Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.5% → 99.8%

Time: 20.4s

Alternatives: 17

Speedup: 3.0×

• 30.1% 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: 59.5% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (- lambda1 lambda2)
    (log1p
     (expm1
      (-
       (* (cos (* phi1 0.5)) (cos (* 0.5 phi2)))
       (* (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) * log1p(expm1(((cos((phi1 * 0.5)) * cos((0.5 * phi2))) - (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.log1p(Math.expm1(((Math.cos((phi1 * 0.5)) * Math.cos((0.5 * phi2))) - (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.log1p(math.expm1(((math.cos((phi1 * 0.5)) * math.cos((0.5 * phi2))) - (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) * log1p(expm1(Float64(Float64(cos(Float64(phi1 * 0.5)) * cos(Float64(0.5 * phi2))) - Float64(sin(Float64(phi1 * 0.5)) * sin(Float64(0.5 * phi2))))))), Float64(phi1 - phi2)))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Log[1 + N[(Exp[N[(N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

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

   1. hypot-define 96.5%

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

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. Step-by-step derivation

   1. log1p-expm1-u 96.5%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)}, \phi_1 - \phi_2\right) \]

   2. div-inv 96.5%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}\right)\right), \phi_1 - \phi_2\right) \]

   3. metadata-eval 96.5%

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

6. Applied egg-rr 96.5%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\right)}, \phi_1 - \phi_2\right) \]
7. Step-by-step derivation

   1. *-commutative 96.5%

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

   2. distribute-rgt-in 96.5%

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

   3. cos-sum 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)\right), \phi_1 - \phi_2\right) \]
10. Add Preprocessing

Alternative 2: 95.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.2 \cdot 10^{+80}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 2.2e+80)
   (*
    R
    (hypot (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (- phi1 phi2)))
   (*
    R
    (hypot
     (*
      lambda2
      (-
       (* (sin (* phi1 0.5)) (sin (* 0.5 phi2)))
       (* (cos (* phi1 0.5)) (cos (* 0.5 phi2)))))
     (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 2.2e+80) {
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * hypot((lambda2 * ((sin((phi1 * 0.5)) * sin((0.5 * phi2))) - (cos((phi1 * 0.5)) * 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 (lambda2 <= 2.2e+80) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((lambda2 * ((Math.sin((phi1 * 0.5)) * Math.sin((0.5 * phi2))) - (Math.cos((phi1 * 0.5)) * Math.cos((0.5 * phi2))))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 2.2e+80:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))), (phi1 - phi2))
	else:
		tmp = R * math.hypot((lambda2 * ((math.sin((phi1 * 0.5)) * math.sin((0.5 * phi2))) - (math.cos((phi1 * 0.5)) * math.cos((0.5 * phi2))))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 2.2e+80)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(lambda2 * Float64(Float64(sin(Float64(phi1 * 0.5)) * sin(Float64(0.5 * phi2))) - Float64(cos(Float64(phi1 * 0.5)) * 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 (lambda2 <= 2.2e+80)
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))), (phi1 - phi2));
	else
		tmp = R * hypot((lambda2 * ((sin((phi1 * 0.5)) * sin((0.5 * phi2))) - (cos((phi1 * 0.5)) * cos((0.5 * phi2))))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 2.2e+80], 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], N[(R * N[Sqrt[N[(lambda2 * N[(N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 2.20000000000000003e80

   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 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

   if 2.20000000000000003e80 < lambda2

   1. Initial program 55.4%

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

      1. hypot-define 95.0%

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

   3. Simplified 95.0%

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

      1. log1p-expm1-u 95.0%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)}, \phi_1 - \phi_2\right) \]

      2. div-inv 95.0%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}\right)\right), \phi_1 - \phi_2\right) \]

      3. metadata-eval 95.0%

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

   6. Applied egg-rr 95.0%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\right)}, \phi_1 - \phi_2\right) \]
   7. Step-by-step derivation

      1. *-commutative 95.0%

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

      2. distribute-rgt-in 95.0%

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

      3. cos-sum 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in lambda1 around 0 96.0%

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

       1. mul-1-neg 96.0%

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

   11. Simplified 96.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.2 \cdot 10^{+80}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.2 \cdot 10^{+25}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 7.2e+25)
   (* 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 <= 7.2e+25) {
		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 <= 7.2e+25) {
		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 <= 7.2e+25:
		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 <= 7.2e+25)
		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 <= 7.2e+25)
		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, 7.2e+25], 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 < 7.20000000000000031e25

   1. Initial program 65.1%

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

      1. hypot-define 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. Taylor expanded in phi2 around 0 93.9%

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

   if 7.20000000000000031e25 < phi2

   1. Initial program 49.0%

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

      1. hypot-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in phi1 around 0 93.1%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.2 \cdot 10^{+25}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.1 \cdot 10^{+40}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.1e+40)
   (* R (hypot (* (- lambda1 lambda2) (cos (* phi1 0.5))) (- phi1 phi2)))
   (* R (hypot (* lambda1 (cos (* 0.5 phi2))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.1e+40) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * hypot((lambda1 * 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 <= 2.1e+40) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((lambda1 * Math.cos((0.5 * phi2))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.1e+40:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi1 * 0.5))), (phi1 - phi2))
	else:
		tmp = R * math.hypot((lambda1 * math.cos((0.5 * phi2))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.1e+40)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(lambda1 * 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 <= 2.1e+40)
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
	else
		tmp = R * hypot((lambda1 * cos((0.5 * phi2))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.1e+40], 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[(lambda1 * 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 < 2.1000000000000001e40

   1. Initial program 64.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. Taylor expanded in phi2 around 0 93.9%

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

   if 2.1000000000000001e40 < phi2

   1. Initial program 48.9%

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

      1. hypot-define 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in phi1 around 0 92.9%

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

   6. Taylor expanded in lambda1 around inf 89.4%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.1 \cdot 10^{+40}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4 \cdot 10^{-37}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 4e-37)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (hypot (* lambda1 (cos (* 0.5 phi2))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4e-37) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * hypot((lambda1 * 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 <= 4e-37) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * Math.hypot((lambda1 * Math.cos((0.5 * phi2))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 4e-37:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * math.hypot((lambda1 * math.cos((0.5 * phi2))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 4e-37)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * hypot(Float64(lambda1 * 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 <= 4e-37)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * hypot((lambda1 * cos((0.5 * phi2))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 4e-37], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda1 * 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 < 4.00000000000000027e-37

   1. Initial program 65.9%

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

      1. hypot-define 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in phi1 around 0 89.9%

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

   6. Taylor expanded in phi2 around 0 56.4%

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

      1. unpow2 56.4%

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

      2. unpow2 56.4%

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

      3. hypot-define 73.0%

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

   8. Simplified 73.0%

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

   if 4.00000000000000027e-37 < phi2

   1. Initial program 49.0%

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

      1. hypot-define 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in phi1 around 0 90.9%

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

   6. Taylor expanded in lambda1 around inf 86.3%

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

Alternative 6: 96.3% 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 61.0%

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

   1. hypot-define 96.5%

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

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. Add Preprocessing

Alternative 7: 70.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.2 \cdot 10^{+27}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3.2e+27)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (* phi2 (- 1.0 (/ phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.2e+27) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} 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 <= 3.2e+27) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3.2e+27:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3.2e+27)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 3.2e+27)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3.2e+27], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 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 < 3.20000000000000015e27

   1. Initial program 65.1%

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

      1. hypot-define 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. Taylor expanded in phi1 around 0 89.2%

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

   6. Taylor expanded in phi2 around 0 56.0%

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

      1. unpow2 56.0%

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

      2. unpow2 56.0%

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

      3. hypot-define 73.1%

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

   8. Simplified 73.1%

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

   if 3.20000000000000015e27 < phi2

   1. Initial program 49.0%

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

   3. Taylor expanded in phi2 around inf 70.0%

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

      1. mul-1-neg 70.0%

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

      2. unsub-neg 70.0%

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

   5. Simplified 70.0%

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

Alternative 8: 45.8% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6 \cdot 10^{+152}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq -1.55 \cdot 10^{-49}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{elif}\;\phi_1 \leq 2.2 \cdot 10^{-266}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{R \cdot \lambda_2}{\lambda_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R + \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -6e+152)
   (* phi1 (- (* R (/ phi2 phi1)) R))
   (if (<= phi1 -1.55e-49)
     (* phi2 (- R (/ (* R phi1) phi2)))
     (if (<= phi1 2.2e-266)
       (* lambda1 (- (/ (* R lambda2) lambda1) R))
       (* phi2 (+ R (* phi1 (/ R phi2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -6e+152) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= -1.55e-49) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else if (phi1 <= 2.2e-266) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} 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) :: tmp
    if (phi1 <= (-6d+152)) then
        tmp = phi1 * ((r * (phi2 / phi1)) - r)
    else if (phi1 <= (-1.55d-49)) then
        tmp = phi2 * (r - ((r * phi1) / phi2))
    else if (phi1 <= 2.2d-266) then
        tmp = lambda1 * (((r * lambda2) / lambda1) - r)
    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 tmp;
	if (phi1 <= -6e+152) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= -1.55e-49) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else if (phi1 <= 2.2e-266) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} else {
		tmp = phi2 * (R + (phi1 * (R / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -6e+152:
		tmp = phi1 * ((R * (phi2 / phi1)) - R)
	elif phi1 <= -1.55e-49:
		tmp = phi2 * (R - ((R * phi1) / phi2))
	elif phi1 <= 2.2e-266:
		tmp = lambda1 * (((R * lambda2) / lambda1) - R)
	else:
		tmp = phi2 * (R + (phi1 * (R / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -6e+152)
		tmp = Float64(phi1 * Float64(Float64(R * Float64(phi2 / phi1)) - R));
	elseif (phi1 <= -1.55e-49)
		tmp = Float64(phi2 * Float64(R - Float64(Float64(R * phi1) / phi2)));
	elseif (phi1 <= 2.2e-266)
		tmp = Float64(lambda1 * Float64(Float64(Float64(R * lambda2) / lambda1) - R));
	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)
	tmp = 0.0;
	if (phi1 <= -6e+152)
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	elseif (phi1 <= -1.55e-49)
		tmp = phi2 * (R - ((R * phi1) / phi2));
	elseif (phi1 <= 2.2e-266)
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	else
		tmp = phi2 * (R + (phi1 * (R / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -6e+152], N[(phi1 * N[(N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -1.55e-49], N[(phi2 * N[(R - N[(N[(R * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 2.2e-266], N[(lambda1 * N[(N[(N[(R * lambda2), $MachinePrecision] / lambda1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(phi2 * N[(R + N[(phi1 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -5.99999999999999981e152

   1. Initial program 41.7%

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

      1. hypot-define 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in phi2 around inf 73.7%

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

      1. associate-*r/ 73.7%

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

      2. mul-1-neg 73.7%

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

      3. *-commutative 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in phi1 around inf 84.9%

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

      1. +-commutative 84.9%

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

      2. mul-1-neg 84.9%

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

      3. unsub-neg 84.9%

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

      4. associate-/l* 84.9%

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

   10. Simplified 84.9%

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

   if -5.99999999999999981e152 < phi1 < -1.55e-49

   1. Initial program 67.0%

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

      1. hypot-define 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in phi2 around inf 60.6%

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

      1. associate-*r/ 60.6%

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

      2. mul-1-neg 60.6%

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

      3. *-commutative 60.6%

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

   7. Simplified 60.6%

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

      1. *-un-lft-identity 60.6%

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

      2. fma-define 60.6%

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

      3. distribute-frac-neg 60.6%

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

      4. add-sqr-sqrt 38.2%

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

      5. times-frac 38.1%

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

      6. add-sqr-sqrt 24.9%

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

      7. sqrt-unprod 25.1%

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

      8. sqr-neg 25.1%

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

      9. sqrt-unprod 2.5%

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

      10. add-sqr-sqrt 20.3%

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

      11. times-frac 15.9%

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

      12. distribute-rgt-neg-in 15.9%

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

      13. add-sqr-sqrt 16.4%

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

      14. fmm-def 16.4%

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

      15. *-un-lft-identity 16.4%

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

      16. distribute-rgt-neg-in 16.4%

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

      17. add-sqr-sqrt 15.9%

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

      18. times-frac 20.3%

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

   9. Applied egg-rr 60.6%

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

   if -1.55e-49 < phi1 < 2.2e-266

   1. Initial program 63.8%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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 31.2%

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

      1. mul-1-neg 31.2%

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

      2. distribute-rgt-neg-in 31.2%

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

      3. +-commutative 31.2%

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

      4. mul-1-neg 31.2%

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

      5. unsub-neg 31.2%

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

      6. *-commutative 31.2%

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

      7. associate-*r* 31.2%

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

   7. Simplified 31.2%

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

   8. Taylor expanded in phi1 around 0 31.2%

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

   9. Taylor expanded in phi2 around 0 34.6%

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

   if 2.2e-266 < phi1

   1. Initial program 62.8%

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

      1. hypot-define 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in phi2 around inf 13.3%

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

      1. associate-*r/ 13.3%

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

      2. mul-1-neg 13.3%

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

      3. *-commutative 13.3%

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

   7. Simplified 13.3%

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

      1. distribute-lft-in 13.3%

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

      2. distribute-rgt-neg-in 13.3%

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

      3. add-sqr-sqrt 12.8%

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

      4. times-frac 12.8%

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

      5. add-sqr-sqrt 9.1%

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

      6. sqrt-unprod 20.5%

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

      7. sqr-neg 20.5%

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

      8. sqrt-unprod 13.1%

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

      9. add-sqr-sqrt 29.2%

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

      10. times-frac 31.6%

          \[\leadsto \phi_2 \cdot R + \phi_2 \cdot \color{blue}{\frac{\phi_1 \cdot R}{\sqrt{\phi_2} \cdot \sqrt{\phi_2}}} \]

      11. add-sqr-sqrt 43.4%

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

   9. Applied egg-rr 43.4%

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

       1. distribute-lft-in 45.9%

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

       2. associate-*r/ 42.6%

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

   11. Simplified 42.6%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6 \cdot 10^{+152}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq -1.55 \cdot 10^{-49}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{elif}\;\phi_1 \leq 2.2 \cdot 10^{-266}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{R \cdot \lambda_2}{\lambda_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R + \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.4% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \phi_1 \cdot \frac{R}{\phi_2}\\ \mathbf{if}\;\phi_1 \leq -4.1 \cdot 10^{+151}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq -9.5 \cdot 10^{-14}:\\ \;\;\;\;\phi_2 \cdot \left(R - t\_0\right)\\ \mathbf{elif}\;\phi_1 \leq 2.2 \cdot 10^{-268}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{R \cdot \lambda_2}{\lambda_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* phi1 (/ R phi2))))
   (if (<= phi1 -4.1e+151)
     (* phi1 (- (* R (/ phi2 phi1)) R))
     (if (<= phi1 -9.5e-14)
       (* phi2 (- R t_0))
       (if (<= phi1 2.2e-268)
         (* lambda1 (- (/ (* R lambda2) lambda1) R))
         (* phi2 (+ R t_0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = phi1 * (R / phi2);
	double tmp;
	if (phi1 <= -4.1e+151) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= -9.5e-14) {
		tmp = phi2 * (R - t_0);
	} else if (phi1 <= 2.2e-268) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} else {
		tmp = phi2 * (R + t_0);
	}
	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 = phi1 * (r / phi2)
    if (phi1 <= (-4.1d+151)) then
        tmp = phi1 * ((r * (phi2 / phi1)) - r)
    else if (phi1 <= (-9.5d-14)) then
        tmp = phi2 * (r - t_0)
    else if (phi1 <= 2.2d-268) then
        tmp = lambda1 * (((r * lambda2) / lambda1) - r)
    else
        tmp = phi2 * (r + t_0)
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = phi1 * (R / phi2);
	double tmp;
	if (phi1 <= -4.1e+151) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= -9.5e-14) {
		tmp = phi2 * (R - t_0);
	} else if (phi1 <= 2.2e-268) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} else {
		tmp = phi2 * (R + t_0);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = phi1 * (R / phi2)
	tmp = 0
	if phi1 <= -4.1e+151:
		tmp = phi1 * ((R * (phi2 / phi1)) - R)
	elif phi1 <= -9.5e-14:
		tmp = phi2 * (R - t_0)
	elif phi1 <= 2.2e-268:
		tmp = lambda1 * (((R * lambda2) / lambda1) - R)
	else:
		tmp = phi2 * (R + t_0)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(phi1 * Float64(R / phi2))
	tmp = 0.0
	if (phi1 <= -4.1e+151)
		tmp = Float64(phi1 * Float64(Float64(R * Float64(phi2 / phi1)) - R));
	elseif (phi1 <= -9.5e-14)
		tmp = Float64(phi2 * Float64(R - t_0));
	elseif (phi1 <= 2.2e-268)
		tmp = Float64(lambda1 * Float64(Float64(Float64(R * lambda2) / lambda1) - R));
	else
		tmp = Float64(phi2 * Float64(R + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = phi1 * (R / phi2);
	tmp = 0.0;
	if (phi1 <= -4.1e+151)
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	elseif (phi1 <= -9.5e-14)
		tmp = phi2 * (R - t_0);
	elseif (phi1 <= 2.2e-268)
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	else
		tmp = phi2 * (R + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(phi1 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -4.1e+151], N[(phi1 * N[(N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -9.5e-14], N[(phi2 * N[(R - t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 2.2e-268], N[(lambda1 * N[(N[(N[(R * lambda2), $MachinePrecision] / lambda1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(phi2 * N[(R + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -4.0999999999999998e151

   1. Initial program 41.7%

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

      1. hypot-define 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in phi2 around inf 73.7%

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

      1. associate-*r/ 73.7%

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

      2. mul-1-neg 73.7%

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

      3. *-commutative 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in phi1 around inf 84.9%

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

      1. +-commutative 84.9%

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

      2. mul-1-neg 84.9%

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

      3. unsub-neg 84.9%

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

      4. associate-/l* 84.9%

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

   10. Simplified 84.9%

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

   if -4.0999999999999998e151 < phi1 < -9.4999999999999999e-14

   1. Initial program 69.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. Taylor expanded in phi2 around inf 65.1%

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

      1. associate-*r/ 65.1%

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

      2. mul-1-neg 65.1%

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

      3. *-commutative 65.1%

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

   7. Simplified 65.1%

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

   8. Taylor expanded in phi2 around inf 65.1%

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

      1. mul-1-neg 65.1%

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

      2. *-commutative 65.1%

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

      3. sub-neg 65.1%

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

      4. associate-*r/ 62.2%

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

   10. Simplified 62.2%

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

   if -9.4999999999999999e-14 < phi1 < 2.20000000000000004e-268

   1. Initial program 61.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 31.2%

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

      1. mul-1-neg 31.2%

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

      2. distribute-rgt-neg-in 31.2%

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

      3. +-commutative 31.2%

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

      4. mul-1-neg 31.2%

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

      5. unsub-neg 31.2%

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

      6. *-commutative 31.2%

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

      7. associate-*r* 31.2%

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

   7. Simplified 31.2%

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

   8. Taylor expanded in phi1 around 0 31.2%

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

   9. Taylor expanded in phi2 around 0 34.2%

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

   if 2.20000000000000004e-268 < phi1

   1. Initial program 63.1%

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

      1. hypot-define 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in phi2 around inf 13.2%

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

      1. associate-*r/ 13.2%

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

      2. mul-1-neg 13.2%

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

      3. *-commutative 13.2%

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

   7. Simplified 13.2%

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

      1. distribute-lft-in 13.2%

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

      2. distribute-rgt-neg-in 13.2%

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

      3. add-sqr-sqrt 12.7%

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

      4. times-frac 12.7%

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

      5. add-sqr-sqrt 9.0%

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

      6. sqrt-unprod 21.2%

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

      7. sqr-neg 21.2%

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

      8. sqrt-unprod 13.9%

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

      9. add-sqr-sqrt 29.8%

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

      10. times-frac 31.4%

          \[\leadsto \phi_2 \cdot R + \phi_2 \cdot \color{blue}{\frac{\phi_1 \cdot R}{\sqrt{\phi_2} \cdot \sqrt{\phi_2}}} \]

      11. add-sqr-sqrt 43.1%

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

   9. Applied egg-rr 43.1%

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

       1. distribute-lft-in 45.6%

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

       2. associate-*r/ 43.1%

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

   11. Simplified 43.1%

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

Alternative 10: 45.3% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.25 \cdot 10^{+152}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq -6.4 \cdot 10^{-52}:\\ \;\;\;\;\phi_2 \cdot \left(R - R \cdot \frac{\phi_1}{\phi_2}\right)\\ \mathbf{elif}\;\phi_1 \leq 4.5 \cdot 10^{-265}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{R \cdot \lambda_2}{\lambda_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R + \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.25e+152)
   (* phi1 (- (* R (/ phi2 phi1)) R))
   (if (<= phi1 -6.4e-52)
     (* phi2 (- R (* R (/ phi1 phi2))))
     (if (<= phi1 4.5e-265)
       (* lambda1 (- (/ (* R lambda2) lambda1) R))
       (* phi2 (+ R (* phi1 (/ R phi2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.25e+152) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= -6.4e-52) {
		tmp = phi2 * (R - (R * (phi1 / phi2)));
	} else if (phi1 <= 4.5e-265) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} 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) :: tmp
    if (phi1 <= (-1.25d+152)) then
        tmp = phi1 * ((r * (phi2 / phi1)) - r)
    else if (phi1 <= (-6.4d-52)) then
        tmp = phi2 * (r - (r * (phi1 / phi2)))
    else if (phi1 <= 4.5d-265) then
        tmp = lambda1 * (((r * lambda2) / lambda1) - r)
    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 tmp;
	if (phi1 <= -1.25e+152) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= -6.4e-52) {
		tmp = phi2 * (R - (R * (phi1 / phi2)));
	} else if (phi1 <= 4.5e-265) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} else {
		tmp = phi2 * (R + (phi1 * (R / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.25e+152:
		tmp = phi1 * ((R * (phi2 / phi1)) - R)
	elif phi1 <= -6.4e-52:
		tmp = phi2 * (R - (R * (phi1 / phi2)))
	elif phi1 <= 4.5e-265:
		tmp = lambda1 * (((R * lambda2) / lambda1) - R)
	else:
		tmp = phi2 * (R + (phi1 * (R / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.25e+152)
		tmp = Float64(phi1 * Float64(Float64(R * Float64(phi2 / phi1)) - R));
	elseif (phi1 <= -6.4e-52)
		tmp = Float64(phi2 * Float64(R - Float64(R * Float64(phi1 / phi2))));
	elseif (phi1 <= 4.5e-265)
		tmp = Float64(lambda1 * Float64(Float64(Float64(R * lambda2) / lambda1) - R));
	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)
	tmp = 0.0;
	if (phi1 <= -1.25e+152)
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	elseif (phi1 <= -6.4e-52)
		tmp = phi2 * (R - (R * (phi1 / phi2)));
	elseif (phi1 <= 4.5e-265)
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	else
		tmp = phi2 * (R + (phi1 * (R / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.25e+152], N[(phi1 * N[(N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -6.4e-52], N[(phi2 * N[(R - N[(R * N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4.5e-265], N[(lambda1 * N[(N[(N[(R * lambda2), $MachinePrecision] / lambda1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(phi2 * N[(R + N[(phi1 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -1.25e152

   1. Initial program 41.7%

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

      1. hypot-define 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in phi2 around inf 73.7%

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

      1. associate-*r/ 73.7%

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

      2. mul-1-neg 73.7%

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

      3. *-commutative 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in phi1 around inf 84.9%

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

      1. +-commutative 84.9%

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

      2. mul-1-neg 84.9%

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

      3. unsub-neg 84.9%

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

      4. associate-/l* 84.9%

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

   10. Simplified 84.9%

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

   if -1.25e152 < phi1 < -6.4000000000000002e-52

   1. Initial program 67.7%

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

      1. hypot-define 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in phi2 around inf 59.3%

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

      1. mul-1-neg 59.3%

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

      2. unsub-neg 59.3%

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

      3. associate-/l* 54.9%

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

   7. Simplified 54.9%

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

   if -6.4000000000000002e-52 < phi1 < 4.5000000000000003e-265

   1. Initial program 63.1%

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

      1. hypot-define 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 31.8%

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

      1. mul-1-neg 31.8%

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

      2. distribute-rgt-neg-in 31.8%

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

      3. +-commutative 31.8%

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

      4. mul-1-neg 31.8%

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

      5. unsub-neg 31.8%

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

      6. *-commutative 31.8%

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

      7. associate-*r* 31.8%

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

   7. Simplified 31.8%

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

   8. Taylor expanded in phi1 around 0 31.8%

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

   9. Taylor expanded in phi2 around 0 33.5%

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

   if 4.5000000000000003e-265 < phi1

   1. Initial program 62.8%

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

      1. hypot-define 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in phi2 around inf 13.3%

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

      1. associate-*r/ 13.3%

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

      2. mul-1-neg 13.3%

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

      3. *-commutative 13.3%

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

   7. Simplified 13.3%

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

      1. distribute-lft-in 13.3%

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

      2. distribute-rgt-neg-in 13.3%

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

      3. add-sqr-sqrt 12.8%

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

      4. times-frac 12.8%

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

      5. add-sqr-sqrt 9.1%

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

      6. sqrt-unprod 20.5%

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

      7. sqr-neg 20.5%

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

      8. sqrt-unprod 13.1%

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

      9. add-sqr-sqrt 29.2%

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

      10. times-frac 31.6%

          \[\leadsto \phi_2 \cdot R + \phi_2 \cdot \color{blue}{\frac{\phi_1 \cdot R}{\sqrt{\phi_2} \cdot \sqrt{\phi_2}}} \]

      11. add-sqr-sqrt 43.4%

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

   9. Applied egg-rr 43.4%

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

       1. distribute-lft-in 45.9%

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

       2. associate-*r/ 42.6%

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

   11. Simplified 42.6%

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

Alternative 11: 45.6% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.1 \cdot 10^{-12}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq 1.3 \cdot 10^{-266}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{R \cdot \lambda_2}{\lambda_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R + \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -5.1e-12)
   (* phi1 (- (* R (/ phi2 phi1)) R))
   (if (<= phi1 1.3e-266)
     (* lambda1 (- (/ (* R lambda2) lambda1) R))
     (* phi2 (+ R (* phi1 (/ R phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5.1e-12) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= 1.3e-266) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} 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) :: tmp
    if (phi1 <= (-5.1d-12)) then
        tmp = phi1 * ((r * (phi2 / phi1)) - r)
    else if (phi1 <= 1.3d-266) then
        tmp = lambda1 * (((r * lambda2) / lambda1) - r)
    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 tmp;
	if (phi1 <= -5.1e-12) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= 1.3e-266) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} else {
		tmp = phi2 * (R + (phi1 * (R / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -5.1e-12:
		tmp = phi1 * ((R * (phi2 / phi1)) - R)
	elif phi1 <= 1.3e-266:
		tmp = lambda1 * (((R * lambda2) / lambda1) - R)
	else:
		tmp = phi2 * (R + (phi1 * (R / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -5.1e-12)
		tmp = Float64(phi1 * Float64(Float64(R * Float64(phi2 / phi1)) - R));
	elseif (phi1 <= 1.3e-266)
		tmp = Float64(lambda1 * Float64(Float64(Float64(R * lambda2) / lambda1) - R));
	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)
	tmp = 0.0;
	if (phi1 <= -5.1e-12)
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	elseif (phi1 <= 1.3e-266)
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	else
		tmp = phi2 * (R + (phi1 * (R / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -5.1e-12], N[(phi1 * N[(N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 1.3e-266], N[(lambda1 * N[(N[(N[(R * lambda2), $MachinePrecision] / lambda1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(phi2 * N[(R + N[(phi1 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -5.09999999999999968e-12

   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 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in phi2 around inf 69.2%

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

      1. associate-*r/ 69.2%

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

      2. mul-1-neg 69.2%

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

      3. *-commutative 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in phi1 around inf 69.1%

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

      1. +-commutative 69.1%

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

      2. mul-1-neg 69.1%

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

      3. unsub-neg 69.1%

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

      4. associate-/l* 69.1%

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

   10. Simplified 69.1%

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

   if -5.09999999999999968e-12 < phi1 < 1.3e-266

   1. Initial program 62.5%

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

      1. 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 32.2%

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

      1. mul-1-neg 32.2%

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

      2. distribute-rgt-neg-in 32.2%

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

      3. +-commutative 32.2%

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

      4. mul-1-neg 32.2%

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

      5. unsub-neg 32.2%

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

      6. *-commutative 32.2%

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

      7. associate-*r* 32.2%

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

   7. Simplified 32.2%

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

   8. Taylor expanded in phi1 around 0 32.2%

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

   9. Taylor expanded in phi2 around 0 35.2%

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

   if 1.3e-266 < phi1

   1. Initial program 62.8%

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

      1. hypot-define 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in phi2 around inf 13.3%

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

      1. associate-*r/ 13.3%

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

      2. mul-1-neg 13.3%

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

      3. *-commutative 13.3%

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

   7. Simplified 13.3%

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

      1. distribute-lft-in 13.3%

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

      2. distribute-rgt-neg-in 13.3%

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

      3. add-sqr-sqrt 12.8%

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

      4. times-frac 12.8%

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

      5. add-sqr-sqrt 9.1%

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

      6. sqrt-unprod 20.5%

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

      7. sqr-neg 20.5%

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

      8. sqrt-unprod 13.1%

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

      9. add-sqr-sqrt 29.2%

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

      10. times-frac 31.6%

          \[\leadsto \phi_2 \cdot R + \phi_2 \cdot \color{blue}{\frac{\phi_1 \cdot R}{\sqrt{\phi_2} \cdot \sqrt{\phi_2}}} \]

      11. add-sqr-sqrt 43.4%

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

   9. Applied egg-rr 43.4%

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

       1. distribute-lft-in 45.9%

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

       2. associate-*r/ 42.6%

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

   11. Simplified 42.6%

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

Alternative 12: 33.4% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.45 \cdot 10^{-116}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_2 \leq 1.35 \cdot 10^{-31}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{R \cdot \lambda_2}{\lambda_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -1.45e-116)
   (* phi1 (- (* R (/ phi2 phi1)) R))
   (if (<= phi2 1.35e-31)
     (* lambda1 (- (/ (* R lambda2) lambda1) R))
     (* R (* phi2 (- 1.0 (/ phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -1.45e-116) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi2 <= 1.35e-31) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= (-1.45d-116)) then
        tmp = phi1 * ((r * (phi2 / phi1)) - r)
    else if (phi2 <= 1.35d-31) then
        tmp = lambda1 * (((r * lambda2) / lambda1) - r)
    else
        tmp = r * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -1.45e-116) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi2 <= 1.35e-31) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -1.45e-116:
		tmp = phi1 * ((R * (phi2 / phi1)) - R)
	elif phi2 <= 1.35e-31:
		tmp = lambda1 * (((R * lambda2) / lambda1) - R)
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -1.45e-116)
		tmp = Float64(phi1 * Float64(Float64(R * Float64(phi2 / phi1)) - R));
	elseif (phi2 <= 1.35e-31)
		tmp = Float64(lambda1 * Float64(Float64(Float64(R * lambda2) / lambda1) - R));
	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 <= -1.45e-116)
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	elseif (phi2 <= 1.35e-31)
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -1.45e-116], N[(phi1 * N[(N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.35e-31], N[(lambda1 * N[(N[(N[(R * lambda2), $MachinePrecision] / lambda1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.4499999999999999e-116

   1. Initial program 66.4%

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

      1. hypot-define 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 17.3%

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

      1. associate-*r/ 17.3%

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

      2. mul-1-neg 17.3%

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

      3. *-commutative 17.3%

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

   7. Simplified 17.3%

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

   8. Taylor expanded in phi1 around inf 16.2%

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

      1. +-commutative 16.2%

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

      2. mul-1-neg 16.2%

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

      3. unsub-neg 16.2%

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

      4. associate-/l* 16.1%

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

   10. Simplified 16.1%

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

   if -1.4499999999999999e-116 < phi2 < 1.35000000000000007e-31

   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 99.9%

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

   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 36.4%

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

      1. mul-1-neg 36.4%

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

      2. distribute-rgt-neg-in 36.4%

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

      3. +-commutative 36.4%

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

      4. mul-1-neg 36.4%

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

      5. unsub-neg 36.4%

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

      6. *-commutative 36.4%

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

      7. associate-*r* 36.4%

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

   7. Simplified 36.4%

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

   8. Taylor expanded in phi1 around 0 34.8%

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

   9. Taylor expanded in phi2 around 0 34.8%

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

   if 1.35000000000000007e-31 < phi2

   1. Initial program 48.8%

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

   3. Taylor expanded in phi2 around inf 67.5%

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

      1. mul-1-neg 67.5%

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

      2. unsub-neg 67.5%

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

   5. Simplified 67.5%

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

Alternative 13: 34.4% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-32}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{R \cdot \lambda_2}{\lambda_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -8.5e-115)
   (* R (- phi1))
   (if (<= phi2 7.5e-32)
     (* lambda1 (- (/ (* R lambda2) lambda1) R))
     (* R (* phi2 (- 1.0 (/ phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -8.5e-115) {
		tmp = R * -phi1;
	} else if (phi2 <= 7.5e-32) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= (-8.5d-115)) then
        tmp = r * -phi1
    else if (phi2 <= 7.5d-32) then
        tmp = lambda1 * (((r * lambda2) / lambda1) - r)
    else
        tmp = r * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -8.5e-115) {
		tmp = R * -phi1;
	} else if (phi2 <= 7.5e-32) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -8.5e-115:
		tmp = R * -phi1
	elif phi2 <= 7.5e-32:
		tmp = lambda1 * (((R * lambda2) / lambda1) - R)
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -8.5e-115)
		tmp = Float64(R * Float64(-phi1));
	elseif (phi2 <= 7.5e-32)
		tmp = Float64(lambda1 * Float64(Float64(Float64(R * lambda2) / lambda1) - R));
	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 <= -8.5e-115)
		tmp = R * -phi1;
	elseif (phi2 <= 7.5e-32)
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -8.5e-115], N[(R * (-phi1)), $MachinePrecision], If[LessEqual[phi2, 7.5e-32], N[(lambda1 * N[(N[(N[(R * lambda2), $MachinePrecision] / lambda1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -8.49999999999999953e-115

   1. Initial program 66.4%

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

      1. hypot-define 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 phi1 around -inf 18.3%

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

      1. mul-1-neg 18.3%

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

      2. *-commutative 18.3%

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

      3. distribute-rgt-neg-in 18.3%

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

   7. Simplified 18.3%

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

   if -8.49999999999999953e-115 < phi2 < 7.49999999999999953e-32

   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 99.9%

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

   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 36.4%

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

      1. mul-1-neg 36.4%

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

      2. distribute-rgt-neg-in 36.4%

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

      3. +-commutative 36.4%

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

      4. mul-1-neg 36.4%

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

      5. unsub-neg 36.4%

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

      6. *-commutative 36.4%

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

      7. associate-*r* 36.4%

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

   7. Simplified 36.4%

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

   8. Taylor expanded in phi1 around 0 34.8%

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

   9. Taylor expanded in phi2 around 0 34.8%

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

   if 7.49999999999999953e-32 < phi2

   1. Initial program 48.8%

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

   3. Taylor expanded in phi2 around inf 67.5%

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

      1. mul-1-neg 67.5%

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

      2. unsub-neg 67.5%

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

   5. Simplified 67.5%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-32}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{R \cdot \lambda_2}{\lambda_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 34.4% accurate, 27.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3 \cdot 10^{+120}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 3e+120) (- (* R phi2) (* R phi1)) (* R lambda2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 3e+120) {
		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 <= 3d+120) 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 <= 3e+120) {
		tmp = (R * phi2) - (R * phi1);
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 3e+120:
		tmp = (R * phi2) - (R * phi1)
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 3e+120)
		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 <= 3e+120)
		tmp = (R * phi2) - (R * phi1);
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 3e+120], N[(N[(R * phi2), $MachinePrecision] - N[(R * phi1), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 3e120

   1. Initial program 61.6%

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

      1. hypot-define 97.0%

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

   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 phi2 around inf 33.4%

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

      1. associate-*r/ 33.4%

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

      2. mul-1-neg 33.4%

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

      3. *-commutative 33.4%

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

   7. Simplified 33.4%

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

   8. Taylor expanded in phi2 around 0 33.4%

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

      1. +-commutative 33.4%

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

      2. *-commutative 33.4%

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

      3. mul-1-neg 33.4%

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

      4. *-commutative 33.4%

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

      5. unsub-neg 33.4%

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

   10. Simplified 33.4%

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

   if 3e120 < lambda2

   1. Initial program 58.0%

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

      1. hypot-define 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 40.3%

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

      1. mul-1-neg 40.3%

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

      2. *-commutative 40.3%

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

      3. distribute-rgt-neg-in 40.3%

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

   7. Simplified 40.3%

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

   8. Taylor expanded in phi2 around 0 25.8%

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

      1. *-commutative 25.8%

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

   10. Simplified 25.8%

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

   11. Taylor expanded in phi1 around 0 0.5%

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

       1. associate-*r* 0.5%

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

       2. neg-mul-1 0.5%

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

   13. Simplified 0.5%

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

       1. add-sqr-sqrt 0.3%

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

       2. sqrt-unprod 34.4%

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

       3. sqr-neg 34.4%

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

       4. sqrt-unprod 35.2%

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

       5. add-sqr-sqrt 62.6%

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

       6. *-un-lft-identity 62.6%

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

   15. Applied egg-rr 62.6%

       \[\leadsto \color{blue}{\left(1 \cdot R\right)} \cdot \lambda_2 \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3 \cdot 10^{+120}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 28.7% accurate, 36.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-46}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.7e-46) (* R (- phi1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.7e-46) {
		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 <= (-3.7d-46)) 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 <= -3.7e-46) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.7e-46:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.7e-46)
		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 <= -3.7e-46)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3.7e-46], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.69999999999999983e-46

   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 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in phi1 around -inf 58.2%

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

      1. mul-1-neg 58.2%

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

      2. *-commutative 58.2%

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

      3. distribute-rgt-neg-in 58.2%

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

   7. Simplified 58.2%

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

   if -3.69999999999999983e-46 < phi1

   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 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 phi2 around inf 18.6%

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

      1. *-commutative 18.6%

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

   7. Simplified 18.6%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{-46}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 17.9% 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 61.0%

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

   1. hypot-define 96.5%

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

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 17.2%

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

   1. *-commutative 17.2%

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

7. Simplified 17.2%

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

8. Final simplification 17.2%

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

Alternative 17: 13.3% 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 61.0%

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

   1. hypot-define 96.5%

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

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 phi1 around 0 90.2%

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

6. Taylor expanded in lambda1 around inf 13.8%

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

7. Taylor expanded in phi2 around 0 11.7%

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

   1. *-commutative 11.7%

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

9. Simplified 11.7%

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

10. Final simplification 11.7%

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

Reproduce

herbie shell --seed 2024131 
(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))))))