Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.6% → 99.9%

Time: 10.2s

Alternatives: 17

Speedup: 2.6×

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

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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: 60.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 60.6

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

3. Applied rewrites 96.0%

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

5. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 60.6

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

3. Applied rewrites 96.0%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. metadata-eval N/A

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

   9. mult-flip N/A

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

   10. sin-+PI/2-rev N/A

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

   11. cos-neg-rev N/A

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

   12. distribute-neg-frac2 N/A

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

   13. distribute-frac-neg N/A

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

   14. frac-2neg N/A

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

   15. div-add N/A

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

   16. mult-flip-rev N/A

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

   17. metadata-eval N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. add-flip N/A

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

   21. cos-diff N/A

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

5. Applied rewrites 99.9%

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

Alternative 3: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+226}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_2, \mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(-0.5 \cdot \phi_2\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_2 - \phi_1, \cos \left(\left(\phi_2 + \phi_1\right) \cdot -0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -1e+226)
   (*
    (hypot
     phi2
     (*
      (fma
       (sin (* phi1 0.5))
       (sin (* -0.5 phi2))
       (* (cos (* phi1 0.5)) (cos (* -0.5 phi2))))
      (- lambda1 lambda2)))
    R)
   (*
    (hypot (- phi2 phi1) (* (cos (* (+ phi2 phi1) -0.5)) (- lambda1 lambda2)))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -1e+226) {
		tmp = hypot(phi2, (fma(sin((phi1 * 0.5)), sin((-0.5 * phi2)), (cos((phi1 * 0.5)) * cos((-0.5 * phi2)))) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi2 - phi1), (cos(((phi2 + phi1) * -0.5)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -1e+226)
		tmp = Float64(hypot(phi2, Float64(fma(sin(Float64(phi1 * 0.5)), sin(Float64(-0.5 * phi2)), Float64(cos(Float64(phi1 * 0.5)) * cos(Float64(-0.5 * phi2)))) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(phi2 - phi1), Float64(cos(Float64(Float64(phi2 + phi1) * -0.5)) * Float64(lambda1 - lambda2))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -1e+226], N[(N[Sqrt[phi2 ^ 2 + N[(N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi2 - phi1), $MachinePrecision] ^ 2 + N[(N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -9.99999999999999961e225

   1. Initial program 60.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.6

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

   3. Applied rewrites 96.0%

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2}, \mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_2\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   7. Step-by-step derivation

   8. Applied rewrites 79.5%

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

   if -9.99999999999999961e225 < (-.f64 lambda1 lambda2)

   1. Initial program 60.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.6

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

   3. Applied rewrites 96.0%

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

Alternative 4: 96.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 60.6

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

3. Applied rewrites 96.0%

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

Alternative 5: 93.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_2 - \phi_1, \cos \left(-0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_2 - \phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -6.8e-7)
   (* (hypot (- phi2 phi1) (* (cos (* -0.5 phi1)) (- lambda1 lambda2))) R)
   (* (hypot (- phi2 phi1) (* (cos (* -0.5 phi2)) (- lambda1 lambda2))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -6.8e-7) {
		tmp = hypot((phi2 - phi1), (cos((-0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = hypot((phi2 - phi1), (cos((-0.5 * phi2)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -6.8e-7) {
		tmp = Math.hypot((phi2 - phi1), (Math.cos((-0.5 * phi1)) * (lambda1 - lambda2))) * R;
	} else {
		tmp = Math.hypot((phi2 - phi1), (Math.cos((-0.5 * phi2)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -6.8e-7:
		tmp = math.hypot((phi2 - phi1), (math.cos((-0.5 * phi1)) * (lambda1 - lambda2))) * R
	else:
		tmp = math.hypot((phi2 - phi1), (math.cos((-0.5 * phi2)) * (lambda1 - lambda2))) * R
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -6.8e-7)
		tmp = Float64(hypot(Float64(phi2 - phi1), Float64(cos(Float64(-0.5 * phi1)) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(hypot(Float64(phi2 - phi1), Float64(cos(Float64(-0.5 * phi2)) * Float64(lambda1 - lambda2))) * R);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -6.8e-7)
		tmp = hypot((phi2 - phi1), (cos((-0.5 * phi1)) * (lambda1 - lambda2))) * R;
	else
		tmp = hypot((phi2 - phi1), (cos((-0.5 * phi2)) * (lambda1 - lambda2))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -6.8e-7], N[(N[Sqrt[N[(phi2 - phi1), $MachinePrecision] ^ 2 + N[(N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(phi2 - phi1), $MachinePrecision] ^ 2 + N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -6.79999999999999948e-7

   1. Initial program 60.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.6

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

   3. Applied rewrites 96.0%

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

   4. Taylor expanded in phi1 around inf

      \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \cos \color{blue}{\left(\frac{-1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   5. Step-by-step derivation

      1. lower-*.f64 90.5

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

   6. Applied rewrites 90.5%

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

   if -6.79999999999999948e-7 < phi1

   1. Initial program 60.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.6

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

   3. Applied rewrites 96.0%

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

   4. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \cos \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   5. Step-by-step derivation

      1. lower-*.f64 90.8

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

   6. Applied rewrites 90.8%

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

Alternative 6: 90.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 60.6

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

3. Applied rewrites 96.0%

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

4. Taylor expanded in phi1 around inf

   \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \cos \color{blue}{\left(\frac{-1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
5. Step-by-step derivation

   1. lower-*.f64 90.5

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

6. Applied rewrites 90.5%

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

Alternative 7: 75.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.02 \cdot 10^{+93}:\\ \;\;\;\;-\mathsf{fma}\left(-R, \phi_2, R \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_2, \cos \left(-0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.02e+93)
   (- (fma (- R) phi2 (* R phi1)))
   (* (hypot phi2 (* (cos (* -0.5 phi1)) (- lambda1 lambda2))) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.02e+93) {
		tmp = -fma(-R, phi2, (R * phi1));
	} else {
		tmp = hypot(phi2, (cos((-0.5 * phi1)) * (lambda1 - lambda2))) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.02e+93)
		tmp = Float64(-fma(Float64(-R), phi2, Float64(R * phi1)));
	else
		tmp = Float64(hypot(phi2, Float64(cos(Float64(-0.5 * phi1)) * Float64(lambda1 - lambda2))) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.02e+93], (-N[((-R) * phi2 + N[(R * phi1), $MachinePrecision]), $MachinePrecision]), N[(N[Sqrt[phi2 ^ 2 + N[(N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.0200000000000001e93

   1. Initial program 60.6%

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

   2. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 30.2

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

   4. Applied rewrites 30.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 30.2

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

   6. Applied rewrites 30.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 30.2

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. inv-pow N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

          \[\leadsto -\left(\left(\left(-R\right) \cdot \phi_2\right) \cdot {\phi_1}^{0} + R \cdot \phi_1\right) \]

      12. metadata-eval N/A

          \[\leadsto -\left(\left(\left(-R\right) \cdot \phi_2\right) \cdot 1 + R \cdot \phi_1\right) \]

      13. *-rgt-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 30.7

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

   8. Applied rewrites 30.7%

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

   if -1.0200000000000001e93 < phi1

   1. Initial program 60.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.6

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

   3. Applied rewrites 96.0%

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

   4. Taylor expanded in phi1 around inf

      \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \cos \color{blue}{\left(\frac{-1}{2} \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   5. Step-by-step derivation

      1. lower-*.f64 90.5

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

   6. Applied rewrites 90.5%

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

   7. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{hypot}\left(\color{blue}{\phi_2}, \cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   8. Step-by-step derivation

   9. Applied rewrites 70.3%

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

Alternative 8: 74.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.8 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_2 - \phi_1, \left(1 + -0.125 \cdot {\phi_2}^{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \mathsf{fma}\left(\phi_1, R, \left(-R\right) \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 6.8e+36)
   (*
    (hypot
     (- phi2 phi1)
     (* (+ 1.0 (* -0.125 (pow phi2 2.0))) (- lambda1 lambda2)))
    R)
   (* -1.0 (fma phi1 R (* (- R) phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 6.8e+36) {
		tmp = hypot((phi2 - phi1), ((1.0 + (-0.125 * pow(phi2, 2.0))) * (lambda1 - lambda2))) * R;
	} else {
		tmp = -1.0 * fma(phi1, R, (-R * phi2));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 6.8e+36)
		tmp = Float64(hypot(Float64(phi2 - phi1), Float64(Float64(1.0 + Float64(-0.125 * (phi2 ^ 2.0))) * Float64(lambda1 - lambda2))) * R);
	else
		tmp = Float64(-1.0 * fma(phi1, R, Float64(Float64(-R) * phi2)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 6.8e+36], N[(N[Sqrt[N[(phi2 - phi1), $MachinePrecision] ^ 2 + N[(N[(1.0 + N[(-0.125 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(-1.0 * N[(phi1 * R + N[((-R) * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 6.7999999999999996e36

   1. Initial program 60.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.6

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

   3. Applied rewrites 96.0%

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) + \phi_2 \cdot \left(\frac{-1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \frac{-1}{8} \cdot \left(\phi_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) + \color{blue}{\phi_2 \cdot \left(\frac{-1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \frac{-1}{8} \cdot \left(\phi_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) + \color{blue}{\phi_2} \cdot \left(\frac{-1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \frac{-1}{8} \cdot \left(\phi_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) + \phi_2 \cdot \left(\frac{-1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \frac{-1}{8} \cdot \left(\phi_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) + \phi_2 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \frac{-1}{8} \cdot \left(\phi_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) + \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}, \frac{-1}{8} \cdot \left(\phi_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. lower-sin.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) + \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \frac{-1}{8} \cdot \left(\phi_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) + \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \frac{-1}{8} \cdot \left(\phi_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) + \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \frac{-1}{8} \cdot \left(\phi_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) + \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \frac{-1}{8} \cdot \left(\phi_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      10. lower-cos.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) + \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \frac{-1}{8} \cdot \left(\phi_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      11. lower-*.f64 73.5

          \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(\cos \left(0.5 \cdot \phi_1\right) + \phi_2 \cdot \mathsf{fma}\left(-0.5, \sin \left(0.5 \cdot \phi_1\right), -0.125 \cdot \left(\phi_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   8. Applied rewrites 73.5%

      \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) + \phi_2 \cdot \mathsf{fma}\left(-0.5, \sin \left(0.5 \cdot \phi_1\right), -0.125 \cdot \left(\phi_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(1 + \color{blue}{\frac{-1}{8} \cdot {\phi_2}^{2}}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(1 + \frac{-1}{8} \cdot \color{blue}{{\phi_2}^{2}}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(1 + \frac{-1}{8} \cdot {\phi_2}^{\color{blue}{2}}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

       3. lower-pow.f64 68.2

          \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(1 + -0.125 \cdot {\phi_2}^{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   11. Applied rewrites 68.2%

       \[\leadsto \mathsf{hypot}\left(\phi_2 - \phi_1, \left(1 + \color{blue}{-0.125 \cdot {\phi_2}^{2}}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   if 6.7999999999999996e36 < phi2

   1. Initial program 60.6%

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

   2. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 30.2

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

   4. Applied rewrites 30.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 30.2

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

   6. Applied rewrites 30.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. inv-pow N/A

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

      10. pow-plus N/A

          \[\leadsto -1 \cdot \mathsf{fma}\left(\phi_1, R, \left(\left(-R\right) \cdot \phi_2\right) \cdot {\phi_1}^{\left(-1 + 1\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto -1 \cdot \mathsf{fma}\left(\phi_1, R, \left(\left(-R\right) \cdot \phi_2\right) \cdot {\phi_1}^{0}\right) \]

      12. metadata-eval N/A

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

      13. *-rgt-identity 30.7

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

   8. Applied rewrites 30.7%

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

Alternative 9: 60.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.5 \cdot 10^{+98}:\\ \;\;\;\;-\mathsf{fma}\left(-R, \phi_2, R \cdot \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{hypot}\left(\left(1 + -0.25 \cdot \left(\phi_1 \cdot \phi_2\right)\right) \cdot \left(\lambda_2 - \lambda_1\right), \phi_1 - \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(R - \frac{R \cdot \phi_2}{\phi_1}\right) \cdot \left(-\phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.5e+98)
   (- (fma (- R) phi2 (* R phi1)))
   (if (<= phi1 3.2e-48)
     (*
      (hypot
       (* (+ 1.0 (* -0.25 (* phi1 phi2))) (- lambda2 lambda1))
       (- phi1 phi2))
      R)
     (* (- R (/ (* R phi2) phi1)) (- phi1)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.5e+98) {
		tmp = -fma(-R, phi2, (R * phi1));
	} else if (phi1 <= 3.2e-48) {
		tmp = hypot(((1.0 + (-0.25 * (phi1 * phi2))) * (lambda2 - lambda1)), (phi1 - phi2)) * R;
	} else {
		tmp = (R - ((R * phi2) / phi1)) * -phi1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.5e+98)
		tmp = Float64(-fma(Float64(-R), phi2, Float64(R * phi1)));
	elseif (phi1 <= 3.2e-48)
		tmp = Float64(hypot(Float64(Float64(1.0 + Float64(-0.25 * Float64(phi1 * phi2))) * Float64(lambda2 - lambda1)), Float64(phi1 - phi2)) * R);
	else
		tmp = Float64(Float64(R - Float64(Float64(R * phi2) / phi1)) * Float64(-phi1));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.5e+98], (-N[((-R) * phi2 + N[(R * phi1), $MachinePrecision]), $MachinePrecision]), If[LessEqual[phi1, 3.2e-48], N[(N[Sqrt[N[(N[(1.0 + N[(-0.25 * N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[(R - N[(N[(R * phi2), $MachinePrecision] / phi1), $MachinePrecision]), $MachinePrecision] * (-phi1)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.4999999999999999e98

   1. Initial program 60.6%

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

   2. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 30.2

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

   4. Applied rewrites 30.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 30.2

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

   6. Applied rewrites 30.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 30.2

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. inv-pow N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

          \[\leadsto -\left(\left(\left(-R\right) \cdot \phi_2\right) \cdot {\phi_1}^{0} + R \cdot \phi_1\right) \]

      12. metadata-eval N/A

          \[\leadsto -\left(\left(\left(-R\right) \cdot \phi_2\right) \cdot 1 + R \cdot \phi_1\right) \]

      13. *-rgt-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 30.7

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

   8. Applied rewrites 30.7%

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

   if -2.4999999999999999e98 < phi1 < 3.1999999999999998e-48

   1. Initial program 60.6%

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

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

      7. lower-*.f64 52.0

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

      7. lower-*.f64 58.4

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

   7. Applied rewrites 58.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 58.4

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

   9. Applied rewrites 83.4%

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

   10. Taylor expanded in phi2 around 0

       \[\leadsto \mathsf{hypot}\left(\left(1 + \color{blue}{\frac{-1}{4} \cdot \left(\phi_1 \cdot \phi_2\right)}\right) \cdot \left(\lambda_2 - \lambda_1\right), \phi_1 - \phi_2\right) \cdot R \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\left(1 + \frac{-1}{4} \cdot \color{blue}{\left(\phi_1 \cdot \phi_2\right)}\right) \cdot \left(\lambda_2 - \lambda_1\right), \phi_1 - \phi_2\right) \cdot R \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{hypot}\left(\left(1 + \frac{-1}{4} \cdot \left(\phi_1 \cdot \color{blue}{\phi_2}\right)\right) \cdot \left(\lambda_2 - \lambda_1\right), \phi_1 - \phi_2\right) \cdot R \]

       3. lower-*.f64 72.0

          \[\leadsto \mathsf{hypot}\left(\left(1 + -0.25 \cdot \left(\phi_1 \cdot \phi_2\right)\right) \cdot \left(\lambda_2 - \lambda_1\right), \phi_1 - \phi_2\right) \cdot R \]

   12. Applied rewrites 72.0%

       \[\leadsto \mathsf{hypot}\left(\left(1 + \color{blue}{-0.25 \cdot \left(\phi_1 \cdot \phi_2\right)}\right) \cdot \left(\lambda_2 - \lambda_1\right), \phi_1 - \phi_2\right) \cdot R \]

   if 3.1999999999999998e-48 < phi1

   1. Initial program 60.6%

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

   2. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 30.2

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

   4. Applied rewrites 30.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. add-flip N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lift-/.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-frac-neg2 N/A

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

      15. frac-2neg N/A

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

      16. lift-/.f64 N/A

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

      17. lower-neg.f64 30.2

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

   6. Applied rewrites 30.2%

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

Alternative 10: 48.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\\ t_1 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_2 := \left(\lambda_1 - \lambda_2\right) \cdot 1\\ \mathbf{if}\;\sqrt{t\_1 \cdot t\_1 + t\_0} \leq 2 \cdot 10^{+154}:\\ \;\;\;\;R \cdot \sqrt{t\_2 \cdot t\_2 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\phi_1 \cdot \left(\phi_2 \cdot \mathsf{fma}\left(-1, \frac{R}{\phi_1}, \frac{R}{\phi_2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- phi1 phi2) (- phi1 phi2)))
        (t_1 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))))
        (t_2 (* (- lambda1 lambda2) 1.0)))
   (if (<= (sqrt (+ (* t_1 t_1) t_0)) 2e+154)
     (* R (sqrt (+ (* t_2 t_2) t_0)))
     (* -1.0 (* phi1 (* phi2 (fma -1.0 (/ R phi1) (/ R phi2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (phi1 - phi2) * (phi1 - phi2);
	double t_1 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	double t_2 = (lambda1 - lambda2) * 1.0;
	double tmp;
	if (sqrt(((t_1 * t_1) + t_0)) <= 2e+154) {
		tmp = R * sqrt(((t_2 * t_2) + t_0));
	} else {
		tmp = -1.0 * (phi1 * (phi2 * fma(-1.0, (R / phi1), (R / phi2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2))
	t_1 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	t_2 = Float64(Float64(lambda1 - lambda2) * 1.0)
	tmp = 0.0
	if (sqrt(Float64(Float64(t_1 * t_1) + t_0)) <= 2e+154)
		tmp = Float64(R * sqrt(Float64(Float64(t_2 * t_2) + t_0)));
	else
		tmp = Float64(-1.0 * Float64(phi1 * Float64(phi2 * fma(-1.0, Float64(R / phi1), Float64(R / phi2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(lambda1 - lambda2), $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision], 2e+154], N[(R * N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(phi1 * N[(phi2 * N[(-1.0 * N[(R / phi1), $MachinePrecision] + N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) #s(literal 2 binary64)))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) #s(literal 2 binary64))))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2)))) < 2.00000000000000007e154

   1. Initial program 60.6%

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

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

      7. lower-*.f64 52.0

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

      7. lower-*.f64 58.4

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

   7. Applied rewrites 58.4%

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

   8. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot 1\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 49.0%

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

   11. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot 1\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 1\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 57.8%

       \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot 1\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 1\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

   if 2.00000000000000007e154 < (sqrt.f64 (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) #s(literal 2 binary64)))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) #s(literal 2 binary64))))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2))))

   1. Initial program 60.6%

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

   2. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 30.2

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

   4. Applied rewrites 30.2%

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

   5. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 29.4

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

   7. Applied rewrites 29.4%

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

Alternative 11: 31.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -3.8 \cdot 10^{+227}:\\ \;\;\;\;-1 \cdot \left(\phi_1 \cdot \left(\phi_2 \cdot \mathsf{fma}\left(-1, \frac{R}{\phi_1}, \frac{R}{\phi_2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \mathsf{fma}\left(\phi_1, R, \left(-R\right) \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -3.8e+227)
   (* -1.0 (* phi1 (* phi2 (fma -1.0 (/ R phi1) (/ R phi2)))))
   (* -1.0 (fma phi1 R (* (- R) phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -3.8e+227) {
		tmp = -1.0 * (phi1 * (phi2 * fma(-1.0, (R / phi1), (R / phi2))));
	} else {
		tmp = -1.0 * fma(phi1, R, (-R * phi2));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -3.8e+227)
		tmp = Float64(-1.0 * Float64(phi1 * Float64(phi2 * fma(-1.0, Float64(R / phi1), Float64(R / phi2)))));
	else
		tmp = Float64(-1.0 * fma(phi1, R, Float64(Float64(-R) * phi2)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -3.8e+227], N[(-1.0 * N[(phi1 * N[(phi2 * N[(-1.0 * N[(R / phi1), $MachinePrecision] + N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(phi1 * R + N[((-R) * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -3.80000000000000035e227

   1. Initial program 60.6%

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

   2. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 30.2

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

   4. Applied rewrites 30.2%

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

   5. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 29.4

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

   7. Applied rewrites 29.4%

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

   if -3.80000000000000035e227 < (-.f64 lambda1 lambda2)

   1. Initial program 60.6%

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

   2. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 30.2

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

   4. Applied rewrites 30.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 30.2

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

   6. Applied rewrites 30.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. inv-pow N/A

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

      10. pow-plus N/A

          \[\leadsto -1 \cdot \mathsf{fma}\left(\phi_1, R, \left(\left(-R\right) \cdot \phi_2\right) \cdot {\phi_1}^{\left(-1 + 1\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto -1 \cdot \mathsf{fma}\left(\phi_1, R, \left(\left(-R\right) \cdot \phi_2\right) \cdot {\phi_1}^{0}\right) \]

      12. metadata-eval N/A

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

      13. *-rgt-identity 30.7

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

   8. Applied rewrites 30.7%

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

Alternative 12: 31.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+228}:\\ \;\;\;\;\phi_2 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \mathsf{fma}\left(\phi_1, R, \left(-R\right) \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -5e+228)
   (* phi2 (+ R (* -1.0 (/ (* R phi1) phi2))))
   (* -1.0 (fma phi1 R (* (- R) phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -5e+228) {
		tmp = phi2 * (R + (-1.0 * ((R * phi1) / phi2)));
	} else {
		tmp = -1.0 * fma(phi1, R, (-R * phi2));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -5e+228)
		tmp = Float64(phi2 * Float64(R + Float64(-1.0 * Float64(Float64(R * phi1) / phi2))));
	else
		tmp = Float64(-1.0 * fma(phi1, R, Float64(Float64(-R) * phi2)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -5e+228], N[(phi2 * N[(R + N[(-1.0 * N[(N[(R * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(phi1 * R + N[((-R) * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -5e228

   1. Initial program 60.6%

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

   2. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 29.9

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

   4. Applied rewrites 29.9%

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

   if -5e228 < (-.f64 lambda1 lambda2)

   1. Initial program 60.6%

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

   2. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 30.2

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

   4. Applied rewrites 30.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 30.2

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

   6. Applied rewrites 30.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. inv-pow N/A

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

      10. pow-plus N/A

          \[\leadsto -1 \cdot \mathsf{fma}\left(\phi_1, R, \left(\left(-R\right) \cdot \phi_2\right) \cdot {\phi_1}^{\left(-1 + 1\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto -1 \cdot \mathsf{fma}\left(\phi_1, R, \left(\left(-R\right) \cdot \phi_2\right) \cdot {\phi_1}^{0}\right) \]

      12. metadata-eval N/A

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

      13. *-rgt-identity 30.7

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

   8. Applied rewrites 30.7%

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

Alternative 13: 30.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ -1 \cdot \mathsf{fma}\left(\phi_1, R, \left(-R\right) \cdot \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* -1.0 (fma phi1 R (* (- R) phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return -1.0 * fma(phi1, R, (-R * phi2));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(-1.0 * fma(phi1, R, Float64(Float64(-R) * phi2)))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(-1.0 * N[(phi1 * R + N[((-R) * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.6%

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

2. Taylor expanded in phi1 around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 30.2

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

4. Applied rewrites 30.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. associate-*r* N/A

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

   7. mul-1-neg N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-lft-neg-in N/A

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

   11. lower-*.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 30.2

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

6. Applied rewrites 30.2%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-/.f64 N/A

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

   9. inv-pow N/A

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

   10. pow-plus N/A

       \[\leadsto -1 \cdot \mathsf{fma}\left(\phi_1, R, \left(\left(-R\right) \cdot \phi_2\right) \cdot {\phi_1}^{\left(-1 + 1\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto -1 \cdot \mathsf{fma}\left(\phi_1, R, \left(\left(-R\right) \cdot \phi_2\right) \cdot {\phi_1}^{0}\right) \]

   12. metadata-eval N/A

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

   13. *-rgt-identity 30.7

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

8. Applied rewrites 30.7%

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

Alternative 14: 30.7% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.6%

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

2. Taylor expanded in phi1 around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 30.2

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

4. Applied rewrites 30.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. associate-*r* N/A

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

   7. mul-1-neg N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-lft-neg-in N/A

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

   11. lower-*.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 30.2

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

6. Applied rewrites 30.2%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 30.2

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. associate-*l* N/A

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

   8. lift-/.f64 N/A

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

   9. inv-pow N/A

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

   10. pow-plus N/A

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

   11. metadata-eval N/A

       \[\leadsto -\left(\left(\left(-R\right) \cdot \phi_2\right) \cdot {\phi_1}^{0} + R \cdot \phi_1\right) \]

   12. metadata-eval N/A

       \[\leadsto -\left(\left(\left(-R\right) \cdot \phi_2\right) \cdot 1 + R \cdot \phi_1\right) \]

   13. *-rgt-identity N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 30.7

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

8. Applied rewrites 30.7%

   \[\leadsto \color{blue}{-\mathsf{fma}\left(-R, \phi_2, R \cdot \phi_1\right)} \]
9. Add Preprocessing

Alternative 15: 29.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.8 \cdot 10^{+22}:\\ \;\;\;\;R \cdot \left(-1 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.8e+22) (* R (* -1.0 phi1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.8e+22) {
		tmp = R * (-1.0 * phi1);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-3.8d+22)) then
        tmp = r * ((-1.0d0) * phi1)
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.8e+22) {
		tmp = R * (-1.0 * phi1);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.8e+22:
		tmp = R * (-1.0 * phi1)
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.8e+22)
		tmp = Float64(R * Float64(-1.0 * phi1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3.8e+22)
		tmp = R * (-1.0 * phi1);
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3.8e+22], N[(R * N[(-1.0 * phi1), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.8000000000000004e22

   1. Initial program 60.6%

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

   2. Taylor expanded in phi1 around -inf

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

      1. lower-*.f64 18.1

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

   4. Applied rewrites 18.1%

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

   if -3.8000000000000004e22 < phi1

   1. Initial program 60.6%

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

   2. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 18.1

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

   4. Applied rewrites 18.1%

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

Alternative 16: 18.1% accurate, 27.0× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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 60.6%

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

2. Taylor expanded in phi2 around inf

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

   1. lower-*.f64 18.1

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

4. Applied rewrites 18.1%

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
5. Add Preprocessing

Alternative 17: 17.6% accurate, 27.0× speedup

Math

\[\begin{array}{l} \\ R \cdot \phi_1 \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    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 * phi1
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.6%

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

2. Taylor expanded in phi1 around inf

   \[\leadsto \color{blue}{R \cdot \phi_1} \]
3. Step-by-step derivation

   1. lower-*.f64 17.6

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

4. Applied rewrites 17.6%

   \[\leadsto \color{blue}{R \cdot \phi_1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025150 
(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))))))