		tmp = lambda1 * (-R * cos((phi1 * 0.5)));
	elseif (lambda2 <= 1.95e+187)
		tmp = phi1 * (((R * phi2) / phi1) - R);
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -2.3e-85], N[(lambda1 * N[((-R) * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 1.95e+187], N[(phi1 * N[(N[(N[(R * phi2), $MachinePrecision] / phi1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -2.3e-85

   1. Initial program 53.5%

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

      1. hypot-define 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in lambda1 around -inf 29.7%

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

      1. mul-1-neg 29.7%

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

      2. *-commutative 29.7%

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

      3. distribute-rgt-neg-in 29.7%

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

      4. +-commutative 29.7%

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

      5. mul-1-neg 29.7%

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

      6. unsub-neg 29.7%

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

      7. associate-*r* 29.7%

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

      8. associate-/l* 29.7%

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

   7. Simplified 29.7%

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

   8. Taylor expanded in lambda2 around 0 19.6%

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

   9. Taylor expanded in phi2 around 0 18.5%

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

   if -2.3e-85 < lambda2 < 1.94999999999999991e187

   1. Initial program 67.4%

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

      1. hypot-define 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in phi1 around -inf 28.6%

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

      1. associate-*r* 28.6%

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

      2. mul-1-neg 28.6%

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

      3. associate-*r/ 28.6%

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

      4. mul-1-neg 28.6%

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

   7. Simplified 28.6%

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

   if 1.94999999999999991e187 < lambda2

   1. Initial program 41.4%

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

      1. hypot-define 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in lambda2 around -inf 29.0%

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

      1. mul-1-neg 29.0%

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

      2. associate-*r* 28.9%

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

   7. Simplified 28.9%

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

   8. Taylor expanded in phi2 around 0 24.7%

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

      1. associate-*r* 24.7%

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

   10. Simplified 24.7%

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

       1. add-sqr-sqrt 16.2%

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

       2. sqrt-unprod 37.1%

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

       3. sqr-neg 37.1%

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

       4. sqrt-unprod 24.7%

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

       5. add-sqr-sqrt 43.5%

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

       6. *-commutative 43.5%

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

   12. Applied egg-rr 43.5%

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

   13. Taylor expanded in phi1 around 0 61.9%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.3 \cdot 10^{-85}:\\ \;\;\;\;\lambda_1 \cdot \left(\left(-R\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.95 \cdot 10^{+187}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -7.3 \cdot 10^{-87}:\\ \;\;\;\;\left(R \cdot \lambda_1\right) \cdot \left(-\cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.05 \cdot 10^{+186}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -7.3e-87)
   (* (* R lambda1) (- (cos (* phi1 0.5))))
   (if (<= lambda2 1.05e+186)
     (* phi1 (- (/ (* R phi2) phi1) R))
     (* R lambda2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -7.3e-87) {
		tmp = (R * lambda1) * -cos((phi1 * 0.5));
	} else if (lambda2 <= 1.05e+186) {
		tmp = phi1 * (((R * phi2) / phi1) - R);
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= (-7.3d-87)) then
        tmp = (r * lambda1) * -cos((phi1 * 0.5d0))
    else if (lambda2 <= 1.05d+186) then
        tmp = phi1 * (((r * phi2) / phi1) - r)
    else
        tmp = r * lambda2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -7.3e-87) {
		tmp = (R * lambda1) * -Math.cos((phi1 * 0.5));
	} else if (lambda2 <= 1.05e+186) {
		tmp = phi1 * (((R * phi2) / phi1) - R);
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= -7.3e-87:
		tmp = (R * lambda1) * -math.cos((phi1 * 0.5))
	elif lambda2 <= 1.05e+186:
		tmp = phi1 * (((R * phi2) / phi1) - R)
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -7.3e-87)
		tmp = Float64(Float64(R * lambda1) * Float64(-cos(Float64(phi1 * 0.5))));
	elseif (lambda2 <= 1.05e+186)
		tmp = Float64(phi1 * Float64(Float64(Float64(R * phi2) / phi1) - R));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= -7.3e-87)
		tmp = (R * lambda1) * -cos((phi1 * 0.5));
	elseif (lambda2 <= 1.05e+186)
		tmp = phi1 * (((R * phi2) / phi1) - R);
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -7.3e-87], N[(N[(R * lambda1), $MachinePrecision] * (-N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[lambda2, 1.05e+186], N[(phi1 * N[(N[(N[(R * phi2), $MachinePrecision] / phi1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -7.29999999999999967e-87

   1. Initial program 53.5%

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

      1. hypot-define 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in lambda1 around -inf 29.7%

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

      1. mul-1-neg 29.7%

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

      2. *-commutative 29.7%

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

      3. distribute-rgt-neg-in 29.7%

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

      4. +-commutative 29.7%

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

      5. mul-1-neg 29.7%

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

      6. unsub-neg 29.7%

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

      7. associate-*r* 29.7%

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

      8. associate-/l* 29.7%

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

   7. Simplified 29.7%

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

   8. Taylor expanded in lambda2 around 0 19.6%

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

   9. Taylor expanded in phi2 around 0 18.5%

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

       1. mul-1-neg 18.5%

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

       2. associate-*r* 18.5%

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

       3. *-commutative 18.5%

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

       4. distribute-rgt-neg-in 18.5%

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

       5. *-commutative 18.5%

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

   11. Simplified 18.5%

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

   if -7.29999999999999967e-87 < lambda2 < 1.05e186

   1. Initial program 67.4%

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

      1. hypot-define 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in phi1 around -inf 28.6%

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

      1. associate-*r* 28.6%

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

      2. mul-1-neg 28.6%

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

      3. associate-*r/ 28.6%

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

      4. mul-1-neg 28.6%

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

   7. Simplified 28.6%

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

   if 1.05e186 < lambda2

   1. Initial program 41.4%

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

      1. hypot-define 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in lambda2 around -inf 29.0%

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

      1. mul-1-neg 29.0%

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

      2. associate-*r* 28.9%

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

   7. Simplified 28.9%

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

   8. Taylor expanded in phi2 around 0 24.7%

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

      1. associate-*r* 24.7%

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

   10. Simplified 24.7%

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

       1. add-sqr-sqrt 16.2%

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

       2. sqrt-unprod 37.1%

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

       3. sqr-neg 37.1%

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

       4. sqrt-unprod 24.7%

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

       5. add-sqr-sqrt 43.5%

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

       6. *-commutative 43.5%

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

   12. Applied egg-rr 43.5%

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

   13. Taylor expanded in phi1 around 0 61.9%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -7.3 \cdot 10^{-87}:\\ \;\;\;\;\left(R \cdot \lambda_1\right) \cdot \left(-\cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.05 \cdot 10^{+186}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.4% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.2 \cdot 10^{+185}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 5.2e+185) (* phi1 (- (/ (* R phi2) phi1) R)) (* R lambda2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 5.2e+185) {
		tmp = phi1 * (((R * phi2) / phi1) - R);
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 5.2d+185) then
        tmp = phi1 * (((r * phi2) / phi1) - r)
    else
        tmp = r * lambda2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 5.2e+185) {
		tmp = phi1 * (((R * phi2) / phi1) - R);
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 5.2e+185:
		tmp = phi1 * (((R * phi2) / phi1) - R)
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 5.2e+185)
		tmp = Float64(phi1 * Float64(Float64(Float64(R * phi2) / phi1) - R));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 5.2e+185)
		tmp = phi1 * (((R * phi2) / phi1) - R);
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 5.2e+185], N[(phi1 * N[(N[(N[(R * phi2), $MachinePrecision] / phi1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 5.20000000000000001e185

   1. Initial program 62.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 24.2%

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

      1. associate-*r* 24.2%

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

      2. mul-1-neg 24.2%

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

      3. associate-*r/ 24.2%

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

      4. mul-1-neg 24.2%

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

   7. Simplified 24.2%

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

   if 5.20000000000000001e185 < lambda2

   1. Initial program 41.4%

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

      1. hypot-define 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in lambda2 around -inf 29.0%

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

      1. mul-1-neg 29.0%

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

      2. associate-*r* 28.9%

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

   7. Simplified 28.9%

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

   8. Taylor expanded in phi2 around 0 24.7%

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

      1. associate-*r* 24.7%

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

   10. Simplified 24.7%

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

       1. add-sqr-sqrt 16.2%

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

       2. sqrt-unprod 37.1%

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

       3. sqr-neg 37.1%

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

       4. sqrt-unprod 24.7%

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

       5. add-sqr-sqrt 43.5%

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

       6. *-commutative 43.5%

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

   12. Applied egg-rr 43.5%

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

   13. Taylor expanded in phi1 around 0 61.9%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.2 \cdot 10^{+185}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 26.4% accurate, 25.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -4.5 \cdot 10^{-204}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 4.9 \cdot 10^{+84}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -4.5e-204)
   (* R (- phi1))
   (if (<= phi2 4.9e+84) (* R lambda2) (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -4.5e-204) {
		tmp = R * -phi1;
	} else if (phi2 <= 4.9e+84) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= (-4.5d-204)) then
        tmp = r * -phi1
    else if (phi2 <= 4.9d+84) then
        tmp = r * lambda2
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -4.5e-204) {
		tmp = R * -phi1;
	} else if (phi2 <= 4.9e+84) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -4.5e-204:
		tmp = R * -phi1
	elif phi2 <= 4.9e+84:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -4.5e-204)
		tmp = Float64(R * Float64(-phi1));
	elseif (phi2 <= 4.9e+84)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= -4.5e-204)
		tmp = R * -phi1;
	elseif (phi2 <= 4.9e+84)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -4.5e-204], N[(R * (-phi1)), $MachinePrecision], If[LessEqual[phi2, 4.9e+84], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -4.49999999999999974e-204

   1. Initial program 59.5%

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

      1. hypot-define 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in phi1 around -inf 11.2%

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

      1. mul-1-neg 11.2%

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

      2. distribute-rgt-neg-in 11.2%

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

   7. Simplified 11.2%

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

   if -4.49999999999999974e-204 < phi2 < 4.9e84

   1. Initial program 64.7%

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

      1. hypot-define 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in lambda2 around -inf 24.4%

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

      1. mul-1-neg 24.4%

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

      2. associate-*r* 24.4%

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

   7. Simplified 24.4%

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

   8. Taylor expanded in phi2 around 0 22.7%

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

      1. associate-*r* 22.7%

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

   10. Simplified 22.7%

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

       1. add-sqr-sqrt 13.8%

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

       2. sqrt-unprod 16.5%

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

       3. sqr-neg 16.5%

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

       4. sqrt-unprod 8.3%

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

       5. add-sqr-sqrt 18.4%

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

       6. *-commutative 18.4%

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

   12. Applied egg-rr 18.4%

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

   13. Taylor expanded in phi1 around 0 19.2%

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

   if 4.9e84 < phi2

   1. Initial program 51.1%

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

      1. hypot-define 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in phi2 around inf 67.6%

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

Alternative 9: 34.5% accurate, 27.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.8 \cdot 10^{+110}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 3.8e+110) (- (* R phi2) (* R phi1)) (* R lambda2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 3.8e+110) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 3.8d+110) then
        tmp = (r * phi2) - (r * phi1)
    else
        tmp = r * lambda2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 3.8e+110) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 3.8e+110:
		tmp = (R * phi2) - (R * phi1)
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 3.8e+110)
		tmp = Float64(Float64(R * phi2) - Float64(R * phi1));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 3.8e+110)
		tmp = (R * phi2) - (R * phi1);
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 3.8e+110], N[(N[(R * phi2), $MachinePrecision] - N[(R * phi1), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 3.79999999999999989e110

   1. Initial program 62.6%

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

      1. 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 inf 24.0%

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

      1. associate-*r/ 24.0%

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

      2. mul-1-neg 24.0%

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

   7. Simplified 24.0%

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

   8. Taylor expanded in phi2 around 0 24.1%

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

      1. neg-mul-1 24.1%

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

      2. +-commutative 24.1%

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

      3. unsub-neg 24.1%

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

   10. Simplified 24.1%

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

   if 3.79999999999999989e110 < lambda2

   1. Initial program 49.5%

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

      1. hypot-define 93.3%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in lambda2 around -inf 30.4%

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

      1. mul-1-neg 30.4%

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

      2. associate-*r* 30.4%

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

   7. Simplified 30.4%

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

   8. Taylor expanded in phi2 around 0 23.5%

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

      1. associate-*r* 23.5%

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

   10. Simplified 23.5%

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

       1. add-sqr-sqrt 13.9%

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

       2. sqrt-unprod 36.4%

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

       3. sqr-neg 36.4%

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

       4. sqrt-unprod 20.8%

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

       5. add-sqr-sqrt 38.6%

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

       6. *-commutative 38.6%

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

   12. Applied egg-rr 38.6%

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

   13. Taylor expanded in phi1 around 0 52.9%

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

Alternative 10: 25.4% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5 \cdot 10^{+84}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 5e+84) (* R lambda2) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5e+84) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 5d+84) then
        tmp = r * lambda2
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5e+84) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 5e+84:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5e+84)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 5e+84)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 5e+84], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5.0000000000000001e84

   1. Initial program 62.0%

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

      1. hypot-define 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in lambda2 around -inf 21.6%

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

      1. mul-1-neg 21.6%

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

      2. associate-*r* 21.6%

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

   7. Simplified 21.6%

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

   8. Taylor expanded in phi2 around 0 19.7%

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

      1. associate-*r* 19.7%

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

   10. Simplified 19.7%

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

       1. add-sqr-sqrt 12.4%

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

       2. sqrt-unprod 18.3%

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

       3. sqr-neg 18.3%

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

       4. sqrt-unprod 8.0%

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

       5. add-sqr-sqrt 16.3%

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

       6. *-commutative 16.3%

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

   12. Applied egg-rr 16.3%

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

   13. Taylor expanded in phi1 around 0 16.5%

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

   if 5.0000000000000001e84 < phi2

   1. Initial program 51.1%

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

      1. hypot-define 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in phi2 around inf 67.6%

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

Alternative 11: 14.2% accurate, 109.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

   1. hypot-define 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 lambda2 around -inf 20.3%

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

   1. mul-1-neg 20.3%

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

   2. associate-*r* 20.2%

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

7. Simplified 20.2%

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

8. Taylor expanded in phi2 around 0 17.2%

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

   1. associate-*r* 17.2%

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

10. Simplified 17.2%

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

    1. add-sqr-sqrt 10.8%

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

    2. sqrt-unprod 17.0%

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

    3. sqr-neg 17.0%

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

    4. sqrt-unprod 7.4%

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

    5. add-sqr-sqrt 15.0%

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

    6. *-commutative 15.0%

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

12. Applied egg-rr 15.0%

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

13. Taylor expanded in phi1 around 0 15.2%

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

Reproduce

herbie shell --seed 2024169 
(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))))))