Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.1% → 99.8%

Time: 18.8s

Alternatives: 12

Speedup: 1.6×

• Unsound rule application detected in e-graph. Results may not be sound.

• 31.5% 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: 60.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

   1. hypot-def 96.8%

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

3. Simplified 96.8%

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

   1. add-sqr-sqrt 63.4%

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

   2. sqrt-unprod 96.8%

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

   3. sqr-cos-a 96.8%

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

   4. cos-2 96.8%

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

   5. cos-sum 96.8%

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

   6. add-log-exp 26.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\color{blue}{\log \left(e^{\frac{\phi_1 + \phi_2}{2}}\right)} + \frac{\phi_1 + \phi_2}{2}\right)}, \phi_1 - \phi_2\right) \]

   7. add-log-exp 26.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\log \left(e^{\frac{\phi_1 + \phi_2}{2}}\right) + \color{blue}{\log \left(e^{\frac{\phi_1 + \phi_2}{2}}\right)}\right)}, \phi_1 - \phi_2\right) \]

   8. sum-log 26.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \color{blue}{\log \left(e^{\frac{\phi_1 + \phi_2}{2}} \cdot e^{\frac{\phi_1 + \phi_2}{2}}\right)}}, \phi_1 - \phi_2\right) \]

   9. exp-sqrt 26.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \log \left(\color{blue}{\sqrt{e^{\phi_1 + \phi_2}}} \cdot e^{\frac{\phi_1 + \phi_2}{2}}\right)}, \phi_1 - \phi_2\right) \]

   10. exp-sqrt 26.2%

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

   11. add-sqr-sqrt 26.2%

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

   12. add-log-exp 96.8%

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

5. Applied egg-rr 96.8%

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

   1. +-commutative 96.8%

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

7. Simplified 96.8%

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

   1. cos-sum 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 88.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4.5 \cdot 10^{-35}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \sqrt{0.5 + 0.5 \cdot \left(\cos \phi_2 \cdot \cos \phi_1 - \sin \phi_2 \cdot \sin \phi_1\right)}, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 4.5e-35)
   (*
    R
    (hypot
     (*
      lambda1
      (sqrt
       (+
        0.5
        (* 0.5 (- (* (cos phi2) (cos phi1)) (* (sin phi2) (sin phi1)))))))
     (- 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 (lambda2 <= 4.5e-35) {
		tmp = R * hypot((lambda1 * sqrt((0.5 + (0.5 * ((cos(phi2) * cos(phi1)) - (sin(phi2) * sin(phi1))))))), (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 (lambda2 <= 4.5e-35) {
		tmp = R * Math.hypot((lambda1 * Math.sqrt((0.5 + (0.5 * ((Math.cos(phi2) * Math.cos(phi1)) - (Math.sin(phi2) * Math.sin(phi1))))))), (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 lambda2 <= 4.5e-35:
		tmp = R * math.hypot((lambda1 * math.sqrt((0.5 + (0.5 * ((math.cos(phi2) * math.cos(phi1)) - (math.sin(phi2) * math.sin(phi1))))))), (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 (lambda2 <= 4.5e-35)
		tmp = Float64(R * hypot(Float64(lambda1 * sqrt(Float64(0.5 + Float64(0.5 * Float64(Float64(cos(phi2) * cos(phi1)) - Float64(sin(phi2) * sin(phi1))))))), 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 (lambda2 <= 4.5e-35)
		tmp = R * hypot((lambda1 * sqrt((0.5 + (0.5 * ((cos(phi2) * cos(phi1)) - (sin(phi2) * sin(phi1))))))), (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[lambda2, 4.5e-35], N[(R * N[Sqrt[N[(lambda1 * N[Sqrt[N[(0.5 + N[(0.5 * N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $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 lambda2 < 4.5000000000000001e-35

   1. Initial program 61.2%

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

      1. add-sqr-sqrt 61.9%

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

      2. sqrt-unprod 96.2%

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

      3. sqr-cos-a 96.2%

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

      4. cos-2 96.1%

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

      5. cos-sum 96.2%

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

      6. add-log-exp 24.2%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\color{blue}{\log \left(e^{\frac{\phi_1 + \phi_2}{2}}\right)} + \frac{\phi_1 + \phi_2}{2}\right)}, \phi_1 - \phi_2\right) \]

      7. add-log-exp 24.2%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\log \left(e^{\frac{\phi_1 + \phi_2}{2}}\right) + \color{blue}{\log \left(e^{\frac{\phi_1 + \phi_2}{2}}\right)}\right)}, \phi_1 - \phi_2\right) \]

      8. sum-log 24.2%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \color{blue}{\log \left(e^{\frac{\phi_1 + \phi_2}{2}} \cdot e^{\frac{\phi_1 + \phi_2}{2}}\right)}}, \phi_1 - \phi_2\right) \]

      9. exp-sqrt 24.2%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \log \left(\color{blue}{\sqrt{e^{\phi_1 + \phi_2}}} \cdot e^{\frac{\phi_1 + \phi_2}{2}}\right)}, \phi_1 - \phi_2\right) \]

      10. exp-sqrt 24.2%

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

      11. add-sqr-sqrt 24.2%

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

      12. add-log-exp 96.2%

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

   5. Applied egg-rr 96.2%

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

      1. +-commutative 96.2%

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

   7. Simplified 96.2%

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

      1. cos-sum 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in lambda1 around inf 84.4%

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

   if 4.5000000000000001e-35 < lambda2

   1. Initial program 62.3%

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

      1. hypot-def 98.7%

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

   3. Simplified 98.7%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4.5 \cdot 10^{-35}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \sqrt{0.5 + 0.5 \cdot \left(\cos \phi_2 \cdot \cos \phi_1 - \sin \phi_2 \cdot \sin \phi_1\right)}, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 3: 93.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.6 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3.6e-7)
   (* 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 (phi2 <= 3.6e-7) {
		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 (phi2 <= 3.6e-7) {
		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 phi2 <= 3.6e-7:
		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 (phi2 <= 3.6e-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(0.5 * phi2))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 3.6e-7)
		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[phi2, 3.6e-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[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.59999999999999994e-7

   1. Initial program 64.6%

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

      1. hypot-def 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in phi2 around 0 93.6%

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

   if 3.59999999999999994e-7 < phi2

   1. Initial program 53.0%

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

      1. hypot-def 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in phi1 around 0 94.2%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.6 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 4: 95.9% accurate, 1.5× speedup

Math

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

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 61.5%

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

   1. hypot-def 96.8%

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

3. Simplified 96.8%

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

4. Final simplification 96.8%

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

Alternative 5: 90.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := 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. Initial program 61.5%

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

   1. hypot-def 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in phi2 around 0 89.6%

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

5. Final simplification 89.6%

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

Alternative 6: 29.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \mathbf{if}\;\lambda_1 \leq -8.8 \cdot 10^{+119}:\\ \;\;\;\;t_0 \cdot \left(-R \cdot \lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.45 \cdot 10^{-230} \lor \neg \left(\lambda_1 \leq 3.8 \cdot 10^{-173}\right) \land \lambda_1 \leq 5.8 \cdot 10^{-79}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 (+ phi2 phi1)))))
   (if (<= lambda1 -8.8e+119)
     (* t_0 (- (* R lambda1)))
     (if (or (<= lambda1 1.45e-230)
             (and (not (<= lambda1 3.8e-173)) (<= lambda1 5.8e-79)))
       (* R (- phi2 phi1))
       (* R (* lambda2 t_0))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * (phi2 + phi1)));
	double tmp;
	if (lambda1 <= -8.8e+119) {
		tmp = t_0 * -(R * lambda1);
	} else if ((lambda1 <= 1.45e-230) || (!(lambda1 <= 3.8e-173) && (lambda1 <= 5.8e-79))) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (lambda2 * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((0.5d0 * (phi2 + phi1)))
    if (lambda1 <= (-8.8d+119)) then
        tmp = t_0 * -(r * lambda1)
    else if ((lambda1 <= 1.45d-230) .or. (.not. (lambda1 <= 3.8d-173)) .and. (lambda1 <= 5.8d-79)) then
        tmp = r * (phi2 - phi1)
    else
        tmp = r * (lambda2 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((0.5 * (phi2 + phi1)));
	double tmp;
	if (lambda1 <= -8.8e+119) {
		tmp = t_0 * -(R * lambda1);
	} else if ((lambda1 <= 1.45e-230) || (!(lambda1 <= 3.8e-173) && (lambda1 <= 5.8e-79))) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (lambda2 * t_0);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((0.5 * (phi2 + phi1)))
	tmp = 0
	if lambda1 <= -8.8e+119:
		tmp = t_0 * -(R * lambda1)
	elif (lambda1 <= 1.45e-230) or (not (lambda1 <= 3.8e-173) and (lambda1 <= 5.8e-79)):
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * (lambda2 * t_0)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * Float64(phi2 + phi1)))
	tmp = 0.0
	if (lambda1 <= -8.8e+119)
		tmp = Float64(t_0 * Float64(-Float64(R * lambda1)));
	elseif ((lambda1 <= 1.45e-230) || (!(lambda1 <= 3.8e-173) && (lambda1 <= 5.8e-79)))
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * Float64(lambda2 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((0.5 * (phi2 + phi1)));
	tmp = 0.0;
	if (lambda1 <= -8.8e+119)
		tmp = t_0 * -(R * lambda1);
	elseif ((lambda1 <= 1.45e-230) || (~((lambda1 <= 3.8e-173)) && (lambda1 <= 5.8e-79)))
		tmp = R * (phi2 - phi1);
	else
		tmp = R * (lambda2 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda1, -8.8e+119], N[(t$95$0 * (-N[(R * lambda1), $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[lambda1, 1.45e-230], And[N[Not[LessEqual[lambda1, 3.8e-173]], $MachinePrecision], LessEqual[lambda1, 5.8e-79]]], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -8.8000000000000005e119

   1. Initial program 53.1%

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

      1. hypot-def 94.1%

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

   3. Simplified 94.1%

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

      1. add-cube-cbrt 93.1%

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

      2. pow3 93.1%

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

      3. *-commutative 93.1%

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

      4. div-inv 93.1%

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

      5. metadata-eval 93.1%

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

   5. Applied egg-rr 93.1%

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

      1. expm1-log1p-u 88.7%

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

      2. rem-cube-cbrt 88.7%

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

      3. *-commutative 88.7%

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

      4. +-commutative 88.7%

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

   7. Applied egg-rr 88.7%

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

   8. Taylor expanded in lambda1 around -inf 58.9%

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

      1. mul-1-neg 58.9%

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

      2. *-commutative 58.9%

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

      3. *-commutative 58.9%

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

      4. +-commutative 58.9%

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

      5. associate-*r* 58.8%

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

      6. distribute-rgt-neg-in 58.8%

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

      7. +-commutative 58.8%

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

      8. *-commutative 58.8%

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

   10. Simplified 58.8%

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

   if -8.8000000000000005e119 < lambda1 < 1.45000000000000003e-230 or 3.8000000000000003e-173 < lambda1 < 5.8000000000000001e-79

   1. Initial program 63.4%

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

      1. hypot-def 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in phi1 around -inf 37.4%

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

      1. *-commutative 37.4%

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

      2. associate-*r* 37.4%

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

      3. distribute-rgt-out 38.1%

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

      4. mul-1-neg 38.1%

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

      5. unsub-neg 38.1%

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

   6. Simplified 38.1%

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

   if 1.45000000000000003e-230 < lambda1 < 3.8000000000000003e-173 or 5.8000000000000001e-79 < lambda1

   1. Initial program 62.7%

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

      1. hypot-def 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 lambda2 around inf 19.8%

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

      1. *-commutative 19.8%

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

      2. +-commutative 19.8%

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

   6. Simplified 19.8%

      \[\leadsto R \cdot \color{blue}{\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -8.8 \cdot 10^{+119}:\\ \;\;\;\;\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(-R \cdot \lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.45 \cdot 10^{-230} \lor \neg \left(\lambda_1 \leq 3.8 \cdot 10^{-173}\right) \land \lambda_1 \leq 5.8 \cdot 10^{-79}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\ \end{array} \]

Alternative 7: 29.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\\ \mathbf{if}\;\lambda_1 \leq -2.35 \cdot 10^{+120}:\\ \;\;\;\;\left(\lambda_1 \cdot t_0\right) \cdot \left(-R\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.45 \cdot 10^{-230} \lor \neg \left(\lambda_1 \leq 2.2 \cdot 10^{-173}\right) \land \lambda_1 \leq 5.8 \cdot 10^{-79}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 (+ phi2 phi1)))))
   (if (<= lambda1 -2.35e+120)
     (* (* lambda1 t_0) (- R))
     (if (or (<= lambda1 1.45e-230)
             (and (not (<= lambda1 2.2e-173)) (<= lambda1 5.8e-79)))
       (* R (- phi2 phi1))
       (* R (* lambda2 t_0))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * (phi2 + phi1)));
	double tmp;
	if (lambda1 <= -2.35e+120) {
		tmp = (lambda1 * t_0) * -R;
	} else if ((lambda1 <= 1.45e-230) || (!(lambda1 <= 2.2e-173) && (lambda1 <= 5.8e-79))) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (lambda2 * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((0.5d0 * (phi2 + phi1)))
    if (lambda1 <= (-2.35d+120)) then
        tmp = (lambda1 * t_0) * -r
    else if ((lambda1 <= 1.45d-230) .or. (.not. (lambda1 <= 2.2d-173)) .and. (lambda1 <= 5.8d-79)) then
        tmp = r * (phi2 - phi1)
    else
        tmp = r * (lambda2 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((0.5 * (phi2 + phi1)));
	double tmp;
	if (lambda1 <= -2.35e+120) {
		tmp = (lambda1 * t_0) * -R;
	} else if ((lambda1 <= 1.45e-230) || (!(lambda1 <= 2.2e-173) && (lambda1 <= 5.8e-79))) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (lambda2 * t_0);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((0.5 * (phi2 + phi1)))
	tmp = 0
	if lambda1 <= -2.35e+120:
		tmp = (lambda1 * t_0) * -R
	elif (lambda1 <= 1.45e-230) or (not (lambda1 <= 2.2e-173) and (lambda1 <= 5.8e-79)):
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * (lambda2 * t_0)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * Float64(phi2 + phi1)))
	tmp = 0.0
	if (lambda1 <= -2.35e+120)
		tmp = Float64(Float64(lambda1 * t_0) * Float64(-R));
	elseif ((lambda1 <= 1.45e-230) || (!(lambda1 <= 2.2e-173) && (lambda1 <= 5.8e-79)))
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * Float64(lambda2 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((0.5 * (phi2 + phi1)));
	tmp = 0.0;
	if (lambda1 <= -2.35e+120)
		tmp = (lambda1 * t_0) * -R;
	elseif ((lambda1 <= 1.45e-230) || (~((lambda1 <= 2.2e-173)) && (lambda1 <= 5.8e-79)))
		tmp = R * (phi2 - phi1);
	else
		tmp = R * (lambda2 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda1, -2.35e+120], N[(N[(lambda1 * t$95$0), $MachinePrecision] * (-R)), $MachinePrecision], If[Or[LessEqual[lambda1, 1.45e-230], And[N[Not[LessEqual[lambda1, 2.2e-173]], $MachinePrecision], LessEqual[lambda1, 5.8e-79]]], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -2.34999999999999997e120

   1. Initial program 53.1%

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

      1. hypot-def 94.1%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in lambda1 around -inf 58.9%

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

      1. mul-1-neg 58.9%

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

      2. *-commutative 58.9%

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

      3. distribute-rgt-neg-in 58.9%

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

      4. +-commutative 58.9%

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

   6. Simplified 58.9%

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

   if -2.34999999999999997e120 < lambda1 < 1.45000000000000003e-230 or 2.1999999999999999e-173 < lambda1 < 5.8000000000000001e-79

   1. Initial program 63.4%

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

      1. hypot-def 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in phi1 around -inf 37.4%

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

      1. *-commutative 37.4%

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

      2. associate-*r* 37.4%

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

      3. distribute-rgt-out 38.1%

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

      4. mul-1-neg 38.1%

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

      5. unsub-neg 38.1%

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

   6. Simplified 38.1%

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

   if 1.45000000000000003e-230 < lambda1 < 2.1999999999999999e-173 or 5.8000000000000001e-79 < lambda1

   1. Initial program 62.7%

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

      1. hypot-def 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 lambda2 around inf 19.8%

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

      1. *-commutative 19.8%

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

      2. +-commutative 19.8%

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

   6. Simplified 19.8%

      \[\leadsto R \cdot \color{blue}{\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.35 \cdot 10^{+120}:\\ \;\;\;\;\left(\lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right) \cdot \left(-R\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.45 \cdot 10^{-230} \lor \neg \left(\lambda_1 \leq 2.2 \cdot 10^{-173}\right) \land \lambda_1 \leq 5.8 \cdot 10^{-79}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\ \end{array} \]

Alternative 8: 32.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.4 \cdot 10^{+42} \lor \neg \left(\lambda_2 \leq 3.8 \cdot 10^{+146}\right) \land \lambda_2 \leq 3.5 \cdot 10^{+195}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda2 1.4e+42)
         (and (not (<= lambda2 3.8e+146)) (<= lambda2 3.5e+195)))
   (* R (- phi2 phi1))
   (* R (* lambda2 (cos (* 0.5 (+ phi2 phi1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= 1.4e+42) || (!(lambda2 <= 3.8e+146) && (lambda2 <= 3.5e+195))) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (lambda2 * cos((0.5 * (phi2 + phi1))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((lambda2 <= 1.4d+42) .or. (.not. (lambda2 <= 3.8d+146)) .and. (lambda2 <= 3.5d+195)) then
        tmp = r * (phi2 - phi1)
    else
        tmp = r * (lambda2 * cos((0.5d0 * (phi2 + phi1))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= 1.4e+42) || (!(lambda2 <= 3.8e+146) && (lambda2 <= 3.5e+195))) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (lambda2 * Math.cos((0.5 * (phi2 + phi1))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda2 <= 1.4e+42) or (not (lambda2 <= 3.8e+146) and (lambda2 <= 3.5e+195)):
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * (lambda2 * math.cos((0.5 * (phi2 + phi1))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda2 <= 1.4e+42) || (!(lambda2 <= 3.8e+146) && (lambda2 <= 3.5e+195)))
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * Float64(lambda2 * cos(Float64(0.5 * Float64(phi2 + phi1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda2 <= 1.4e+42) || (~((lambda2 <= 3.8e+146)) && (lambda2 <= 3.5e+195)))
		tmp = R * (phi2 - phi1);
	else
		tmp = R * (lambda2 * cos((0.5 * (phi2 + phi1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda2, 1.4e+42], And[N[Not[LessEqual[lambda2, 3.8e+146]], $MachinePrecision], LessEqual[lambda2, 3.5e+195]]], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 * N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.4e42 or 3.79999999999999979e146 < lambda2 < 3.5000000000000002e195

   1. Initial program 61.0%

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

      1. hypot-def 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in phi1 around -inf 30.2%

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

      1. *-commutative 30.2%

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

      2. associate-*r* 30.2%

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

      3. distribute-rgt-out 31.1%

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

      4. mul-1-neg 31.1%

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

      5. unsub-neg 31.1%

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

   6. Simplified 31.1%

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

   if 1.4e42 < lambda2 < 3.79999999999999979e146 or 3.5000000000000002e195 < lambda2

   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 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in lambda2 around inf 55.7%

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

      1. *-commutative 55.7%

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

      2. +-commutative 55.7%

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

   6. Simplified 55.7%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.4 \cdot 10^{+42} \lor \neg \left(\lambda_2 \leq 3.8 \cdot 10^{+146}\right) \land \lambda_2 \leq 3.5 \cdot 10^{+195}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\ \end{array} \]

Alternative 9: 35.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.06 \cdot 10^{+18} \lor \neg \left(\phi_1 \leq 3.8 \cdot 10^{-105}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -1.06e+18) (not (<= phi1 3.8e-105)))
   (* R (- phi2 phi1))
   (* R (* (cos (* 0.5 (+ phi2 phi1))) (- lambda2 lambda1)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -1.06e+18) || !(phi1 <= 3.8e-105)) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (cos((0.5 * (phi2 + phi1))) * (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 <= (-1.06d+18)) .or. (.not. (phi1 <= 3.8d-105))) then
        tmp = r * (phi2 - phi1)
    else
        tmp = r * (cos((0.5d0 * (phi2 + phi1))) * (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 <= -1.06e+18) || !(phi1 <= 3.8e-105)) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (Math.cos((0.5 * (phi2 + phi1))) * (lambda2 - lambda1));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi1 <= -1.06e+18) or not (phi1 <= 3.8e-105):
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * (math.cos((0.5 * (phi2 + phi1))) * (lambda2 - lambda1))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -1.06e+18) || !(phi1 <= 3.8e-105))
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * Float64(cos(Float64(0.5 * Float64(phi2 + phi1))) * Float64(lambda2 - lambda1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi1 <= -1.06e+18) || ~((phi1 <= 3.8e-105)))
		tmp = R * (phi2 - phi1);
	else
		tmp = R * (cos((0.5 * (phi2 + phi1))) * (lambda2 - lambda1));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -1.06e+18], N[Not[LessEqual[phi1, 3.8e-105]], $MachinePrecision]], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * N[(N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.06e18 or 3.7999999999999998e-105 < phi1

   1. Initial program 59.8%

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

      1. hypot-def 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 35.0%

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

      1. *-commutative 35.0%

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

      2. associate-*r* 35.0%

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

      3. distribute-rgt-out 36.5%

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

      4. mul-1-neg 36.5%

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

      5. unsub-neg 36.5%

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

   6. Simplified 36.5%

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

   if -1.06e18 < phi1 < 3.7999999999999998e-105

   1. Initial program 63.3%

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

      1. hypot-def 98.8%

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

   3. Simplified 98.8%

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

      1. add-cube-cbrt 97.4%

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

      2. pow3 97.4%

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

      3. *-commutative 97.4%

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

      4. div-inv 97.4%

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

      5. metadata-eval 97.4%

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

   5. Applied egg-rr 97.4%

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

      1. expm1-log1p-u 92.6%

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

      2. rem-cube-cbrt 92.7%

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

      3. *-commutative 92.7%

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

      4. +-commutative 92.7%

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

   7. Applied egg-rr 92.7%

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

   8. Taylor expanded in lambda1 around -inf 34.6%

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

      1. mul-1-neg 34.6%

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

      2. distribute-rgt-neg-in 34.6%

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

      3. mul-1-neg 34.6%

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

      4. *-commutative 34.6%

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

      5. *-commutative 34.6%

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

      6. +-commutative 34.6%

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

      7. associate-*r* 34.6%

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

      8. +-commutative 34.6%

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

      9. *-commutative 34.6%

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

      10. *-commutative 34.6%

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

      11. distribute-lft-in 35.5%

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

      12. *-lft-identity 35.5%

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

   10. Simplified 35.5%

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

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.06 \cdot 10^{+18} \lor \neg \left(\phi_1 \leq 3.8 \cdot 10^{-105}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]

Alternative 10: 28.8% accurate, 54.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -9.4 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -9.4e-6) (* R (- phi1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -9.4e-6) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-9.4d-6)) then
        tmp = r * -phi1
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -9.4e-6) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -9.4e-6:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -9.4e-6)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -9.4e-6)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -9.4e-6], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -9.39999999999999979e-6

   1. Initial program 57.9%

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

      1. hypot-def 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 phi1 around -inf 61.1%

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

      1. associate-*r* 61.1%

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

      2. mul-1-neg 61.1%

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

   6. Simplified 61.1%

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

   if -9.39999999999999979e-6 < phi1

   1. Initial program 62.7%

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

      1. hypot-def 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 18.5%

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

      1. *-commutative 18.5%

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

   6. Simplified 18.5%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -9.4 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 11: 29.6% 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 61.5%

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

   1. hypot-def 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in phi1 around -inf 27.6%

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

   1. *-commutative 27.6%

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

   2. associate-*r* 27.6%

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

   3. distribute-rgt-out 28.3%

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

   4. mul-1-neg 28.3%

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

   5. unsub-neg 28.3%

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

6. Simplified 28.3%

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

7. Final simplification 28.3%

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

Alternative 12: 17.7% accurate, 109.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

   1. hypot-def 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in phi2 around inf 16.7%

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

   1. *-commutative 16.7%

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

6. Simplified 16.7%

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

7. Final simplification 16.7%

   \[\leadsto R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023187 
(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))))))