Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.7% → 99.8%

Time: 20.4s

Alternatives: 15

Speedup: 1.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.7% 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, 0.4× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (- lambda1 lambda2)
    (expm1
     (log1p
      (-
       (* (cos (* 0.5 phi1)) (cos (* 0.5 phi2)))
       (expm1 (log1p (* (sin (* 0.5 phi2)) (sin (* 0.5 phi1)))))))))
   (- 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((0.5 * phi2))) - expm1(log1p((sin((0.5 * phi2)) * sin((0.5 * phi1))))))))), (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((0.5 * phi2))) - Math.expm1(Math.log1p((Math.sin((0.5 * phi2)) * Math.sin((0.5 * phi1))))))))), (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((0.5 * phi2))) - math.expm1(math.log1p((math.sin((0.5 * phi2)) * math.sin((0.5 * phi1))))))))), (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(0.5 * phi2))) - expm1(log1p(Float64(sin(Float64(0.5 * phi2)) * sin(Float64(0.5 * phi1))))))))), 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[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(Exp[N[Log[1 + N[(N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.3%

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

   1. hypot-def 96.3%

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

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. Step-by-step derivation

   1. add-cbrt-cube 96.3%

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

   2. pow3 96.3%

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

   3. div-inv 96.3%

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

   4. metadata-eval 96.3%

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

5. Applied egg-rr 96.3%

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

   1. *-commutative 96.3%

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

   2. +-commutative 96.3%

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

   3. distribute-rgt-in 96.3%

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

   4. cos-sum 99.8%

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

7. Applied egg-rr 99.8%

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

   1. expm1-log1p-u 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{3}}\right)\right)}, \phi_1 - \phi_2\right) \]

   2. rem-cbrt-cube 99.9%

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

   3. *-commutative 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

9. Applied egg-rr 99.9%

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

    1. expm1-log1p-u 99.9%

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

    2. *-commutative 99.9%

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

11. Applied egg-rr 99.9%

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

12. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (- lambda1 lambda2)
    (expm1
     (log1p
      (-
       (* (cos (* 0.5 phi1)) (cos (* 0.5 phi2)))
       (* (sin (* 0.5 phi2)) (sin (* 0.5 phi1)))))))
   (- 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((0.5 * phi2))) - (sin((0.5 * phi2)) * sin((0.5 * phi1))))))), (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((0.5 * phi2))) - (Math.sin((0.5 * phi2)) * Math.sin((0.5 * phi1))))))), (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((0.5 * phi2))) - (math.sin((0.5 * phi2)) * math.sin((0.5 * phi1))))))), (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(0.5 * phi2))) - Float64(sin(Float64(0.5 * phi2)) * sin(Float64(0.5 * phi1))))))), 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[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.3%

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

   1. hypot-def 96.3%

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

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. Step-by-step derivation

   1. add-cbrt-cube 96.3%

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

   2. pow3 96.3%

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

   3. div-inv 96.3%

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

   4. metadata-eval 96.3%

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

5. Applied egg-rr 96.3%

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

   1. *-commutative 96.3%

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

   2. +-commutative 96.3%

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

   3. distribute-rgt-in 96.3%

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

   4. cos-sum 99.8%

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

7. Applied egg-rr 99.8%

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

   1. expm1-log1p-u 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{3}}\right)\right)}, \phi_1 - \phi_2\right) \]

   2. rem-cbrt-cube 99.9%

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

   3. *-commutative 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 3: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.3%

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

   1. hypot-def 96.3%

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

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. Step-by-step derivation

   1. add-cbrt-cube 96.3%

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

   2. pow3 96.3%

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

   3. div-inv 96.3%

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

   4. metadata-eval 96.3%

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

5. Applied egg-rr 96.3%

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

   1. *-commutative 96.3%

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

   2. +-commutative 96.3%

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

   3. distribute-rgt-in 96.3%

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

   4. cos-sum 99.8%

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

7. Applied egg-rr 99.8%

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

   1. expm1-log1p-u 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{3}}\right)\right)}, \phi_1 - \phi_2\right) \]

   2. rem-cbrt-cube 99.9%

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

   3. *-commutative 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

9. Applied egg-rr 99.9%

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

    1. expm1-udef 99.8%

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

    2. log1p-expm1-u 99.8%

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

    3. log1p-udef 99.8%

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

    4. add-exp-log 99.8%

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

    5. expm1-log1p-u 99.8%

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

    6. *-commutative 99.8%

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

    7. *-commutative 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 4: 82.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.8 \cdot 10^{-42}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.8e-42)
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) (- phi1 phi2)))
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* 0.5 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.8e-42) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((0.5 * phi2))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.8e-42) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((0.5 * phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.8e-42:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((0.5 * phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.8e-42)
		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(0.5 * phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.8e-42)
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((0.5 * phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.8e-42], 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[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.8000000000000001e-42

   1. Initial program 52.7%

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

      1. hypot-def 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in phi2 around 0 91.1%

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

   if -1.8000000000000001e-42 < phi1

   1. Initial program 65.2%

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

      1. hypot-def 98.5%

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

   3. Simplified 98.5%

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

      1. expm1-log1p-u 62.7%

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

      2. *-commutative 62.7%

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

      3. div-inv 62.7%

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

      4. metadata-eval 62.7%

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

   5. Applied egg-rr 62.7%

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

   6. Taylor expanded in phi1 around 0 57.7%

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

      1. *-commutative 57.7%

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

      2. unpow2 57.7%

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

      3. unpow2 57.7%

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

      4. unpow2 57.7%

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

      5. unswap-sqr 57.7%

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

      6. hypot-def 79.9%

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

   8. Simplified 79.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.8 \cdot 10^{-42}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \]

Alternative 5: 93.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -8e-6)
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi2))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -8e-6) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi2))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -8e-6) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi2))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -8e-6:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi2))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -8e-6)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi2))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -8e-6)
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi2))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -8e-6], 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[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -7.99999999999999964e-6

   1. Initial program 48.6%

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

      1. hypot-def 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in phi2 around 0 90.8%

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

   if -7.99999999999999964e-6 < phi1

   1. Initial program 66.0%

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

      1. hypot-def 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in phi1 around 0 94.4%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 6: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.3%

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

   1. hypot-def 96.3%

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

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. Final simplification 96.3%

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

Alternative 7: 74.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.35 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.35e-25)
   (* 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 <= 1.35e-25) {
		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 <= 1.35e-25) {
		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 <= 1.35e-25:
		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 <= 1.35e-25)
		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 <= 1.35e-25)
		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, 1.35e-25], 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 < 1.35000000000000008e-25

   1. Initial program 62.8%

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

      1. hypot-def 97.3%

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

   3. Simplified 97.3%

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

      1. expm1-log1p-u 65.7%

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

      2. *-commutative 65.7%

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

      3. div-inv 65.7%

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

      4. metadata-eval 65.7%

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

   5. Applied egg-rr 65.7%

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

   6. Taylor expanded in phi2 around 0 51.1%

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

      1. *-commutative 51.1%

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

      2. +-commutative 51.1%

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

      3. unpow2 51.1%

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

      4. unpow2 51.1%

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

      5. unpow2 51.1%

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

      6. unswap-sqr 51.1%

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

      7. hypot-def 77.1%

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

   8. Simplified 77.1%

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

   if 1.35000000000000008e-25 < phi2

   1. Initial program 56.3%

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

      1. hypot-def 93.0%

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

   3. Simplified 93.0%

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

   4. Taylor expanded in phi1 around -inf 65.0%

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

      1. *-commutative 65.0%

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

      2. associate-*r* 65.0%

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

      3. distribute-rgt-out 66.7%

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

      4. mul-1-neg 66.7%

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

      5. unsub-neg 66.7%

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

   6. Simplified 66.7%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.35 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 8: 78.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.4 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -5.4e-17)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* 0.5 phi1)))))
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* 0.5 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5.4e-17) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((0.5 * phi2))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5.4e-17) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((0.5 * phi1))));
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((0.5 * phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -5.4e-17:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((0.5 * phi1))))
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((0.5 * phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -5.4e-17)
		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(0.5 * phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -5.4e-17)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((0.5 * phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -5.4e-17], 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[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5.4000000000000002e-17

   1. Initial program 51.4%

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

      1. hypot-def 90.8%

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

   3. Simplified 90.8%

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

      1. expm1-log1p-u 57.6%

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

      2. *-commutative 57.6%

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

      3. div-inv 57.6%

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

      4. metadata-eval 57.6%

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

   5. Applied egg-rr 57.6%

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

   6. Taylor expanded in phi2 around 0 45.7%

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

      1. *-commutative 45.7%

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

      2. +-commutative 45.7%

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

      3. unpow2 45.7%

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

      4. unpow2 45.7%

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

      5. unpow2 45.7%

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

      6. unswap-sqr 45.7%

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

      7. hypot-def 71.9%

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

   8. Simplified 71.9%

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

   if -5.4000000000000002e-17 < phi1

   1. Initial program 65.4%

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

      1. hypot-def 98.6%

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

   3. Simplified 98.6%

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

      1. expm1-log1p-u 63.8%

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

      2. *-commutative 63.8%

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

      3. div-inv 63.8%

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

      4. metadata-eval 63.8%

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

   5. Applied egg-rr 63.8%

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

   6. Taylor expanded in phi1 around 0 57.6%

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

      1. *-commutative 57.6%

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

      2. unpow2 57.6%

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

      3. unpow2 57.6%

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

      4. unpow2 57.6%

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

      5. unswap-sqr 57.6%

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

      6. hypot-def 80.0%

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

   8. Simplified 80.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.4 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \]

Alternative 9: 34.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{-26} \lor \neg \left(\phi_1 \leq 5.2 \cdot 10^{-197}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \cos \left(0.5 \cdot \phi_2\right)\right) \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -2.8e-26) (not (<= phi1 5.2e-197)))
   (* R (- phi2 phi1))
   (* (* R (cos (* 0.5 phi2))) (- lambda2 lambda1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -2.8e-26) || !(phi1 <= 5.2e-197)) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = (R * cos((0.5 * phi2))) * (lambda2 - lambda1);
	}
	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-26)) .or. (.not. (phi1 <= 5.2d-197))) then
        tmp = r * (phi2 - phi1)
    else
        tmp = (r * cos((0.5d0 * phi2))) * (lambda2 - lambda1)
    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-26) || !(phi1 <= 5.2e-197)) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = (R * Math.cos((0.5 * phi2))) * (lambda2 - lambda1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi1 <= -2.8e-26) or not (phi1 <= 5.2e-197):
		tmp = R * (phi2 - phi1)
	else:
		tmp = (R * math.cos((0.5 * phi2))) * (lambda2 - lambda1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -2.8e-26) || !(phi1 <= 5.2e-197))
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(Float64(R * cos(Float64(0.5 * phi2))) * Float64(lambda2 - lambda1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi1 <= -2.8e-26) || ~((phi1 <= 5.2e-197)))
		tmp = R * (phi2 - phi1);
	else
		tmp = (R * cos((0.5 * phi2))) * (lambda2 - lambda1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -2.8e-26], N[Not[LessEqual[phi1, 5.2e-197]], $MachinePrecision]], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(N[(R * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.8000000000000001e-26 or 5.2000000000000003e-197 < phi1

   1. Initial program 56.6%

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

      1. hypot-def 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in phi1 around -inf 36.7%

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

      1. *-commutative 36.7%

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

      2. associate-*r* 36.7%

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

      3. distribute-rgt-out 38.4%

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

      4. mul-1-neg 38.4%

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

      5. unsub-neg 38.4%

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

   6. Simplified 38.4%

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

   if -2.8000000000000001e-26 < phi1 < 5.2000000000000003e-197

   1. Initial program 71.1%

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

      1. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in lambda1 around -inf 35.7%

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

      1. +-commutative 35.7%

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

      2. mul-1-neg 35.7%

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

      3. unsub-neg 35.7%

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

      4. *-commutative 35.7%

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

      5. *-commutative 35.7%

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

      6. associate-*l* 35.7%

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

      7. *-commutative 35.7%

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

      8. +-commutative 35.7%

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

      9. *-commutative 35.7%

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

      10. associate-*l* 35.7%

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

      11. +-commutative 35.7%

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

   6. Simplified 35.7%

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

   7. Taylor expanded in phi1 around 0 35.7%

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

      1. *-commutative 35.7%

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

      2. associate-*r* 35.7%

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

      3. distribute-lft-out-- 35.7%

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

      4. *-commutative 35.7%

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

   9. Simplified 35.7%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{-26} \lor \neg \left(\phi_1 \leq 5.2 \cdot 10^{-197}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \cos \left(0.5 \cdot \phi_2\right)\right) \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

Alternative 10: 27.9% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -1.2 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1 \cdot 10^{-101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -6 \cdot 10^{-197}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_1 \leq -4.1 \cdot 10^{-218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -1.75 \cdot 10^{-252}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq -1.05 \cdot 10^{-296}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 2.05 \cdot 10^{-291}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* R (- lambda1))))
   (if (<= phi1 -1.2e-20)
     (* R (- phi1))
     (if (<= phi1 -1e-101)
       t_0
       (if (<= phi1 -6e-197)
         (* R lambda2)
         (if (<= phi1 -4.1e-218)
           t_0
           (if (<= phi1 -1.75e-252)
             (* R phi2)
             (if (<= phi1 -1.05e-296)
               t_0
               (if (<= phi1 2.05e-291) (* R lambda2) (* R phi2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * -lambda1;
	double tmp;
	if (phi1 <= -1.2e-20) {
		tmp = R * -phi1;
	} else if (phi1 <= -1e-101) {
		tmp = t_0;
	} else if (phi1 <= -6e-197) {
		tmp = R * lambda2;
	} else if (phi1 <= -4.1e-218) {
		tmp = t_0;
	} else if (phi1 <= -1.75e-252) {
		tmp = R * phi2;
	} else if (phi1 <= -1.05e-296) {
		tmp = t_0;
	} else if (phi1 <= 2.05e-291) {
		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) :: t_0
    real(8) :: tmp
    t_0 = r * -lambda1
    if (phi1 <= (-1.2d-20)) then
        tmp = r * -phi1
    else if (phi1 <= (-1d-101)) then
        tmp = t_0
    else if (phi1 <= (-6d-197)) then
        tmp = r * lambda2
    else if (phi1 <= (-4.1d-218)) then
        tmp = t_0
    else if (phi1 <= (-1.75d-252)) then
        tmp = r * phi2
    else if (phi1 <= (-1.05d-296)) then
        tmp = t_0
    else if (phi1 <= 2.05d-291) 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 t_0 = R * -lambda1;
	double tmp;
	if (phi1 <= -1.2e-20) {
		tmp = R * -phi1;
	} else if (phi1 <= -1e-101) {
		tmp = t_0;
	} else if (phi1 <= -6e-197) {
		tmp = R * lambda2;
	} else if (phi1 <= -4.1e-218) {
		tmp = t_0;
	} else if (phi1 <= -1.75e-252) {
		tmp = R * phi2;
	} else if (phi1 <= -1.05e-296) {
		tmp = t_0;
	} else if (phi1 <= 2.05e-291) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = R * -lambda1
	tmp = 0
	if phi1 <= -1.2e-20:
		tmp = R * -phi1
	elif phi1 <= -1e-101:
		tmp = t_0
	elif phi1 <= -6e-197:
		tmp = R * lambda2
	elif phi1 <= -4.1e-218:
		tmp = t_0
	elif phi1 <= -1.75e-252:
		tmp = R * phi2
	elif phi1 <= -1.05e-296:
		tmp = t_0
	elif phi1 <= 2.05e-291:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * Float64(-lambda1))
	tmp = 0.0
	if (phi1 <= -1.2e-20)
		tmp = Float64(R * Float64(-phi1));
	elseif (phi1 <= -1e-101)
		tmp = t_0;
	elseif (phi1 <= -6e-197)
		tmp = Float64(R * lambda2);
	elseif (phi1 <= -4.1e-218)
		tmp = t_0;
	elseif (phi1 <= -1.75e-252)
		tmp = Float64(R * phi2);
	elseif (phi1 <= -1.05e-296)
		tmp = t_0;
	elseif (phi1 <= 2.05e-291)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = R * -lambda1;
	tmp = 0.0;
	if (phi1 <= -1.2e-20)
		tmp = R * -phi1;
	elseif (phi1 <= -1e-101)
		tmp = t_0;
	elseif (phi1 <= -6e-197)
		tmp = R * lambda2;
	elseif (phi1 <= -4.1e-218)
		tmp = t_0;
	elseif (phi1 <= -1.75e-252)
		tmp = R * phi2;
	elseif (phi1 <= -1.05e-296)
		tmp = t_0;
	elseif (phi1 <= 2.05e-291)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * (-lambda1)), $MachinePrecision]}, If[LessEqual[phi1, -1.2e-20], N[(R * (-phi1)), $MachinePrecision], If[LessEqual[phi1, -1e-101], t$95$0, If[LessEqual[phi1, -6e-197], N[(R * lambda2), $MachinePrecision], If[LessEqual[phi1, -4.1e-218], t$95$0, If[LessEqual[phi1, -1.75e-252], N[(R * phi2), $MachinePrecision], If[LessEqual[phi1, -1.05e-296], t$95$0, If[LessEqual[phi1, 2.05e-291], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -1.19999999999999996e-20

   1. Initial program 52.1%

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

      1. hypot-def 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in phi1 around -inf 59.0%

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

      1. associate-*r* 59.0%

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

      2. mul-1-neg 59.0%

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

   6. Simplified 59.0%

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

   if -1.19999999999999996e-20 < phi1 < -1.00000000000000005e-101 or -6.00000000000000051e-197 < phi1 < -4.0999999999999998e-218 or -1.74999999999999993e-252 < phi1 < -1.05e-296

   1. Initial program 63.9%

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

      1. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in lambda1 around -inf 23.1%

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

      1. +-commutative 23.1%

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

      2. mul-1-neg 23.1%

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

      3. unsub-neg 23.1%

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

      4. *-commutative 23.1%

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

      5. *-commutative 23.1%

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

      6. associate-*l* 23.1%

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

      7. *-commutative 23.1%

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

      8. +-commutative 23.1%

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

      9. *-commutative 23.1%

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

      10. associate-*l* 23.0%

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

      11. +-commutative 23.0%

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

   6. Simplified 23.0%

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

   7. Taylor expanded in phi1 around 0 23.1%

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

      1. *-commutative 23.1%

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

      2. associate-*r* 23.1%

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

      3. distribute-lft-out-- 23.1%

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

      4. *-commutative 23.1%

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

   9. Simplified 23.1%

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

   10. Taylor expanded in phi2 around 0 17.2%

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

   11. Taylor expanded in lambda2 around 0 10.7%

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

       1. mul-1-neg 10.7%

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

       2. distribute-rgt-neg-in 10.7%

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

   13. Simplified 10.7%

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

   if -1.00000000000000005e-101 < phi1 < -6.00000000000000051e-197 or -1.05e-296 < phi1 < 2.05e-291

   1. Initial program 74.8%

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

      1. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in lambda1 around -inf 47.0%

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

      1. +-commutative 47.0%

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

      2. mul-1-neg 47.0%

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

      3. unsub-neg 47.0%

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

      4. *-commutative 47.0%

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

      5. *-commutative 47.0%

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

      6. associate-*l* 47.0%

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

      7. *-commutative 47.0%

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

      8. +-commutative 47.0%

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

      9. *-commutative 47.0%

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

      10. associate-*l* 47.0%

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

      11. +-commutative 47.0%

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

   6. Simplified 47.0%

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

   7. Taylor expanded in phi1 around 0 47.0%

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

      1. *-commutative 47.0%

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

      2. associate-*r* 47.0%

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

      3. distribute-lft-out-- 47.0%

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

      4. *-commutative 47.0%

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

   9. Simplified 47.0%

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

   10. Taylor expanded in phi2 around 0 35.5%

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

   11. Taylor expanded in lambda2 around inf 28.2%

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

   if -4.0999999999999998e-218 < phi1 < -1.74999999999999993e-252 or 2.05e-291 < phi1

   1. Initial program 63.5%

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

      1. hypot-def 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in phi2 around inf 19.3%

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

      1. *-commutative 19.3%

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

   6. Simplified 19.3%

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

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.2 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1 \cdot 10^{-101}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq -6 \cdot 10^{-197}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_1 \leq -4.1 \cdot 10^{-218}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.75 \cdot 10^{-252}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq -1.05 \cdot 10^{-296}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq 2.05 \cdot 10^{-291}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 11: 23.4% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ \mathbf{if}\;\lambda_2 \leq -3.7 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq -4 \cdot 10^{-259}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\lambda_2 \leq 1.2 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 2.8 \cdot 10^{-258}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\lambda_2 \leq 2.3 \cdot 10^{-183}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 1.8 \cdot 10^{+39}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* R (- lambda1))))
   (if (<= lambda2 -3.7e-145)
     t_0
     (if (<= lambda2 -4e-259)
       (* R phi2)
       (if (<= lambda2 1.2e-287)
         t_0
         (if (<= lambda2 2.8e-258)
           (* R phi2)
           (if (<= lambda2 2.3e-183)
             t_0
             (if (<= lambda2 1.8e+39) (* R phi2) (* R lambda2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * -lambda1;
	double tmp;
	if (lambda2 <= -3.7e-145) {
		tmp = t_0;
	} else if (lambda2 <= -4e-259) {
		tmp = R * phi2;
	} else if (lambda2 <= 1.2e-287) {
		tmp = t_0;
	} else if (lambda2 <= 2.8e-258) {
		tmp = R * phi2;
	} else if (lambda2 <= 2.3e-183) {
		tmp = t_0;
	} else if (lambda2 <= 1.8e+39) {
		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) :: t_0
    real(8) :: tmp
    t_0 = r * -lambda1
    if (lambda2 <= (-3.7d-145)) then
        tmp = t_0
    else if (lambda2 <= (-4d-259)) then
        tmp = r * phi2
    else if (lambda2 <= 1.2d-287) then
        tmp = t_0
    else if (lambda2 <= 2.8d-258) then
        tmp = r * phi2
    else if (lambda2 <= 2.3d-183) then
        tmp = t_0
    else if (lambda2 <= 1.8d+39) 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 t_0 = R * -lambda1;
	double tmp;
	if (lambda2 <= -3.7e-145) {
		tmp = t_0;
	} else if (lambda2 <= -4e-259) {
		tmp = R * phi2;
	} else if (lambda2 <= 1.2e-287) {
		tmp = t_0;
	} else if (lambda2 <= 2.8e-258) {
		tmp = R * phi2;
	} else if (lambda2 <= 2.3e-183) {
		tmp = t_0;
	} else if (lambda2 <= 1.8e+39) {
		tmp = R * phi2;
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = R * -lambda1
	tmp = 0
	if lambda2 <= -3.7e-145:
		tmp = t_0
	elif lambda2 <= -4e-259:
		tmp = R * phi2
	elif lambda2 <= 1.2e-287:
		tmp = t_0
	elif lambda2 <= 2.8e-258:
		tmp = R * phi2
	elif lambda2 <= 2.3e-183:
		tmp = t_0
	elif lambda2 <= 1.8e+39:
		tmp = R * phi2
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * Float64(-lambda1))
	tmp = 0.0
	if (lambda2 <= -3.7e-145)
		tmp = t_0;
	elseif (lambda2 <= -4e-259)
		tmp = Float64(R * phi2);
	elseif (lambda2 <= 1.2e-287)
		tmp = t_0;
	elseif (lambda2 <= 2.8e-258)
		tmp = Float64(R * phi2);
	elseif (lambda2 <= 2.3e-183)
		tmp = t_0;
	elseif (lambda2 <= 1.8e+39)
		tmp = Float64(R * phi2);
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = R * -lambda1;
	tmp = 0.0;
	if (lambda2 <= -3.7e-145)
		tmp = t_0;
	elseif (lambda2 <= -4e-259)
		tmp = R * phi2;
	elseif (lambda2 <= 1.2e-287)
		tmp = t_0;
	elseif (lambda2 <= 2.8e-258)
		tmp = R * phi2;
	elseif (lambda2 <= 2.3e-183)
		tmp = t_0;
	elseif (lambda2 <= 1.8e+39)
		tmp = R * phi2;
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * (-lambda1)), $MachinePrecision]}, If[LessEqual[lambda2, -3.7e-145], t$95$0, If[LessEqual[lambda2, -4e-259], N[(R * phi2), $MachinePrecision], If[LessEqual[lambda2, 1.2e-287], t$95$0, If[LessEqual[lambda2, 2.8e-258], N[(R * phi2), $MachinePrecision], If[LessEqual[lambda2, 2.3e-183], t$95$0, If[LessEqual[lambda2, 1.8e+39], N[(R * phi2), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -3.70000000000000013e-145 or -4.0000000000000003e-259 < lambda2 < 1.2e-287 or 2.8000000000000002e-258 < lambda2 < 2.30000000000000016e-183

   1. Initial program 60.7%

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

      1. hypot-def 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 lambda1 around -inf 22.2%

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

      1. +-commutative 22.2%

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

      2. mul-1-neg 22.2%

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

      3. unsub-neg 22.2%

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

      4. *-commutative 22.2%

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

      5. *-commutative 22.2%

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

      6. associate-*l* 22.2%

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

      7. *-commutative 22.2%

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

      8. +-commutative 22.2%

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

      9. *-commutative 22.2%

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

      10. associate-*l* 22.2%

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

      11. +-commutative 22.2%

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

   6. Simplified 22.2%

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

   7. Taylor expanded in phi1 around 0 20.5%

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

      1. *-commutative 20.5%

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

      2. associate-*r* 20.5%

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

      3. distribute-lft-out-- 21.4%

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

      4. *-commutative 21.4%

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

   9. Simplified 21.4%

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

   10. Taylor expanded in phi2 around 0 10.4%

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

   11. Taylor expanded in lambda2 around 0 12.0%

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

       1. mul-1-neg 12.0%

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

       2. distribute-rgt-neg-in 12.0%

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

   13. Simplified 12.0%

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

   if -3.70000000000000013e-145 < lambda2 < -4.0000000000000003e-259 or 1.2e-287 < lambda2 < 2.8000000000000002e-258 or 2.30000000000000016e-183 < lambda2 < 1.79999999999999992e39

   1. Initial program 69.2%

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

      1. hypot-def 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in phi2 around inf 22.7%

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

      1. *-commutative 22.7%

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

   6. Simplified 22.7%

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

   if 1.79999999999999992e39 < lambda2

   1. Initial program 44.2%

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

      1. hypot-def 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in lambda1 around -inf 50.1%

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

      1. +-commutative 50.1%

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

      2. mul-1-neg 50.1%

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

      3. unsub-neg 50.1%

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

      4. *-commutative 50.1%

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

      5. *-commutative 50.1%

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

      6. associate-*l* 50.1%

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

      7. *-commutative 50.1%

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

      8. +-commutative 50.1%

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

      9. *-commutative 50.1%

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

      10. associate-*l* 50.1%

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

      11. +-commutative 50.1%

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

   6. Simplified 50.1%

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

   7. Taylor expanded in phi1 around 0 46.5%

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

      1. *-commutative 46.5%

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

      2. associate-*r* 46.5%

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

      3. distribute-lft-out-- 48.9%

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

      4. *-commutative 48.9%

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

   9. Simplified 48.9%

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

   10. Taylor expanded in phi2 around 0 49.7%

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

   11. Taylor expanded in lambda2 around inf 49.1%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.7 \cdot 10^{-145}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq -4 \cdot 10^{-259}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\lambda_2 \leq 1.2 \cdot 10^{-287}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 2.8 \cdot 10^{-258}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\lambda_2 \leq 2.3 \cdot 10^{-183}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.8 \cdot 10^{+39}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]

Alternative 12: 31.8% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.5 \cdot 10^{-24} \lor \neg \left(\phi_1 \leq -4.6 \cdot 10^{-224} \lor \neg \left(\phi_1 \leq -1.1 \cdot 10^{-250}\right) \land \phi_1 \leq 5.3 \cdot 10^{-197}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -2.5e-24)
         (not
          (or (<= phi1 -4.6e-224)
              (and (not (<= phi1 -1.1e-250)) (<= phi1 5.3e-197)))))
   (* R (- phi2 phi1))
   (* R (- lambda2 lambda1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -2.5e-24) || !((phi1 <= -4.6e-224) || (!(phi1 <= -1.1e-250) && (phi1 <= 5.3e-197)))) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (lambda2 - lambda1);
	}
	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 <= (-2.5d-24)) .or. (.not. (phi1 <= (-4.6d-224)) .or. (.not. (phi1 <= (-1.1d-250))) .and. (phi1 <= 5.3d-197))) then
        tmp = r * (phi2 - phi1)
    else
        tmp = r * (lambda2 - lambda1)
    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 <= -2.5e-24) || !((phi1 <= -4.6e-224) || (!(phi1 <= -1.1e-250) && (phi1 <= 5.3e-197)))) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (lambda2 - lambda1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi1 <= -2.5e-24) or not ((phi1 <= -4.6e-224) or (not (phi1 <= -1.1e-250) and (phi1 <= 5.3e-197))):
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * (lambda2 - lambda1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -2.5e-24) || !((phi1 <= -4.6e-224) || (!(phi1 <= -1.1e-250) && (phi1 <= 5.3e-197))))
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * Float64(lambda2 - lambda1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi1 <= -2.5e-24) || ~(((phi1 <= -4.6e-224) || (~((phi1 <= -1.1e-250)) && (phi1 <= 5.3e-197)))))
		tmp = R * (phi2 - phi1);
	else
		tmp = R * (lambda2 - lambda1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -2.5e-24], N[Not[Or[LessEqual[phi1, -4.6e-224], And[N[Not[LessEqual[phi1, -1.1e-250]], $MachinePrecision], LessEqual[phi1, 5.3e-197]]]], $MachinePrecision]], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.4999999999999999e-24 or -4.59999999999999975e-224 < phi1 < -1.1e-250 or 5.29999999999999972e-197 < phi1

   1. Initial program 57.0%

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

      1. hypot-def 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in phi1 around -inf 36.6%

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

      1. *-commutative 36.6%

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

      2. associate-*r* 36.6%

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

      3. distribute-rgt-out 38.3%

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

      4. mul-1-neg 38.3%

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

      5. unsub-neg 38.3%

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

   6. Simplified 38.3%

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

   if -2.4999999999999999e-24 < phi1 < -4.59999999999999975e-224 or -1.1e-250 < phi1 < 5.29999999999999972e-197

   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 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in lambda1 around -inf 34.5%

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

      1. +-commutative 34.5%

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

      2. mul-1-neg 34.5%

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

      3. unsub-neg 34.5%

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

      4. *-commutative 34.5%

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

      5. *-commutative 34.5%

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

      6. associate-*l* 34.5%

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

      7. *-commutative 34.5%

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

      8. +-commutative 34.5%

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

      9. *-commutative 34.5%

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

      10. associate-*l* 34.5%

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

      11. +-commutative 34.5%

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

   6. Simplified 34.5%

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

   7. Taylor expanded in phi1 around 0 34.5%

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

      1. *-commutative 34.5%

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

      2. associate-*r* 34.5%

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

      3. distribute-lft-out-- 34.5%

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

      4. *-commutative 34.5%

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

   9. Simplified 34.5%

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

   10. Taylor expanded in phi2 around 0 23.7%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.5 \cdot 10^{-24} \lor \neg \left(\phi_1 \leq -4.6 \cdot 10^{-224} \lor \neg \left(\phi_1 \leq -1.1 \cdot 10^{-250}\right) \land \phi_1 \leq 5.3 \cdot 10^{-197}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

Alternative 13: 31.2% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.55 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -4.4 \cdot 10^{-226} \lor \neg \left(\phi_1 \leq -3 \cdot 10^{-252}\right) \land \phi_1 \leq 1.06 \cdot 10^{-196}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.55e-17)
   (* R (- phi1))
   (if (or (<= phi1 -4.4e-226)
           (and (not (<= phi1 -3e-252)) (<= phi1 1.06e-196)))
     (* R (- lambda2 lambda1))
     (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.55e-17) {
		tmp = R * -phi1;
	} else if ((phi1 <= -4.4e-226) || (!(phi1 <= -3e-252) && (phi1 <= 1.06e-196))) {
		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 <= (-1.55d-17)) then
        tmp = r * -phi1
    else if ((phi1 <= (-4.4d-226)) .or. (.not. (phi1 <= (-3d-252))) .and. (phi1 <= 1.06d-196)) 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 <= -1.55e-17) {
		tmp = R * -phi1;
	} else if ((phi1 <= -4.4e-226) || (!(phi1 <= -3e-252) && (phi1 <= 1.06e-196))) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.55e-17:
		tmp = R * -phi1
	elif (phi1 <= -4.4e-226) or (not (phi1 <= -3e-252) and (phi1 <= 1.06e-196)):
		tmp = R * (lambda2 - lambda1)
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.55e-17)
		tmp = Float64(R * Float64(-phi1));
	elseif ((phi1 <= -4.4e-226) || (!(phi1 <= -3e-252) && (phi1 <= 1.06e-196)))
		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 <= -1.55e-17)
		tmp = R * -phi1;
	elseif ((phi1 <= -4.4e-226) || (~((phi1 <= -3e-252)) && (phi1 <= 1.06e-196)))
		tmp = R * (lambda2 - lambda1);
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.55e-17], N[(R * (-phi1)), $MachinePrecision], If[Or[LessEqual[phi1, -4.4e-226], And[N[Not[LessEqual[phi1, -3e-252]], $MachinePrecision], LessEqual[phi1, 1.06e-196]]], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.5499999999999999e-17

   1. Initial program 52.1%

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

      1. hypot-def 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in phi1 around -inf 59.0%

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

      1. associate-*r* 59.0%

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

      2. mul-1-neg 59.0%

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

   6. Simplified 59.0%

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

   if -1.5499999999999999e-17 < phi1 < -4.4e-226 or -2.99999999999999995e-252 < phi1 < 1.05999999999999994e-196

   1. Initial program 70.8%

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

      1. hypot-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in lambda1 around -inf 34.9%

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

      1. +-commutative 34.9%

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

      2. mul-1-neg 34.9%

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

      3. unsub-neg 34.9%

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

      4. *-commutative 34.9%

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

      5. *-commutative 34.9%

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

      6. associate-*l* 34.9%

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

      7. *-commutative 34.9%

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

      8. +-commutative 34.9%

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

      9. *-commutative 34.9%

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

      10. associate-*l* 34.9%

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

      11. +-commutative 34.9%

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

   6. Simplified 34.9%

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

   7. Taylor expanded in phi1 around 0 34.9%

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

      1. *-commutative 34.9%

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

      2. associate-*r* 34.9%

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

      3. distribute-lft-out-- 34.9%

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

      4. *-commutative 34.9%

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

   9. Simplified 34.9%

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

   10. Taylor expanded in phi2 around 0 23.1%

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

   if -4.4e-226 < phi1 < -2.99999999999999995e-252 or 1.05999999999999994e-196 < phi1

   1. Initial program 60.9%

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

      1. hypot-def 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in phi2 around inf 18.2%

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

      1. *-commutative 18.2%

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

   6. Simplified 18.2%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.55 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -4.4 \cdot 10^{-226} \lor \neg \left(\phi_1 \leq -3 \cdot 10^{-252}\right) \land \phi_1 \leq 1.06 \cdot 10^{-196}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 14: 26.6% accurate, 65.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.4 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3.4e-28) (* R lambda2) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.4e-28) {
		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 <= 3.4d-28) 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 <= 3.4e-28) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3.4e-28:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3.4e-28)
		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 <= 3.4e-28)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3.4e-28], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.4000000000000001e-28

   1. Initial program 62.6%

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

      1. hypot-def 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in lambda1 around -inf 24.1%

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

      1. +-commutative 24.1%

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

      2. mul-1-neg 24.1%

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

      3. unsub-neg 24.1%

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

      4. *-commutative 24.1%

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

      5. *-commutative 24.1%

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

      6. associate-*l* 24.1%

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

      7. *-commutative 24.1%

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

      8. +-commutative 24.1%

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

      9. *-commutative 24.1%

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

      10. associate-*l* 24.2%

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

      11. +-commutative 24.2%

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

   6. Simplified 24.2%

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

   7. Taylor expanded in phi1 around 0 22.6%

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

      1. *-commutative 22.6%

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

      2. associate-*r* 22.6%

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

      3. distribute-lft-out-- 23.7%

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

      4. *-commutative 23.7%

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

   9. Simplified 23.7%

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

   10. Taylor expanded in phi2 around 0 21.3%

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

   11. Taylor expanded in lambda2 around inf 13.9%

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

   if 3.4000000000000001e-28 < phi2

   1. Initial program 57.0%

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

      1. hypot-def 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in phi2 around inf 58.7%

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

      1. *-commutative 58.7%

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

   6. Simplified 58.7%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.4 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 15: 14.4% 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 61.3%

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

   1. hypot-def 96.3%

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

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 lambda1 around -inf 23.5%

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

   1. +-commutative 23.5%

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

   2. mul-1-neg 23.5%

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

   3. unsub-neg 23.5%

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

   4. *-commutative 23.5%

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

   5. *-commutative 23.5%

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

   6. associate-*l* 23.5%

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

   7. *-commutative 23.5%

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

   8. +-commutative 23.5%

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

   9. *-commutative 23.5%

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

   10. associate-*l* 23.5%

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

   11. +-commutative 23.5%

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

6. Simplified 23.5%

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

7. Taylor expanded in phi1 around 0 22.0%

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

   1. *-commutative 22.0%

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

   2. associate-*r* 22.0%

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

   3. distribute-lft-out-- 22.8%

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

   4. *-commutative 22.8%

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

9. Simplified 22.8%

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

10. Taylor expanded in phi2 around 0 17.8%

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

11. Taylor expanded in lambda2 around inf 11.7%

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

12. Final simplification 11.7%

    \[\leadsto R \cdot \lambda_2 \]

Reproduce

herbie shell --seed 2023193 
(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))))))