Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.2% → 99.8%

Time: 30.9s

Alternatives: 13

Speedup: 3.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{t_0 \cdot {t_0}^{2}}, \phi_1 - \phi_2\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos (* phi2 0.5))
          (cos (* 0.5 phi1))
          (* (sin (* 0.5 phi1)) (- (sin (* phi2 0.5)))))))
   (*
    R
    (hypot
     (* (- lambda1 lambda2) (cbrt (* t_0 (pow t_0 2.0))))
     (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos((phi2 * 0.5)), cos((0.5 * phi1)), (sin((0.5 * phi1)) * -sin((phi2 * 0.5))));
	return R * hypot(((lambda1 - lambda2) * cbrt((t_0 * pow(t_0, 2.0)))), (phi1 - phi2));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(phi2 * 0.5)), cos(Float64(0.5 * phi1)), Float64(sin(Float64(0.5 * phi1)) * Float64(-sin(Float64(phi2 * 0.5)))))
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cbrt(Float64(t_0 * (t_0 ^ 2.0)))), Float64(phi1 - phi2)))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * (-N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Power[N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 60.5%

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

   1. hypot-def 95.2%

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

3. Simplified 95.2%

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

   1. log1p-expm1-u 95.2%

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

   2. div-inv 95.2%

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

   3. metadata-eval 95.2%

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

5. Applied egg-rr 95.2%

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

   1. *-commutative 95.2%

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

   2. +-commutative 95.2%

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

   3. distribute-lft-in 95.2%

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

   4. cos-sum 99.8%

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

   5. *-commutative 99.8%

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

   6. *-commutative 99.8%

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

   7. *-commutative 99.8%

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

   8. *-commutative 99.8%

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

7. Applied egg-rr 99.8%

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

   1. log1p-expm1-u 99.8%

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

   2. cancel-sign-sub-inv 99.8%

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

   3. fma-def 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

9. Applied egg-rr 99.8%

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

    1. add-cbrt-cube 99.8%

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

11. Applied egg-rr 99.8%

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

    1. associate-*l* 99.8%

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

    2. *-commutative 99.8%

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

13. Simplified 99.8%

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sqrt[3]{\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right) \cdot {\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}}}, \phi_1 - \phi_2\right) \]

14. Final simplification 99.8%

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right) \cdot {\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}}, \phi_1 - \phi_2\right) \]

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.5%

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

   1. hypot-def 95.2%

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

3. Simplified 95.2%

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

   1. log1p-expm1-u 95.2%

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

   2. div-inv 95.2%

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

   3. metadata-eval 95.2%

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

5. Applied egg-rr 95.2%

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

   1. *-commutative 95.2%

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

   2. +-commutative 95.2%

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

   3. distribute-lft-in 95.2%

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

   4. cos-sum 99.8%

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

   5. *-commutative 99.8%

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

   6. *-commutative 99.8%

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

   7. *-commutative 99.8%

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

   8. *-commutative 99.8%

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

7. Applied egg-rr 99.8%

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

   1. log1p-expm1-u 99.8%

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

   2. cancel-sign-sub-inv 99.8%

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

   3. fma-def 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 96.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.1 \cdot 10^{+202}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -4.1e+202)
   (*
    R
    (hypot
     (*
      lambda1
      (-
       (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
       (* (sin (* 0.5 phi1)) (sin (* phi2 0.5)))))
     (- phi1 phi2)))
   (*
    R
    (hypot
     (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0)))
     (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -4.1e+202) {
		tmp = R * hypot((lambda1 * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -4.1e+202) {
		tmp = R * Math.hypot((lambda1 * ((Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - (Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5))))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -4.1e+202:
		tmp = R * math.hypot((lambda1 * ((math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))) - (math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -4.1e+202)
		tmp = Float64(R * hypot(Float64(lambda1 * Float64(Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5))))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -4.1e+202)
		tmp = R * hypot((lambda1 * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -4.1e+202], N[(R * N[Sqrt[N[(lambda1 * N[(N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.0999999999999998e202

   1. Initial program 50.2%

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

      1. hypot-def 83.7%

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

   3. Simplified 83.7%

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

      1. log1p-expm1-u 83.7%

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

      2. div-inv 83.7%

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

      3. metadata-eval 83.7%

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

   5. Applied egg-rr 83.7%

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

      1. *-commutative 83.7%

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

      2. +-commutative 83.7%

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

      3. distribute-lft-in 83.7%

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

      4. cos-sum 99.8%

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

      5. *-commutative 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

      8. *-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in lambda1 around inf 96.3%

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

      1. *-commutative 96.3%

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

   10. Simplified 96.3%

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

   if -4.0999999999999998e202 < lambda1

   1. Initial program 61.6%

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

      1. hypot-def 96.4%

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

   3. Simplified 96.4%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.1 \cdot 10^{+202}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 4: 80.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.2 \cdot 10^{+155}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 3.2e+155)
   (* R (hypot (* lambda1 (cos (* 0.5 (+ phi2 phi1)))) (- phi1 phi2)))
   (* R (hypot phi1 (- lambda1 lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 3.2e+155) {
		tmp = R * hypot((lambda1 * cos((0.5 * (phi2 + phi1)))), (phi1 - phi2));
	} else {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 3.2e+155) {
		tmp = R * Math.hypot((lambda1 * Math.cos((0.5 * (phi2 + phi1)))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 3.2e+155:
		tmp = R * math.hypot((lambda1 * math.cos((0.5 * (phi2 + phi1)))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 3.2e+155)
		tmp = Float64(R * hypot(Float64(lambda1 * cos(Float64(0.5 * Float64(phi2 + phi1)))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 3.2e+155)
		tmp = R * hypot((lambda1 * cos((0.5 * (phi2 + phi1)))), (phi1 - phi2));
	else
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 3.2e+155], N[(R * N[Sqrt[N[(lambda1 * N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 3.20000000000000012e155

   1. Initial program 63.6%

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

      1. hypot-def 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in lambda1 around inf 82.0%

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

      1. *-commutative 82.0%

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

      2. *-lft-identity 82.0%

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

      3. metadata-eval 82.0%

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

      4. cancel-sign-sub-inv 82.0%

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

      5. sub-neg 82.0%

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

      6. mul-1-neg 82.0%

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

      7. remove-double-neg 82.0%

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

      8. +-commutative 82.0%

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

      9. remove-double-neg 82.0%

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

      10. mul-1-neg 82.0%

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

      11. sub-neg 82.0%

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

      12. sub-neg 82.0%

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

      13. mul-1-neg 82.0%

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

      14. remove-double-neg 82.0%

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

   6. Simplified 82.0%

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

   if 3.20000000000000012e155 < lambda2

   1. Initial program 35.3%

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

      1. hypot-def 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in phi1 around 0 79.9%

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

   5. Taylor expanded in phi2 around 0 35.3%

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

      1. unpow2 35.3%

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

      2. unpow2 35.3%

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

      3. hypot-def 72.0%

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

   7. Simplified 72.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.2 \cdot 10^{+155}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \end{array} \]

Alternative 5: 85.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2 \cdot 10^{-125}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 2e-125)
   (* R (hypot (* lambda1 (cos (* 0.5 (+ phi2 phi1)))) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 2e-125) {
		tmp = R * hypot((lambda1 * cos((0.5 * (phi2 + phi1)))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 2e-125) {
		tmp = R * Math.hypot((lambda1 * Math.cos((0.5 * (phi2 + phi1)))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 2e-125:
		tmp = R * math.hypot((lambda1 * math.cos((0.5 * (phi2 + phi1)))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 2e-125)
		tmp = Float64(R * hypot(Float64(lambda1 * cos(Float64(0.5 * Float64(phi2 + phi1)))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 2e-125)
		tmp = R * hypot((lambda1 * cos((0.5 * (phi2 + phi1)))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 2e-125], N[(R * N[Sqrt[N[(lambda1 * N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $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 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 2.00000000000000002e-125

   1. Initial program 58.7%

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

      1. hypot-def 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in lambda1 around inf 81.9%

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

      1. *-commutative 81.9%

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

      2. *-lft-identity 81.9%

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

      3. metadata-eval 81.9%

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

      4. cancel-sign-sub-inv 81.9%

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

      5. sub-neg 81.9%

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

      6. mul-1-neg 81.9%

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

      7. remove-double-neg 81.9%

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

      8. +-commutative 81.9%

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

      9. remove-double-neg 81.9%

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

      10. mul-1-neg 81.9%

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

      11. sub-neg 81.9%

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

      12. sub-neg 81.9%

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

      13. mul-1-neg 81.9%

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

      14. remove-double-neg 81.9%

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

   6. Simplified 81.9%

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

   if 2.00000000000000002e-125 < lambda2

   1. Initial program 64.1%

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

      1. hypot-def 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in phi2 around 0 89.1%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2 \cdot 10^{-125}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 6: 93.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -4.2e-7)
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (* phi2 0.5))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4.2e-7) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4.2e-7) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -4.2e-7:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi2 * 0.5))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -4.2e-7)
		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(phi2 * 0.5))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -4.2e-7)
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -4.2e-7], 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[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.2e-7

   1. Initial program 50.9%

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

      1. hypot-def 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in phi2 around 0 92.6%

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

   if -4.2e-7 < phi1

   1. Initial program 64.2%

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

      1. hypot-def 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in phi1 around 0 93.2%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 7: 96.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.5%

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

   1. hypot-def 95.2%

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

3. Simplified 95.2%

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

4. Final simplification 95.2%

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

Alternative 8: 74.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.3 \cdot 10^{+45}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 5.3e+45)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (hypot (* lambda1 (cos (* phi2 0.5))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.3e+45) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * hypot((lambda1 * cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.3e+45) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * Math.hypot((lambda1 * Math.cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 5.3e+45:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * math.hypot((lambda1 * math.cos((phi2 * 0.5))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5.3e+45)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * hypot(Float64(lambda1 * cos(Float64(phi2 * 0.5))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 5.3e+45)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * hypot((lambda1 * cos((phi2 * 0.5))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 5.3e+45], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda1 * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5.29999999999999991e45

   1. Initial program 64.7%

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

      1. hypot-def 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in phi1 around 0 89.1%

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

   5. Taylor expanded in phi2 around 0 51.7%

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

      1. unpow2 51.7%

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

      2. unpow2 51.7%

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

      3. hypot-def 70.2%

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

   7. Simplified 70.2%

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

   if 5.29999999999999991e45 < phi2

   1. Initial program 46.8%

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

      1. hypot-def 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 0 94.7%

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

   5. Taylor expanded in lambda1 around inf 88.3%

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

      1. *-commutative 88.3%

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

   7. Simplified 88.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.3 \cdot 10^{+45}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 9: 70.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.4 \cdot 10^{+53}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.4e+53)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (- phi2 phi1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.4e+53) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} 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 <= 1.4e+53) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.4e+53:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.4e+53)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	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 <= 1.4e+53)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.4e+53], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.4e53

   1. Initial program 64.7%

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

      1. hypot-def 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in phi1 around 0 89.1%

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

   5. Taylor expanded in phi2 around 0 51.7%

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

      1. unpow2 51.7%

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

      2. unpow2 51.7%

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

      3. hypot-def 70.2%

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

   7. Simplified 70.2%

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

   if 1.4e53 < phi2

   1. Initial program 46.8%

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

      1. hypot-def 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 66.0%

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

      1. mul-1-neg 66.0%

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

      2. unsub-neg 66.0%

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

   6. Simplified 66.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.4 \cdot 10^{+53}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 10: 58.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.18 \cdot 10^{+55}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.18e+55) (* R (hypot phi1 lambda1)) (* R (- phi2 phi1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.18e+55) {
		tmp = R * hypot(phi1, lambda1);
	} 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 <= 1.18e+55) {
		tmp = R * Math.hypot(phi1, lambda1);
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.18e+55:
		tmp = R * math.hypot(phi1, lambda1)
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.18e+55)
		tmp = Float64(R * hypot(phi1, lambda1));
	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 <= 1.18e+55)
		tmp = R * hypot(phi1, lambda1);
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.18e+55], N[(R * N[Sqrt[phi1 ^ 2 + lambda1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.1799999999999999e55

   1. Initial program 64.7%

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

      1. hypot-def 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in phi1 around 0 89.1%

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

   5. Taylor expanded in lambda1 around inf 70.3%

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

      1. *-commutative 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in phi2 around 0 39.6%

      \[\leadsto \color{blue}{R \cdot \sqrt{{\lambda_1}^{2} + {\phi_1}^{2}}} \]
   9. Step-by-step derivation

      1. +-commutative 39.6%

         \[\leadsto R \cdot \sqrt{\color{blue}{{\phi_1}^{2} + {\lambda_1}^{2}}} \]

      2. unpow2 39.6%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_1 \cdot \phi_1} + {\lambda_1}^{2}} \]

      3. unpow2 39.6%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\lambda_1 \cdot \lambda_1}} \]

      4. hypot-def 52.0%

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

   10. Simplified 52.0%

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

   if 1.1799999999999999e55 < phi2

   1. Initial program 46.8%

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

      1. hypot-def 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 66.0%

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

      1. mul-1-neg 66.0%

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

      2. unsub-neg 66.0%

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

   6. Simplified 66.0%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.18 \cdot 10^{+55}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 11: 27.9% accurate, 54.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8 \cdot 10^{+44}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 8e+44) (* R (- phi1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 8e+44) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 8d+44) then
        tmp = r * -phi1
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 8e+44) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 8e+44:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 8e+44)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 8e+44)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 8e+44], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 8.0000000000000007e44

   1. Initial program 65.0%

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

      1. hypot-def 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in phi1 around -inf 19.1%

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

      1. mul-1-neg 19.1%

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

      2. *-commutative 19.1%

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

      3. distribute-rgt-neg-in 19.1%

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

   6. Simplified 19.1%

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

   if 8.0000000000000007e44 < phi2

   1. Initial program 46.1%

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

      1. hypot-def 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in phi2 around inf 60.1%

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

      1. *-commutative 60.1%

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

   6. Simplified 60.1%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8 \cdot 10^{+44}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 12: 29.1% accurate, 65.8× speedup

Math

\[\begin{array}{l} \\ R \cdot \left(\phi_2 - \phi_1\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.5%

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

   1. hypot-def 95.2%

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

3. Simplified 95.2%

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

4. Taylor expanded in phi1 around -inf 29.9%

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

   1. mul-1-neg 29.9%

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

   2. unsub-neg 29.9%

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

6. Simplified 29.9%

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

7. Final simplification 29.9%

   \[\leadsto R \cdot \left(\phi_2 - \phi_1\right) \]

Alternative 13: 17.2% accurate, 109.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.5%

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

   1. hypot-def 95.2%

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

3. Simplified 95.2%

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

4. Taylor expanded in phi2 around inf 16.2%

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

   1. *-commutative 16.2%

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

6. Simplified 16.2%

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

7. Final simplification 16.2%

   \[\leadsto R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023287 
(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))))))