Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.0% → 99.8%

Time: 16.8s

Alternatives: 14

Speedup: 1.6×

• 30.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.0% 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

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (- lambda1 lambda2)
    (expm1
     (log1p
      (-
       (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
       (* (sin (* 0.5 phi1)) (sin (* phi2 0.5)))))))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * expm1(log1p(((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot(((lambda1 - lambda2) * Math.expm1(Math.log1p(((Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - (Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5))))))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot(((lambda1 - lambda2) * math.expm1(math.log1p(((math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))) - (math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))))))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * expm1(log1p(Float64(Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5))))))), Float64(phi1 - phi2)))
end

Wolfram

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

Derivation

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-def 96.2%

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

3. Simplified 96.2%

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

   1. expm1-log1p-u 96.1%

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

   2. div-inv 96.1%

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

   3. metadata-eval 96.1%

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

5. Applied egg-rr 96.1%

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

   1. *-commutative 96.1%

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

   2. +-commutative 96.1%

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

   3. distribute-rgt-in 96.1%

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

   4. cos-sum 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right) + -1\right)\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (- lambda1 lambda2)
    (+
     1.0
     (+
      (-
       (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
       (* (sin (* 0.5 phi1)) (sin (* phi2 0.5))))
      -1.0)))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * (1.0 + (((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5)))) + -1.0))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot(((lambda1 - lambda2) * (1.0 + (((Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - (Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5)))) + -1.0))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot(((lambda1 - lambda2) * (1.0 + (((math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))) - (math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5)))) + -1.0))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * Float64(1.0 + Float64(Float64(Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5)))) + -1.0))), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * (1.0 + (((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5)))) + -1.0))), (phi1 - phi2));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(1.0 + N[(N[(N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

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-def 96.2%

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

3. Simplified 96.2%

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

   1. expm1-log1p-u 96.1%

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

   2. div-inv 96.1%

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

   3. metadata-eval 96.1%

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

5. Applied egg-rr 96.1%

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

   1. *-commutative 96.1%

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

   2. +-commutative 96.1%

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

   3. distribute-rgt-in 96.1%

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

   4. cos-sum 99.9%

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

7. Applied egg-rr 99.9%

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

   1. expm1-udef 99.8%

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

   2. log1p-expm1-u 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)\right)}} - 1\right), \phi_1 - \phi_2\right) \]

   3. log1p-udef 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)\right)}} - 1\right), \phi_1 - \phi_2\right) \]

   4. add-exp-log 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\color{blue}{\left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)\right)} - 1\right), \phi_1 - \phi_2\right) \]

   5. expm1-log1p-u 99.8%

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

   6. *-commutative 99.8%

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

   7. *-commutative 99.8%

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

9. Applied egg-rr 99.8%

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

    1. associate--l+ 99.8%

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

    2. *-commutative 99.8%

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

    3. *-commutative 99.8%

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

    4. *-commutative 99.8%

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

    5. *-commutative 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right) + -1\right)\right), \phi_1 - \phi_2\right) \]

Alternative 3: 96.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4 \cdot 10^{+196}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -4e+196)
   (*
    R
    (hypot
     (*
      lambda1
      (-
       (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
       (* (sin (* 0.5 phi1)) (sin (* phi2 0.5)))))
     (- phi1 phi2)))
   (*
    R
    (hypot
     (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0)))
     (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -4e+196) {
		tmp = R * hypot((lambda1 * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -4e+196) {
		tmp = R * Math.hypot((lambda1 * ((Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - (Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5))))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -4e+196:
		tmp = R * math.hypot((lambda1 * ((math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))) - (math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -4e+196)
		tmp = Float64(R * hypot(Float64(lambda1 * Float64(Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5))))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -4e+196)
		tmp = R * hypot((lambda1 * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -4e+196], N[(R * N[Sqrt[N[(lambda1 * N[(N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -3.9999999999999998e196

   1. Initial program 62.7%

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

      1. hypot-def 93.3%

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

   3. Simplified 93.3%

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

      1. expm1-log1p-u 93.2%

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

      2. div-inv 93.2%

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

      3. metadata-eval 93.2%

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

   5. Applied egg-rr 93.2%

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

      1. *-commutative 93.2%

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

      2. +-commutative 93.2%

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

      3. distribute-rgt-in 93.2%

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

      4. cos-sum 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in lambda1 around inf 95.8%

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

   if -3.9999999999999998e196 < lambda1

   1. Initial program 66.2%

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

      1. hypot-def 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%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4 \cdot 10^{+196}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 4: 83.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.25 \cdot 10^{-58}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.25e-58)
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) (- phi1 phi2)))
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* phi2 0.5)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.25e-58) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.25e-58) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((phi2 * 0.5))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.25e-58:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((phi2 * 0.5))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.25e-58)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.25e-58)
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.25e-58], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.2500000000000001e-58

   1. Initial program 57.7%

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

      1. hypot-def 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in phi2 around 0 90.1%

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

   if -2.2500000000000001e-58 < phi1

   1. Initial program 69.5%

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

      1. hypot-def 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in phi1 around 0 58.3%

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

      1. *-commutative 58.3%

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

      2. unpow2 58.3%

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

      3. unpow2 58.3%

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

      4. unpow2 58.3%

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

      5. unswap-sqr 58.3%

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

      6. hypot-def 79.8%

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

   6. Simplified 79.8%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.25 \cdot 10^{-58}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 95.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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-def 96.2%

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

3. Simplified 96.2%

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

4. Final simplification 96.2%

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

Alternative 6: 74.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.3 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3.3e-9)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* 0.5 phi1)))))
   (* R (- phi2 phi1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.3e-9) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.3e-9) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((0.5 * phi1))));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3.3e-9:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((0.5 * phi1))))
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3.3e-9)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1)))));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 3.3e-9)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3.3e-9], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.30000000000000018e-9

   1. Initial program 68.9%

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

      1. hypot-def 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. Taylor expanded in phi2 around 0 62.0%

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

      1. *-commutative 62.0%

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

      2. +-commutative 62.0%

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

      3. unpow2 62.0%

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

      4. unpow2 62.0%

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

      5. unpow2 62.0%

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

      6. unswap-sqr 62.0%

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

      7. hypot-def 81.4%

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

   6. Simplified 81.4%

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

   if 3.30000000000000018e-9 < phi2

   1. Initial program 57.2%

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

      1. hypot-def 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in phi1 around -inf 63.8%

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

      1. *-commutative 63.8%

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

      2. associate-*r* 63.8%

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

      3. distribute-rgt-out 63.8%

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

      4. mul-1-neg 63.8%

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

   6. Simplified 63.8%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.3 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 7: 77.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8.5 \cdot 10^{-79}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 8.5e-79)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* 0.5 phi1)))))
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* phi2 0.5)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 8.5e-79) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 8.5e-79) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((0.5 * phi1))));
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((phi2 * 0.5))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 8.5e-79:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((0.5 * phi1))))
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((phi2 * 0.5))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 8.5e-79)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1)))));
	else
		tmp = Float64(R * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 8.5e-79)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 8.5e-79], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 8.50000000000000029e-79

   1. Initial program 68.4%

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

      1. hypot-def 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in phi2 around 0 62.1%

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

      1. *-commutative 62.1%

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

      2. +-commutative 62.1%

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

      3. unpow2 62.1%

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

      4. unpow2 62.1%

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

      5. unpow2 62.1%

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

      6. unswap-sqr 62.1%

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

      7. hypot-def 81.2%

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

   6. Simplified 81.2%

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

   if 8.50000000000000029e-79 < phi2

   1. Initial program 60.2%

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

      1. hypot-def 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in phi1 around 0 55.3%

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

      1. *-commutative 55.3%

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

      2. unpow2 55.3%

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

      3. unpow2 55.3%

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

      4. unpow2 55.3%

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

      5. unswap-sqr 55.3%

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

      6. hypot-def 80.5%

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

   6. Simplified 80.5%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8.5 \cdot 10^{-79}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 8: 31.6% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -23000000:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.1 \cdot 10^{-284}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right) + -0.25 \cdot \left(\phi_1 \cdot \left(R \cdot \left(\phi_2 \cdot \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -23000000.0)
   (* R (- phi2 phi1))
   (if (<= phi1 -1.1e-284)
     (+
      (* R (- lambda2 lambda1))
      (* -0.25 (* phi1 (* R (* phi2 (- lambda2 lambda1))))))
     (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -23000000.0) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= -1.1e-284) {
		tmp = (R * (lambda2 - lambda1)) + (-0.25 * (phi1 * (R * (phi2 * (lambda2 - lambda1)))));
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-23000000.0d0)) then
        tmp = r * (phi2 - phi1)
    else if (phi1 <= (-1.1d-284)) then
        tmp = (r * (lambda2 - lambda1)) + ((-0.25d0) * (phi1 * (r * (phi2 * (lambda2 - lambda1)))))
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -23000000.0) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= -1.1e-284) {
		tmp = (R * (lambda2 - lambda1)) + (-0.25 * (phi1 * (R * (phi2 * (lambda2 - lambda1)))));
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -23000000.0:
		tmp = R * (phi2 - phi1)
	elif phi1 <= -1.1e-284:
		tmp = (R * (lambda2 - lambda1)) + (-0.25 * (phi1 * (R * (phi2 * (lambda2 - lambda1)))))
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -23000000.0)
		tmp = Float64(R * Float64(phi2 - phi1));
	elseif (phi1 <= -1.1e-284)
		tmp = Float64(Float64(R * Float64(lambda2 - lambda1)) + Float64(-0.25 * Float64(phi1 * Float64(R * Float64(phi2 * Float64(lambda2 - lambda1))))));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -23000000.0)
		tmp = R * (phi2 - phi1);
	elseif (phi1 <= -1.1e-284)
		tmp = (R * (lambda2 - lambda1)) + (-0.25 * (phi1 * (R * (phi2 * (lambda2 - lambda1)))));
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -23000000.0], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -1.1e-284], N[(N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(phi1 * N[(R * N[(phi2 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.3e7

   1. Initial program 53.7%

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

      1. hypot-def 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in phi1 around -inf 61.2%

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

      1. *-commutative 61.2%

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

      2. associate-*r* 61.2%

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

      3. distribute-rgt-out 63.0%

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

      4. mul-1-neg 63.0%

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

   6. Simplified 63.0%

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

   if -2.3e7 < phi1 < -1.1e-284

   1. Initial program 72.7%

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

      1. hypot-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in lambda1 around -inf 39.8%

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

      1. +-commutative 39.8%

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

      2. mul-1-neg 39.8%

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

      3. unsub-neg 39.8%

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

      4. associate-*r* 39.8%

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

      5. +-commutative 39.8%

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

      6. *-commutative 39.8%

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

      7. *-commutative 39.8%

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

      8. associate-*r* 39.8%

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

      9. *-commutative 39.8%

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

      10. *-commutative 39.8%

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

      11. +-commutative 39.8%

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

   6. Simplified 39.8%

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

   7. Taylor expanded in phi2 around 0 32.1%

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

      1. associate-*r* 32.1%

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

      2. *-commutative 32.1%

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

      3. associate-*r* 32.1%

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

      4. +-commutative 32.1%

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

      5. *-commutative 32.1%

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

      6. associate--l+ 32.1%

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

   9. Simplified 32.2%

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

   10. Taylor expanded in phi1 around 0 31.7%

       \[\leadsto \color{blue}{R \cdot \left(\lambda_2 - \lambda_1\right) + -0.25 \cdot \left(\phi_1 \cdot \left(R \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \phi_2\right)\right)\right)} \]

   if -1.1e-284 < phi1

   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-def 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in phi2 around inf 19.8%

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

      1. *-commutative 19.8%

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

   6. Simplified 19.8%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -23000000:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.1 \cdot 10^{-284}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right) + -0.25 \cdot \left(\phi_1 \cdot \left(R \cdot \left(\phi_2 \cdot \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 9: 27.9% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8500000:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -2.45 \cdot 10^{-185}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_1 \leq -7.8 \cdot 10^{-272}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -8500000.0)
   (* R (- phi1))
   (if (<= phi1 -2.45e-185)
     (* R lambda2)
     (if (<= phi1 -7.8e-272) (* R (- lambda1)) (* R phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -8500000.0) {
		tmp = R * -phi1;
	} else if (phi1 <= -2.45e-185) {
		tmp = R * lambda2;
	} else if (phi1 <= -7.8e-272) {
		tmp = R * -lambda1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-8500000.0d0)) then
        tmp = r * -phi1
    else if (phi1 <= (-2.45d-185)) then
        tmp = r * lambda2
    else if (phi1 <= (-7.8d-272)) then
        tmp = r * -lambda1
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -8500000.0) {
		tmp = R * -phi1;
	} else if (phi1 <= -2.45e-185) {
		tmp = R * lambda2;
	} else if (phi1 <= -7.8e-272) {
		tmp = R * -lambda1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -8500000.0:
		tmp = R * -phi1
	elif phi1 <= -2.45e-185:
		tmp = R * lambda2
	elif phi1 <= -7.8e-272:
		tmp = R * -lambda1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -8500000.0)
		tmp = Float64(R * Float64(-phi1));
	elseif (phi1 <= -2.45e-185)
		tmp = Float64(R * lambda2);
	elseif (phi1 <= -7.8e-272)
		tmp = Float64(R * Float64(-lambda1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -8500000.0)
		tmp = R * -phi1;
	elseif (phi1 <= -2.45e-185)
		tmp = R * lambda2;
	elseif (phi1 <= -7.8e-272)
		tmp = R * -lambda1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -8500000.0], N[(R * (-phi1)), $MachinePrecision], If[LessEqual[phi1, -2.45e-185], N[(R * lambda2), $MachinePrecision], If[LessEqual[phi1, -7.8e-272], N[(R * (-lambda1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -8.5e6

   1. Initial program 54.4%

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

      1. hypot-def 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in phi1 around -inf 57.8%

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

      1. associate-*r* 57.8%

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

      2. mul-1-neg 57.8%

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

   6. Simplified 57.8%

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

   if -8.5e6 < phi1 < -2.4500000000000001e-185

   1. Initial program 72.5%

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

      1. hypot-def 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in lambda1 around -inf 39.2%

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

      1. +-commutative 39.2%

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

      2. mul-1-neg 39.2%

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

      3. unsub-neg 39.2%

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

      4. associate-*r* 39.2%

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

      5. +-commutative 39.2%

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

      6. *-commutative 39.2%

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

      7. *-commutative 39.2%

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

      8. associate-*r* 39.2%

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

      9. *-commutative 39.2%

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

      10. *-commutative 39.2%

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

      11. +-commutative 39.2%

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

   6. Simplified 39.2%

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

   7. Taylor expanded in phi1 around 0 37.1%

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

      1. *-commutative 37.1%

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

      2. associate-*r* 37.1%

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

      3. distribute-lft-out-- 37.2%

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

      4. *-commutative 37.2%

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

   9. Simplified 37.2%

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

   10. Taylor expanded in phi2 around 0 34.7%

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

   11. Taylor expanded in lambda2 around inf 17.5%

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

   if -2.4500000000000001e-185 < phi1 < -7.7999999999999994e-272

   1. Initial program 72.4%

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

      1. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in lambda1 around -inf 52.1%

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

      1. +-commutative 52.1%

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

      2. mul-1-neg 52.1%

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

      3. unsub-neg 52.1%

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

      4. associate-*r* 52.1%

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

      5. +-commutative 52.1%

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

      6. *-commutative 52.1%

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

      7. *-commutative 52.1%

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

      8. associate-*r* 52.1%

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

      9. *-commutative 52.1%

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

      10. *-commutative 52.1%

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

      11. +-commutative 52.1%

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

   6. Simplified 52.1%

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

   7. Taylor expanded in phi1 around 0 52.1%

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

      1. *-commutative 52.1%

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

      2. associate-*r* 52.1%

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

      3. distribute-lft-out-- 52.1%

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

      4. *-commutative 52.1%

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

   9. Simplified 52.1%

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

   10. Taylor expanded in phi2 around 0 52.6%

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

   11. Taylor expanded in lambda2 around 0 31.9%

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

       1. mul-1-neg 31.9%

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

       2. *-commutative 31.9%

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

       3. distribute-rgt-neg-in 31.9%

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

   13. Simplified 31.9%

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

   if -7.7999999999999994e-272 < phi1

   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-def 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in phi2 around inf 19.4%

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

      1. *-commutative 19.4%

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

   6. Simplified 19.4%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8500000:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -2.45 \cdot 10^{-185}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_1 \leq -7.8 \cdot 10^{-272}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 10: 31.3% accurate, 36.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{+85}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 6.6 \cdot 10^{-201}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.8e+85)
   (* R (- phi1))
   (if (<= phi1 6.6e-201) (* R (- lambda2 lambda1)) (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.8e+85) {
		tmp = R * -phi1;
	} else if (phi1 <= 6.6e-201) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-2.8d+85)) then
        tmp = r * -phi1
    else if (phi1 <= 6.6d-201) then
        tmp = r * (lambda2 - lambda1)
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.8e+85) {
		tmp = R * -phi1;
	} else if (phi1 <= 6.6e-201) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.8e+85:
		tmp = R * -phi1
	elif phi1 <= 6.6e-201:
		tmp = R * (lambda2 - lambda1)
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.8e+85)
		tmp = Float64(R * Float64(-phi1));
	elseif (phi1 <= 6.6e-201)
		tmp = Float64(R * Float64(lambda2 - lambda1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.8e+85)
		tmp = R * -phi1;
	elseif (phi1 <= 6.6e-201)
		tmp = R * (lambda2 - lambda1);
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.8e+85], N[(R * (-phi1)), $MachinePrecision], If[LessEqual[phi1, 6.6e-201], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.7999999999999999e85

   1. Initial program 51.5%

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

      1. hypot-def 92.4%

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

   3. Simplified 92.4%

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

   4. Taylor expanded in phi1 around -inf 72.5%

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

      1. associate-*r* 72.5%

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

      2. mul-1-neg 72.5%

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

   6. Simplified 72.5%

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

   if -2.7999999999999999e85 < phi1 < 6.6000000000000007e-201

   1. Initial program 66.1%

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

      1. hypot-def 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. Taylor expanded in lambda1 around -inf 37.1%

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

      1. +-commutative 37.1%

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

      2. mul-1-neg 37.1%

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

      3. unsub-neg 37.1%

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

      4. associate-*r* 37.1%

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

      5. +-commutative 37.1%

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

      6. *-commutative 37.1%

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

      7. *-commutative 37.1%

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

      8. associate-*r* 37.1%

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

      9. *-commutative 37.1%

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

      10. *-commutative 37.1%

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

      11. +-commutative 37.1%

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

   6. Simplified 37.1%

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

   7. Taylor expanded in phi1 around 0 33.9%

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

      1. *-commutative 33.9%

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

      2. associate-*r* 33.9%

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

      3. distribute-lft-out-- 34.8%

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

      4. *-commutative 34.8%

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

   9. Simplified 34.8%

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

   10. Taylor expanded in phi2 around 0 30.2%

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

   if 6.6000000000000007e-201 < phi1

   1. Initial program 71.3%

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

      1. hypot-def 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. Taylor expanded in phi2 around inf 19.1%

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

      1. *-commutative 19.1%

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

   6. Simplified 19.1%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{+85}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 6.6 \cdot 10^{-201}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 11: 33.3% accurate, 36.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -49000000:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 6.5 \cdot 10^{-206}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -49000000.0)
   (* R (- phi2 phi1))
   (if (<= phi1 6.5e-206) (* R (- lambda2 lambda1)) (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -49000000.0) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= 6.5e-206) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-49000000.0d0)) then
        tmp = r * (phi2 - phi1)
    else if (phi1 <= 6.5d-206) then
        tmp = r * (lambda2 - lambda1)
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -49000000.0) {
		tmp = R * (phi2 - phi1);
	} else if (phi1 <= 6.5e-206) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -49000000.0:
		tmp = R * (phi2 - phi1)
	elif phi1 <= 6.5e-206:
		tmp = R * (lambda2 - lambda1)
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -49000000.0)
		tmp = Float64(R * Float64(phi2 - phi1));
	elseif (phi1 <= 6.5e-206)
		tmp = Float64(R * Float64(lambda2 - lambda1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -49000000.0)
		tmp = R * (phi2 - phi1);
	elseif (phi1 <= 6.5e-206)
		tmp = R * (lambda2 - lambda1);
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -49000000.0], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 6.5e-206], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -4.9e7

   1. Initial program 54.5%

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

      1. hypot-def 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in phi1 around -inf 62.3%

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

      1. *-commutative 62.3%

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

      2. associate-*r* 62.3%

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

      3. distribute-rgt-out 64.1%

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

      4. mul-1-neg 64.1%

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

   6. Simplified 64.1%

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

   if -4.9e7 < phi1 < 6.4999999999999996e-206

   1. Initial program 66.6%

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

      1. hypot-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in lambda1 around -inf 38.0%

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

      1. +-commutative 38.0%

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

      2. mul-1-neg 38.0%

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

      3. unsub-neg 38.0%

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

      4. associate-*r* 38.0%

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

      5. +-commutative 38.0%

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

      6. *-commutative 38.0%

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

      7. *-commutative 38.0%

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

      8. associate-*r* 38.0%

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

      9. *-commutative 38.0%

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

      10. *-commutative 38.0%

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

      11. +-commutative 38.0%

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

   6. Simplified 38.0%

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

   7. Taylor expanded in phi1 around 0 37.1%

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

      1. *-commutative 37.1%

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

      2. associate-*r* 37.1%

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

      3. distribute-lft-out-- 38.2%

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

      4. *-commutative 38.2%

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

   9. Simplified 38.2%

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

   10. Taylor expanded in phi2 around 0 33.1%

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

   if 6.4999999999999996e-206 < phi1

   1. Initial program 71.3%

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

      1. hypot-def 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. Taylor expanded in phi2 around inf 19.1%

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

      1. *-commutative 19.1%

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

   6. Simplified 19.1%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -49000000:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 6.5 \cdot 10^{-206}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 12: 23.7% accurate, 46.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.1 \cdot 10^{+67}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 3.7 \cdot 10^{-299}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -2.1e+67)
   (* R (- lambda1))
   (if (<= lambda1 3.7e-299) (* R phi2) (* R lambda2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.1e+67) {
		tmp = R * -lambda1;
	} else if (lambda1 <= 3.7e-299) {
		tmp = R * phi2;
	} 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 (lambda1 <= (-2.1d+67)) then
        tmp = r * -lambda1
    else if (lambda1 <= 3.7d-299) then
        tmp = r * phi2
    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 (lambda1 <= -2.1e+67) {
		tmp = R * -lambda1;
	} else if (lambda1 <= 3.7e-299) {
		tmp = R * phi2;
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -2.1e+67:
		tmp = R * -lambda1
	elif lambda1 <= 3.7e-299:
		tmp = R * phi2
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -2.1e+67)
		tmp = Float64(R * Float64(-lambda1));
	elseif (lambda1 <= 3.7e-299)
		tmp = Float64(R * phi2);
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -2.1e+67)
		tmp = R * -lambda1;
	elseif (lambda1 <= 3.7e-299)
		tmp = R * phi2;
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -2.1e+67], N[(R * (-lambda1)), $MachinePrecision], If[LessEqual[lambda1, 3.7e-299], N[(R * phi2), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -2.1000000000000001e67

   1. Initial program 66.6%

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

      1. hypot-def 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in lambda1 around -inf 49.6%

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

      1. +-commutative 49.6%

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

      2. mul-1-neg 49.6%

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

      3. unsub-neg 49.6%

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

      4. associate-*r* 49.6%

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

      5. +-commutative 49.6%

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

      6. *-commutative 49.6%

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

      7. *-commutative 49.6%

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

      8. associate-*r* 49.6%

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

      9. *-commutative 49.6%

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

      10. *-commutative 49.6%

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

      11. +-commutative 49.6%

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

   6. Simplified 49.6%

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

   7. Taylor expanded in phi1 around 0 51.3%

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

      1. *-commutative 51.3%

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

      2. associate-*r* 51.3%

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

      3. distribute-lft-out-- 55.5%

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

      4. *-commutative 55.5%

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

   9. Simplified 55.5%

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

   10. Taylor expanded in phi2 around 0 64.0%

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

   11. Taylor expanded in lambda2 around 0 61.3%

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

       1. mul-1-neg 61.3%

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

       2. *-commutative 61.3%

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

       3. distribute-rgt-neg-in 61.3%

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

   13. Simplified 61.3%

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

   if -2.1000000000000001e67 < lambda1 < 3.70000000000000014e-299

   1. Initial program 70.1%

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

      1. hypot-def 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in phi2 around inf 31.3%

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

      1. *-commutative 31.3%

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

   6. Simplified 31.3%

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

   if 3.70000000000000014e-299 < lambda1

   1. Initial program 63.3%

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

      1. hypot-def 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. Taylor expanded in lambda1 around -inf 24.6%

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

      1. +-commutative 24.6%

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

      2. mul-1-neg 24.6%

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

      3. unsub-neg 24.6%

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

      4. associate-*r* 24.6%

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

      5. +-commutative 24.6%

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

      6. *-commutative 24.6%

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

      7. *-commutative 24.6%

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

      8. associate-*r* 24.6%

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

      9. *-commutative 24.6%

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

      10. *-commutative 24.6%

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

      11. +-commutative 24.6%

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

   6. Simplified 24.6%

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

   7. Taylor expanded in phi1 around 0 18.1%

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

      1. *-commutative 18.1%

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

      2. associate-*r* 18.1%

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

      3. distribute-lft-out-- 18.1%

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

      4. *-commutative 18.1%

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

   9. Simplified 18.1%

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

   10. Taylor expanded in phi2 around 0 11.6%

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

   11. Taylor expanded in lambda2 around inf 14.1%

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

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.1 \cdot 10^{+67}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 3.7 \cdot 10^{-299}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]

Alternative 13: 25.9% accurate, 65.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.3 \cdot 10^{-45}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 4.3e-45) (* R lambda2) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4.3e-45) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 4.3d-45) then
        tmp = r * lambda2
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4.3e-45) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 4.3e-45:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 4.3e-45)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 4.3e-45)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 4.3e-45], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.2999999999999999e-45

   1. Initial program 68.2%

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

      1. hypot-def 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in lambda1 around -inf 31.8%

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

      1. +-commutative 31.8%

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

      2. mul-1-neg 31.8%

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

      3. unsub-neg 31.8%

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

      4. associate-*r* 31.8%

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

      5. +-commutative 31.8%

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

      6. *-commutative 31.8%

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

      7. *-commutative 31.8%

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

      8. associate-*r* 31.8%

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

      9. *-commutative 31.8%

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

      10. *-commutative 31.8%

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

      11. +-commutative 31.8%

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

   6. Simplified 31.8%

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

   7. Taylor expanded in phi1 around 0 26.3%

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

      1. *-commutative 26.3%

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

      2. associate-*r* 26.3%

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

      3. distribute-lft-out-- 26.9%

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

      4. *-commutative 26.9%

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

   9. Simplified 26.9%

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

   10. Taylor expanded in phi2 around 0 26.8%

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

   11. Taylor expanded in lambda2 around inf 14.5%

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

   if 4.2999999999999999e-45 < phi2

   1. Initial program 59.6%

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

      1. hypot-def 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in phi2 around inf 54.5%

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

      1. *-commutative 54.5%

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

   6. Simplified 54.5%

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

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.3 \cdot 10^{-45}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 14: 13.8% accurate, 109.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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-def 96.2%

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

3. Simplified 96.2%

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

4. Taylor expanded in lambda1 around -inf 29.7%

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

   1. +-commutative 29.7%

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

   2. mul-1-neg 29.7%

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

   3. unsub-neg 29.7%

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

   4. associate-*r* 29.7%

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

   5. +-commutative 29.7%

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

   6. *-commutative 29.7%

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

   7. *-commutative 29.7%

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

   8. associate-*r* 29.7%

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

   9. *-commutative 29.7%

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

   10. *-commutative 29.7%

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

   11. +-commutative 29.7%

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

6. Simplified 29.7%

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

7. Taylor expanded in phi1 around 0 25.7%

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

   1. *-commutative 25.7%

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

   2. associate-*r* 25.7%

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

   3. distribute-lft-out-- 26.5%

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

   4. *-commutative 26.5%

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

9. Simplified 26.5%

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

10. Taylor expanded in phi2 around 0 24.4%

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

11. Taylor expanded in lambda2 around inf 14.8%

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

12. Final simplification 14.8%

    \[\leadsto R \cdot \lambda_2 \]

Reproduce

herbie shell --seed 2023199 
(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))))))