VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.9% → 99.5%

Time: 4.7s

Alternatives: 24

Speedup: 1.6×

Specification

Math

\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (+
 (- (* x (/ 1 (tan B))))
 (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))

C

double code(double F, double B, double x) {
	return -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
end function

Java

public static double code(double F, double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Python

def code(F, B, x):
	return -(x * (1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))

Julia

function code(F, B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
end

Wolfram

code[F_, B_, x_] := N[((-N[(x * N[(1 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2), $MachinePrecision] + N[(2 * x), $MachinePrecision]), $MachinePrecision], (-N[(1 / 2), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.9% accurate, 1.0× speedup

Math

\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (+
 (- (* x (/ 1 (tan B))))
 (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))

C

double code(double F, double B, double x) {
	return -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
end function

Java

public static double code(double F, double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Python

def code(F, B, x):
	return -(x * (1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))

Julia

function code(F, B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
end

Wolfram

code[F_, B_, x_] := N[((-N[(x * N[(1 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2), $MachinePrecision] + N[(2 * x), $MachinePrecision]), $MachinePrecision], (-N[(1 / 2), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\sin B} \cdot \cos B\\ \mathbf{if}\;F \leq -55000000000000002149535471559930195366881323454633330127876789695579674068438989725630022088130560:\\ \;\;\;\;\frac{-1}{\sin B} \cdot \left(\frac{1 + \cos B \cdot x}{F} \cdot F\right)\\ \mathbf{elif}\;F \leq 2:\\ \;\;\;\;t\_0 + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (* (/ (- x) (sin B)) (cos B))))
  (if (<=
       F
       -55000000000000002149535471559930195366881323454633330127876789695579674068438989725630022088130560)
    (* (/ -1 (sin B)) (* (/ (+ 1 (* (cos B) x)) F) F))
    (if (<= F 2)
      (+ t_0 (* (/ (pow (- (+ x x) (- -2 (* F F))) -1/2) (sin B)) F))
      (+ t_0 (/ 1 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = (-x / sin(B)) * cos(B);
	double tmp;
	if (F <= -5.5e+97) {
		tmp = (-1.0 / sin(B)) * (((1.0 + (cos(B) * x)) / F) * F);
	} else if (F <= 2.0) {
		tmp = t_0 + ((pow(((x + x) - (-2.0 - (F * F))), -0.5) / sin(B)) * F);
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-x / sin(b)) * cos(b)
    if (f <= (-5.5d+97)) then
        tmp = ((-1.0d0) / sin(b)) * (((1.0d0 + (cos(b) * x)) / f) * f)
    else if (f <= 2.0d0) then
        tmp = t_0 + (((((x + x) - ((-2.0d0) - (f * f))) ** (-0.5d0)) / sin(b)) * f)
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-x / Math.sin(B)) * Math.cos(B);
	double tmp;
	if (F <= -5.5e+97) {
		tmp = (-1.0 / Math.sin(B)) * (((1.0 + (Math.cos(B) * x)) / F) * F);
	} else if (F <= 2.0) {
		tmp = t_0 + ((Math.pow(((x + x) - (-2.0 - (F * F))), -0.5) / Math.sin(B)) * F);
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-x / math.sin(B)) * math.cos(B)
	tmp = 0
	if F <= -5.5e+97:
		tmp = (-1.0 / math.sin(B)) * (((1.0 + (math.cos(B) * x)) / F) * F)
	elif F <= 2.0:
		tmp = t_0 + ((math.pow(((x + x) - (-2.0 - (F * F))), -0.5) / math.sin(B)) * F)
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(-x) / sin(B)) * cos(B))
	tmp = 0.0
	if (F <= -5.5e+97)
		tmp = Float64(Float64(-1.0 / sin(B)) * Float64(Float64(Float64(1.0 + Float64(cos(B) * x)) / F) * F));
	elseif (F <= 2.0)
		tmp = Float64(t_0 + Float64(Float64((Float64(Float64(x + x) - Float64(-2.0 - Float64(F * F))) ^ -0.5) / sin(B)) * F));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-x / sin(B)) * cos(B);
	tmp = 0.0;
	if (F <= -5.5e+97)
		tmp = (-1.0 / sin(B)) * (((1.0 + (cos(B) * x)) / F) * F);
	elseif (F <= 2.0)
		tmp = t_0 + (((((x + x) - (-2.0 - (F * F))) ^ -0.5) / sin(B)) * F);
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -55000000000000002149535471559930195366881323454633330127876789695579674068438989725630022088130560], N[(N[(-1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1 + N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2], N[(t$95$0 + N[(N[(N[Power[N[(N[(x + x), $MachinePrecision] - N[(-2 - N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1/2], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.5000000000000002e97

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   8. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} \cdot \left(\frac{1 + \cos B \cdot x}{F} \cdot F\right)} \]

   if -5.5000000000000002e97 < F < 2

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. mult-flip N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot F} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot F} \]

   3. Applied rewrites 85.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} \]
   4. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \color{blue}{\frac{1}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      4. lift-tan.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\color{blue}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      5. lift-tan.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\color{blue}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      6. mult-flip-rev N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      7. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{\tan B}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{-x}}{\tan B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      9. lift-tan.f64 N/A

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      10. tan-quot N/A

          \[\leadsto \frac{-x}{\color{blue}{\frac{\sin B}{\cos B}}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{-x}{\frac{\color{blue}{\sin B}}{\cos B}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      12. lift-cos.f64 N/A

          \[\leadsto \frac{-x}{\frac{\sin B}{\color{blue}{\cos B}}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      13. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{-x}{\sin B} \cdot \cos B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-x}{\sin B} \cdot \cos B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      15. lower-/.f64 85.1%

          \[\leadsto \color{blue}{\frac{-x}{\sin B}} \cdot \cos B + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{-x}{\sin B} \cdot \cos B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

   if 2 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. mult-flip N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot F} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot F} \]

   3. Applied rewrites 85.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} \]
   4. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \color{blue}{\frac{1}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      4. lift-tan.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\color{blue}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      5. lift-tan.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\color{blue}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      6. mult-flip-rev N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      7. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{\tan B}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{-x}}{\tan B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      9. lift-tan.f64 N/A

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      10. tan-quot N/A

          \[\leadsto \frac{-x}{\color{blue}{\frac{\sin B}{\cos B}}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{-x}{\frac{\color{blue}{\sin B}}{\cos B}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      12. lift-cos.f64 N/A

          \[\leadsto \frac{-x}{\frac{\sin B}{\color{blue}{\cos B}}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      13. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{-x}{\sin B} \cdot \cos B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-x}{\sin B} \cdot \cos B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      15. lower-/.f64 85.1%

          \[\leadsto \color{blue}{\frac{-x}{\sin B}} \cdot \cos B + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{-x}{\sin B} \cdot \cos B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \color{blue}{\frac{1}{\sin B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 56.5%

         \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \frac{1}{\sin B} \]

   8. Applied rewrites 56.5%

      \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -152000000000000005562493165312671744:\\ \;\;\;\;\frac{-1}{\sin B} \cdot \left(\frac{1 + \cos B \cdot x}{F} \cdot F\right)\\ \mathbf{elif}\;F \leq 1750:\\ \;\;\;\;{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -152000000000000005562493165312671744)
  (* (/ -1 (sin B)) (* (/ (+ 1 (* (cos B) x)) F) F))
  (if (<= F 1750)
    (-
     (* (pow (- (+ x x) (- -2 (* F F))) -1/2) (/ F (sin B)))
     (/ x (tan B)))
    (+ (* (/ (- x) (sin B)) (cos B)) (/ 1 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.52e+35) {
		tmp = (-1.0 / sin(B)) * (((1.0 + (cos(B) * x)) / F) * F);
	} else if (F <= 1750.0) {
		tmp = (pow(((x + x) - (-2.0 - (F * F))), -0.5) * (F / sin(B))) - (x / tan(B));
	} else {
		tmp = ((-x / sin(B)) * cos(B)) + (1.0 / sin(B));
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.52d+35)) then
        tmp = ((-1.0d0) / sin(b)) * (((1.0d0 + (cos(b) * x)) / f) * f)
    else if (f <= 1750.0d0) then
        tmp = ((((x + x) - ((-2.0d0) - (f * f))) ** (-0.5d0)) * (f / sin(b))) - (x / tan(b))
    else
        tmp = ((-x / sin(b)) * cos(b)) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.52e+35) {
		tmp = (-1.0 / Math.sin(B)) * (((1.0 + (Math.cos(B) * x)) / F) * F);
	} else if (F <= 1750.0) {
		tmp = (Math.pow(((x + x) - (-2.0 - (F * F))), -0.5) * (F / Math.sin(B))) - (x / Math.tan(B));
	} else {
		tmp = ((-x / Math.sin(B)) * Math.cos(B)) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.52e+35:
		tmp = (-1.0 / math.sin(B)) * (((1.0 + (math.cos(B) * x)) / F) * F)
	elif F <= 1750.0:
		tmp = (math.pow(((x + x) - (-2.0 - (F * F))), -0.5) * (F / math.sin(B))) - (x / math.tan(B))
	else:
		tmp = ((-x / math.sin(B)) * math.cos(B)) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.52e+35)
		tmp = Float64(Float64(-1.0 / sin(B)) * Float64(Float64(Float64(1.0 + Float64(cos(B) * x)) / F) * F));
	elseif (F <= 1750.0)
		tmp = Float64(Float64((Float64(Float64(x + x) - Float64(-2.0 - Float64(F * F))) ^ -0.5) * Float64(F / sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(Float64(Float64(-x) / sin(B)) * cos(B)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.52e+35)
		tmp = (-1.0 / sin(B)) * (((1.0 + (cos(B) * x)) / F) * F);
	elseif (F <= 1750.0)
		tmp = ((((x + x) - (-2.0 - (F * F))) ^ -0.5) * (F / sin(B))) - (x / tan(B));
	else
		tmp = ((-x / sin(B)) * cos(B)) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -152000000000000005562493165312671744], N[(N[(-1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1 + N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1750], N[(N[(N[Power[N[(N[(x + x), $MachinePrecision] - N[(-2 - N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1/2], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision] + N[(1 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5200000000000001e35

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   8. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} \cdot \left(\frac{1 + \cos B \cdot x}{F} \cdot F\right)} \]

   if -1.5200000000000001e35 < F < 1750

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lower--.f64 76.9%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

   3. Applied rewrites 77.0%

      \[\leadsto \color{blue}{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}} \]

   if 1750 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. mult-flip N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot F} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot F} \]

   3. Applied rewrites 85.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} \]
   4. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \color{blue}{\frac{1}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      4. lift-tan.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\color{blue}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      5. lift-tan.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\color{blue}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      6. mult-flip-rev N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      7. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{\tan B}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{-x}}{\tan B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      9. lift-tan.f64 N/A

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      10. tan-quot N/A

          \[\leadsto \frac{-x}{\color{blue}{\frac{\sin B}{\cos B}}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{-x}{\frac{\color{blue}{\sin B}}{\cos B}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      12. lift-cos.f64 N/A

          \[\leadsto \frac{-x}{\frac{\sin B}{\color{blue}{\cos B}}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      13. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{-x}{\sin B} \cdot \cos B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-x}{\sin B} \cdot \cos B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      15. lower-/.f64 85.1%

          \[\leadsto \color{blue}{\frac{-x}{\sin B}} \cdot \cos B + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{-x}{\sin B} \cdot \cos B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \color{blue}{\frac{1}{\sin B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 56.5%

         \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \frac{1}{\sin B} \]

   8. Applied rewrites 56.5%

      \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -100000000000000004764729344:\\ \;\;\;\;\frac{1 + x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 2:\\ \;\;\;\;\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -100000000000000004764729344)
  (/ (+ 1 (* x (cos B))) (- (sin B)))
  (if (<= F 2)
    (/
     (- (* (pow (- (+ x x) (- -2 (* F F))) -1/2) F) (* (cos B) x))
     (sin B))
    (+ (* (/ (- x) (sin B)) (cos B)) (/ 1 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e+26) {
		tmp = (1.0 + (x * cos(B))) / -sin(B);
	} else if (F <= 2.0) {
		tmp = ((pow(((x + x) - (-2.0 - (F * F))), -0.5) * F) - (cos(B) * x)) / sin(B);
	} else {
		tmp = ((-x / sin(B)) * cos(B)) + (1.0 / sin(B));
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1d+26)) then
        tmp = (1.0d0 + (x * cos(b))) / -sin(b)
    else if (f <= 2.0d0) then
        tmp = (((((x + x) - ((-2.0d0) - (f * f))) ** (-0.5d0)) * f) - (cos(b) * x)) / sin(b)
    else
        tmp = ((-x / sin(b)) * cos(b)) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e+26) {
		tmp = (1.0 + (x * Math.cos(B))) / -Math.sin(B);
	} else if (F <= 2.0) {
		tmp = ((Math.pow(((x + x) - (-2.0 - (F * F))), -0.5) * F) - (Math.cos(B) * x)) / Math.sin(B);
	} else {
		tmp = ((-x / Math.sin(B)) * Math.cos(B)) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1e+26:
		tmp = (1.0 + (x * math.cos(B))) / -math.sin(B)
	elif F <= 2.0:
		tmp = ((math.pow(((x + x) - (-2.0 - (F * F))), -0.5) * F) - (math.cos(B) * x)) / math.sin(B)
	else:
		tmp = ((-x / math.sin(B)) * math.cos(B)) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e+26)
		tmp = Float64(Float64(1.0 + Float64(x * cos(B))) / Float64(-sin(B)));
	elseif (F <= 2.0)
		tmp = Float64(Float64(Float64((Float64(Float64(x + x) - Float64(-2.0 - Float64(F * F))) ^ -0.5) * F) - Float64(cos(B) * x)) / sin(B));
	else
		tmp = Float64(Float64(Float64(Float64(-x) / sin(B)) * cos(B)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1e+26)
		tmp = (1.0 + (x * cos(B))) / -sin(B);
	elseif (F <= 2.0)
		tmp = (((((x + x) - (-2.0 - (F * F))) ^ -0.5) * F) - (cos(B) * x)) / sin(B);
	else
		tmp = ((-x / sin(B)) * cos(B)) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -100000000000000004764729344], N[(N[(1 + N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 2], N[(N[(N[(N[Power[N[(N[(x + x), $MachinePrecision] - N[(-2 - N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1/2], $MachinePrecision] * F), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision] + N[(1 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e26

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{1 + \color{blue}{x \cdot \cos B}}{-\sin B} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1 + x \cdot \color{blue}{\cos B}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1 + x \cdot \cos B}{-\sin B} \]

      3. lower-cos.f64 55.8%

         \[\leadsto \frac{1 + x \cdot \cos B}{-\sin B} \]

   9. Applied rewrites 55.8%

      \[\leadsto \frac{1 + \color{blue}{x \cdot \cos B}}{-\sin B} \]

   if -1e26 < F < 2

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   if 2 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. mult-flip N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot F} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot F} \]

   3. Applied rewrites 85.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} \]
   4. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \color{blue}{\frac{1}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      4. lift-tan.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\color{blue}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      5. lift-tan.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\color{blue}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      6. mult-flip-rev N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      7. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{\tan B}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{-x}}{\tan B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      9. lift-tan.f64 N/A

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      10. tan-quot N/A

          \[\leadsto \frac{-x}{\color{blue}{\frac{\sin B}{\cos B}}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{-x}{\frac{\color{blue}{\sin B}}{\cos B}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      12. lift-cos.f64 N/A

          \[\leadsto \frac{-x}{\frac{\sin B}{\color{blue}{\cos B}}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      13. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{-x}{\sin B} \cdot \cos B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-x}{\sin B} \cdot \cos B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      15. lower-/.f64 85.1%

          \[\leadsto \color{blue}{\frac{-x}{\sin B}} \cdot \cos B + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{-x}{\sin B} \cdot \cos B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \color{blue}{\frac{1}{\sin B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 56.5%

         \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \frac{1}{\sin B} \]

   8. Applied rewrites 56.5%

      \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq \frac{-2050338190066411}{19342813113834066795298816}:\\ \;\;\;\;\frac{1 + x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 2:\\ \;\;\;\;\frac{{\left(\left(x + x\right) - -2\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -2050338190066411/19342813113834066795298816)
  (/ (+ 1 (* x (cos B))) (- (sin B)))
  (if (<= F 2)
    (/ (- (* (pow (- (+ x x) -2) -1/2) F) (* (cos B) x)) (sin B))
    (+ (* (/ (- x) (sin B)) (cos B)) (/ 1 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.06e-10) {
		tmp = (1.0 + (x * cos(B))) / -sin(B);
	} else if (F <= 2.0) {
		tmp = ((pow(((x + x) - -2.0), -0.5) * F) - (cos(B) * x)) / sin(B);
	} else {
		tmp = ((-x / sin(B)) * cos(B)) + (1.0 / sin(B));
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.06d-10)) then
        tmp = (1.0d0 + (x * cos(b))) / -sin(b)
    else if (f <= 2.0d0) then
        tmp = (((((x + x) - (-2.0d0)) ** (-0.5d0)) * f) - (cos(b) * x)) / sin(b)
    else
        tmp = ((-x / sin(b)) * cos(b)) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.06e-10) {
		tmp = (1.0 + (x * Math.cos(B))) / -Math.sin(B);
	} else if (F <= 2.0) {
		tmp = ((Math.pow(((x + x) - -2.0), -0.5) * F) - (Math.cos(B) * x)) / Math.sin(B);
	} else {
		tmp = ((-x / Math.sin(B)) * Math.cos(B)) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.06e-10:
		tmp = (1.0 + (x * math.cos(B))) / -math.sin(B)
	elif F <= 2.0:
		tmp = ((math.pow(((x + x) - -2.0), -0.5) * F) - (math.cos(B) * x)) / math.sin(B)
	else:
		tmp = ((-x / math.sin(B)) * math.cos(B)) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.06e-10)
		tmp = Float64(Float64(1.0 + Float64(x * cos(B))) / Float64(-sin(B)));
	elseif (F <= 2.0)
		tmp = Float64(Float64(Float64((Float64(Float64(x + x) - -2.0) ^ -0.5) * F) - Float64(cos(B) * x)) / sin(B));
	else
		tmp = Float64(Float64(Float64(Float64(-x) / sin(B)) * cos(B)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.06e-10)
		tmp = (1.0 + (x * cos(B))) / -sin(B);
	elseif (F <= 2.0)
		tmp = (((((x + x) - -2.0) ^ -0.5) * F) - (cos(B) * x)) / sin(B);
	else
		tmp = ((-x / sin(B)) * cos(B)) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2050338190066411/19342813113834066795298816], N[(N[(1 + N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 2], N[(N[(N[(N[Power[N[(N[(x + x), $MachinePrecision] - -2), $MachinePrecision], -1/2], $MachinePrecision] * F), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision] + N[(1 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.06e-10

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{1 + \color{blue}{x \cdot \cos B}}{-\sin B} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1 + x \cdot \color{blue}{\cos B}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1 + x \cdot \cos B}{-\sin B} \]

      3. lower-cos.f64 55.8%

         \[\leadsto \frac{1 + x \cdot \cos B}{-\sin B} \]

   9. Applied rewrites 55.8%

      \[\leadsto \frac{1 + \color{blue}{x \cdot \cos B}}{-\sin B} \]

   if -1.06e-10 < F < 2

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{{\left(\left(x + x\right) - \color{blue}{-2}\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.6%

      \[\leadsto \frac{{\left(\left(x + x\right) - \color{blue}{-2}\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B} \]

   if 2 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. mult-flip N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot F} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot F} \]

   3. Applied rewrites 85.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} \]
   4. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \color{blue}{\frac{1}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      4. lift-tan.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\color{blue}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      5. lift-tan.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\color{blue}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      6. mult-flip-rev N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\tan B}}\right)\right) + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      7. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{\tan B}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{-x}}{\tan B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      9. lift-tan.f64 N/A

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      10. tan-quot N/A

          \[\leadsto \frac{-x}{\color{blue}{\frac{\sin B}{\cos B}}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{-x}{\frac{\color{blue}{\sin B}}{\cos B}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      12. lift-cos.f64 N/A

          \[\leadsto \frac{-x}{\frac{\sin B}{\color{blue}{\cos B}}} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      13. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{-x}{\sin B} \cdot \cos B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-x}{\sin B} \cdot \cos B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

      15. lower-/.f64 85.1%

          \[\leadsto \color{blue}{\frac{-x}{\sin B}} \cdot \cos B + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{-x}{\sin B} \cdot \cos B} + \frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \color{blue}{\frac{1}{\sin B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 56.5%

         \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \frac{1}{\sin B} \]

   8. Applied rewrites 56.5%

      \[\leadsto \frac{-x}{\sin B} \cdot \cos B + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq \frac{-2050338190066411}{19342813113834066795298816}:\\ \;\;\;\;\frac{1 + x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 2:\\ \;\;\;\;\frac{{\left(\left(x + x\right) - -2\right)}^{\frac{-1}{2}} \cdot F - t\_0}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (* (cos B) x)))
  (if (<= F -2050338190066411/19342813113834066795298816)
    (/ (+ 1 (* x (cos B))) (- (sin B)))
    (if (<= F 2)
      (/ (- (* (pow (- (+ x x) -2) -1/2) F) t_0) (sin B))
      (/ (- 1 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -1.06e-10) {
		tmp = (1.0 + (x * cos(B))) / -sin(B);
	} else if (F <= 2.0) {
		tmp = ((pow(((x + x) - -2.0), -0.5) * F) - t_0) / sin(B);
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(b) * x
    if (f <= (-1.06d-10)) then
        tmp = (1.0d0 + (x * cos(b))) / -sin(b)
    else if (f <= 2.0d0) then
        tmp = (((((x + x) - (-2.0d0)) ** (-0.5d0)) * f) - t_0) / sin(b)
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.cos(B) * x;
	double tmp;
	if (F <= -1.06e-10) {
		tmp = (1.0 + (x * Math.cos(B))) / -Math.sin(B);
	} else if (F <= 2.0) {
		tmp = ((Math.pow(((x + x) - -2.0), -0.5) * F) - t_0) / Math.sin(B);
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.cos(B) * x
	tmp = 0
	if F <= -1.06e-10:
		tmp = (1.0 + (x * math.cos(B))) / -math.sin(B)
	elif F <= 2.0:
		tmp = ((math.pow(((x + x) - -2.0), -0.5) * F) - t_0) / math.sin(B)
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -1.06e-10)
		tmp = Float64(Float64(1.0 + Float64(x * cos(B))) / Float64(-sin(B)));
	elseif (F <= 2.0)
		tmp = Float64(Float64(Float64((Float64(Float64(x + x) - -2.0) ^ -0.5) * F) - t_0) / sin(B));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = cos(B) * x;
	tmp = 0.0;
	if (F <= -1.06e-10)
		tmp = (1.0 + (x * cos(B))) / -sin(B);
	elseif (F <= 2.0)
		tmp = (((((x + x) - -2.0) ^ -0.5) * F) - t_0) / sin(B);
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -2050338190066411/19342813113834066795298816], N[(N[(1 + N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 2], N[(N[(N[(N[Power[N[(N[(x + x), $MachinePrecision] - -2), $MachinePrecision], -1/2], $MachinePrecision] * F), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.06e-10

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{1 + \color{blue}{x \cdot \cos B}}{-\sin B} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1 + x \cdot \color{blue}{\cos B}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1 + x \cdot \cos B}{-\sin B} \]

      3. lower-cos.f64 55.8%

         \[\leadsto \frac{1 + x \cdot \cos B}{-\sin B} \]

   9. Applied rewrites 55.8%

      \[\leadsto \frac{1 + \color{blue}{x \cdot \cos B}}{-\sin B} \]

   if -1.06e-10 < F < 2

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{{\left(\left(x + x\right) - \color{blue}{-2}\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.6%

      \[\leadsto \frac{{\left(\left(x + x\right) - \color{blue}{-2}\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B} \]

   if 2 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -14000000000000000718667586864996145195776999603682877405921280:\\ \;\;\;\;\frac{-1}{\sin B} \cdot \left(\frac{1 + t\_0}{F} \cdot F\right)\\ \mathbf{elif}\;F \leq \frac{-7742953005213299}{266998379490113760299377713271194014325338065294581596243380200977777465722580068752870260867072}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B}}{\frac{1}{F}}\\ \mathbf{elif}\;F \leq \frac{713053462628379}{158456325028528675187087900672}:\\ \;\;\;\;\frac{1}{\frac{\tan B}{-x}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (* (cos B) x)))
  (if (<=
       F
       -14000000000000000718667586864996145195776999603682877405921280)
    (* (/ -1 (sin B)) (* (/ (+ 1 t_0) F) F))
    (if (<=
         F
         -7742953005213299/266998379490113760299377713271194014325338065294581596243380200977777465722580068752870260867072)
      (+
       (- (/ x B))
       (/ (/ (pow (- (+ x x) (- -2 (* F F))) -1/2) (sin B)) (/ 1 F)))
      (if (<= F 713053462628379/158456325028528675187087900672)
        (+
         (/ 1 (/ (tan B) (- x)))
         (* (/ F B) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2)))))
        (/ (- 1 t_0) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -1.4e+61) {
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -2.9e-80) {
		tmp = -(x / B) + ((pow(((x + x) - (-2.0 - (F * F))), -0.5) / sin(B)) / (1.0 / F));
	} else if (F <= 4.5e-15) {
		tmp = (1.0 / (tan(B) / -x)) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(b) * x
    if (f <= (-1.4d+61)) then
        tmp = ((-1.0d0) / sin(b)) * (((1.0d0 + t_0) / f) * f)
    else if (f <= (-2.9d-80)) then
        tmp = -(x / b) + (((((x + x) - ((-2.0d0) - (f * f))) ** (-0.5d0)) / sin(b)) / (1.0d0 / f))
    else if (f <= 4.5d-15) then
        tmp = (1.0d0 / (tan(b) / -x)) + ((f / b) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.cos(B) * x;
	double tmp;
	if (F <= -1.4e+61) {
		tmp = (-1.0 / Math.sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -2.9e-80) {
		tmp = -(x / B) + ((Math.pow(((x + x) - (-2.0 - (F * F))), -0.5) / Math.sin(B)) / (1.0 / F));
	} else if (F <= 4.5e-15) {
		tmp = (1.0 / (Math.tan(B) / -x)) + ((F / B) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.cos(B) * x
	tmp = 0
	if F <= -1.4e+61:
		tmp = (-1.0 / math.sin(B)) * (((1.0 + t_0) / F) * F)
	elif F <= -2.9e-80:
		tmp = -(x / B) + ((math.pow(((x + x) - (-2.0 - (F * F))), -0.5) / math.sin(B)) / (1.0 / F))
	elif F <= 4.5e-15:
		tmp = (1.0 / (math.tan(B) / -x)) + ((F / B) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -1.4e+61)
		tmp = Float64(Float64(-1.0 / sin(B)) * Float64(Float64(Float64(1.0 + t_0) / F) * F));
	elseif (F <= -2.9e-80)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64((Float64(Float64(x + x) - Float64(-2.0 - Float64(F * F))) ^ -0.5) / sin(B)) / Float64(1.0 / F)));
	elseif (F <= 4.5e-15)
		tmp = Float64(Float64(1.0 / Float64(tan(B) / Float64(-x))) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = cos(B) * x;
	tmp = 0.0;
	if (F <= -1.4e+61)
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	elseif (F <= -2.9e-80)
		tmp = -(x / B) + (((((x + x) - (-2.0 - (F * F))) ^ -0.5) / sin(B)) / (1.0 / F));
	elseif (F <= 4.5e-15)
		tmp = (1.0 / (tan(B) / -x)) + ((F / B) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -14000000000000000718667586864996145195776999603682877405921280], N[(N[(-1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1 + t$95$0), $MachinePrecision] / F), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7742953005213299/266998379490113760299377713271194014325338065294581596243380200977777465722580068752870260867072], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[Power[N[(N[(x + x), $MachinePrecision] - N[(-2 - N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1/2], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[(1 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 713053462628379/158456325028528675187087900672], N[(N[(1 / N[(N[Tan[B], $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2), $MachinePrecision] + N[(2 * x), $MachinePrecision]), $MachinePrecision], (-N[(1 / 2), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.4000000000000001e61

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   8. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} \cdot \left(\frac{1 + \cos B \cdot x}{F} \cdot F\right)} \]

   if -1.4000000000000001e61 < F < -2.9e-80

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{F}{\sin B}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{F}{\sin B}} \]

      4. div-flip-rev N/A

         \[\leadsto \left(-\frac{x}{B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\frac{\sin B}{F}}} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\frac{\sin B}{F}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\color{blue}{\frac{\sin B}{F}}} \]

      8. mult-flip N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\color{blue}{\sin B \cdot \frac{1}{F}}} \]

      9. associate-/r* N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}{\frac{1}{F}}} \]

   6. Applied rewrites 57.3%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B}}{\frac{1}{F}}} \]

   if -2.9e-80 < F < 4.4999999999999998e-15

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 62.4%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 62.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{\tan B}}\right)\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \color{blue}{\frac{1}{\tan B}}\right)\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. lift-tan.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\color{blue}{\tan B}}\right)\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lift-tan.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\color{blue}{\tan B}}\right)\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. mult-flip-rev N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\tan B}}\right)\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{\tan B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{-x}}{\tan B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      9. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\tan B}{-x}}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      10. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\tan B}{-x}}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      11. lower-unsound-/.f64 62.4%

          \[\leadsto \frac{1}{\color{blue}{\frac{\tan B}{-x}}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 62.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\tan B}{-x}}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 4.4999999999999998e-15 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -14000000000000000718667586864996145195776999603682877405921280:\\ \;\;\;\;\frac{-1}{\sin B} \cdot \left(\frac{1 + t\_0}{F} \cdot F\right)\\ \mathbf{elif}\;F \leq \frac{-7742953005213299}{266998379490113760299377713271194014325338065294581596243380200977777465722580068752870260867072}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B}}{\frac{1}{F}}\\ \mathbf{elif}\;F \leq \frac{713053462628379}{158456325028528675187087900672}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (* (cos B) x)))
  (if (<=
       F
       -14000000000000000718667586864996145195776999603682877405921280)
    (* (/ -1 (sin B)) (* (/ (+ 1 t_0) F) F))
    (if (<=
         F
         -7742953005213299/266998379490113760299377713271194014325338065294581596243380200977777465722580068752870260867072)
      (+
       (- (/ x B))
       (/ (/ (pow (- (+ x x) (- -2 (* F F))) -1/2) (sin B)) (/ 1 F)))
      (if (<= F 713053462628379/158456325028528675187087900672)
        (+
         (- (* x (/ 1 (tan B))))
         (* (/ F B) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2)))))
        (/ (- 1 t_0) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -1.4e+61) {
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -2.9e-80) {
		tmp = -(x / B) + ((pow(((x + x) - (-2.0 - (F * F))), -0.5) / sin(B)) / (1.0 / F));
	} else if (F <= 4.5e-15) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(b) * x
    if (f <= (-1.4d+61)) then
        tmp = ((-1.0d0) / sin(b)) * (((1.0d0 + t_0) / f) * f)
    else if (f <= (-2.9d-80)) then
        tmp = -(x / b) + (((((x + x) - ((-2.0d0) - (f * f))) ** (-0.5d0)) / sin(b)) / (1.0d0 / f))
    else if (f <= 4.5d-15) then
        tmp = -(x * (1.0d0 / tan(b))) + ((f / b) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.cos(B) * x;
	double tmp;
	if (F <= -1.4e+61) {
		tmp = (-1.0 / Math.sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -2.9e-80) {
		tmp = -(x / B) + ((Math.pow(((x + x) - (-2.0 - (F * F))), -0.5) / Math.sin(B)) / (1.0 / F));
	} else if (F <= 4.5e-15) {
		tmp = -(x * (1.0 / Math.tan(B))) + ((F / B) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.cos(B) * x
	tmp = 0
	if F <= -1.4e+61:
		tmp = (-1.0 / math.sin(B)) * (((1.0 + t_0) / F) * F)
	elif F <= -2.9e-80:
		tmp = -(x / B) + ((math.pow(((x + x) - (-2.0 - (F * F))), -0.5) / math.sin(B)) / (1.0 / F))
	elif F <= 4.5e-15:
		tmp = -(x * (1.0 / math.tan(B))) + ((F / B) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -1.4e+61)
		tmp = Float64(Float64(-1.0 / sin(B)) * Float64(Float64(Float64(1.0 + t_0) / F) * F));
	elseif (F <= -2.9e-80)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64((Float64(Float64(x + x) - Float64(-2.0 - Float64(F * F))) ^ -0.5) / sin(B)) / Float64(1.0 / F)));
	elseif (F <= 4.5e-15)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = cos(B) * x;
	tmp = 0.0;
	if (F <= -1.4e+61)
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	elseif (F <= -2.9e-80)
		tmp = -(x / B) + (((((x + x) - (-2.0 - (F * F))) ^ -0.5) / sin(B)) / (1.0 / F));
	elseif (F <= 4.5e-15)
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -14000000000000000718667586864996145195776999603682877405921280], N[(N[(-1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1 + t$95$0), $MachinePrecision] / F), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7742953005213299/266998379490113760299377713271194014325338065294581596243380200977777465722580068752870260867072], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[Power[N[(N[(x + x), $MachinePrecision] - N[(-2 - N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1/2], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[(1 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 713053462628379/158456325028528675187087900672], N[((-N[(x * N[(1 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2), $MachinePrecision] + N[(2 * x), $MachinePrecision]), $MachinePrecision], (-N[(1 / 2), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.4000000000000001e61

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   8. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} \cdot \left(\frac{1 + \cos B \cdot x}{F} \cdot F\right)} \]

   if -1.4000000000000001e61 < F < -2.9e-80

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{F}{\sin B}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{F}{\sin B}} \]

      4. div-flip-rev N/A

         \[\leadsto \left(-\frac{x}{B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\frac{\sin B}{F}}} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\frac{\sin B}{F}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\color{blue}{\frac{\sin B}{F}}} \]

      8. mult-flip N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\color{blue}{\sin B \cdot \frac{1}{F}}} \]

      9. associate-/r* N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}{\frac{1}{F}}} \]

   6. Applied rewrites 57.3%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B}}{\frac{1}{F}}} \]

   if -2.9e-80 < F < 4.4999999999999998e-15

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 62.4%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 62.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 4.4999999999999998e-15 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 88.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos B \cdot x\\ t_1 := -\frac{x}{B}\\ t_2 := {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}\\ t_3 := t\_2 \cdot F\\ \mathbf{if}\;F \leq -14000000000000000718667586864996145195776999603682877405921280:\\ \;\;\;\;\frac{-1}{\sin B} \cdot \left(\frac{1 + t\_0}{F} \cdot F\right)\\ \mathbf{elif}\;F \leq \frac{-1196547670217613}{1087770609288739018116276647019455748771006705104961378712461595034426490595025393129804804639189577049885346787832834079429794483512744426310696916513970896780966442670885312576979206144}:\\ \;\;\;\;t\_1 + \frac{\frac{t\_2}{\sin B}}{\frac{1}{F}}\\ \mathbf{elif}\;F \leq \frac{7256757823367339}{93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968}:\\ \;\;\;\;\frac{\frac{-x}{\tan B} \cdot \left(-1 \cdot B\right) - t\_3}{-1 \cdot B}\\ \mathbf{elif}\;F \leq \frac{713053462628379}{158456325028528675187087900672}:\\ \;\;\;\;t\_1 + \frac{1}{\sin B} \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (* (cos B) x))
       (t_1 (- (/ x B)))
       (t_2 (pow (- (+ x x) (- -2 (* F F))) -1/2))
       (t_3 (* t_2 F)))
  (if (<=
       F
       -14000000000000000718667586864996145195776999603682877405921280)
    (* (/ -1 (sin B)) (* (/ (+ 1 t_0) F) F))
    (if (<=
         F
         -1196547670217613/1087770609288739018116276647019455748771006705104961378712461595034426490595025393129804804639189577049885346787832834079429794483512744426310696916513970896780966442670885312576979206144)
      (+ t_1 (/ (/ t_2 (sin B)) (/ 1 F)))
      (if (<=
           F
           7256757823367339/93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968)
        (/ (- (* (/ (- x) (tan B)) (* -1 B)) t_3) (* -1 B))
        (if (<= F 713053462628379/158456325028528675187087900672)
          (+ t_1 (* (/ 1 (sin B)) t_3))
          (/ (- 1 t_0) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double t_1 = -(x / B);
	double t_2 = pow(((x + x) - (-2.0 - (F * F))), -0.5);
	double t_3 = t_2 * F;
	double tmp;
	if (F <= -1.4e+61) {
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -1.1e-171) {
		tmp = t_1 + ((t_2 / sin(B)) / (1.0 / F));
	} else if (F <= 7.8e-122) {
		tmp = (((-x / tan(B)) * (-1.0 * B)) - t_3) / (-1.0 * B);
	} else if (F <= 4.5e-15) {
		tmp = t_1 + ((1.0 / sin(B)) * t_3);
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = cos(b) * x
    t_1 = -(x / b)
    t_2 = ((x + x) - ((-2.0d0) - (f * f))) ** (-0.5d0)
    t_3 = t_2 * f
    if (f <= (-1.4d+61)) then
        tmp = ((-1.0d0) / sin(b)) * (((1.0d0 + t_0) / f) * f)
    else if (f <= (-1.1d-171)) then
        tmp = t_1 + ((t_2 / sin(b)) / (1.0d0 / f))
    else if (f <= 7.8d-122) then
        tmp = (((-x / tan(b)) * ((-1.0d0) * b)) - t_3) / ((-1.0d0) * b)
    else if (f <= 4.5d-15) then
        tmp = t_1 + ((1.0d0 / sin(b)) * t_3)
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.cos(B) * x;
	double t_1 = -(x / B);
	double t_2 = Math.pow(((x + x) - (-2.0 - (F * F))), -0.5);
	double t_3 = t_2 * F;
	double tmp;
	if (F <= -1.4e+61) {
		tmp = (-1.0 / Math.sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -1.1e-171) {
		tmp = t_1 + ((t_2 / Math.sin(B)) / (1.0 / F));
	} else if (F <= 7.8e-122) {
		tmp = (((-x / Math.tan(B)) * (-1.0 * B)) - t_3) / (-1.0 * B);
	} else if (F <= 4.5e-15) {
		tmp = t_1 + ((1.0 / Math.sin(B)) * t_3);
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.cos(B) * x
	t_1 = -(x / B)
	t_2 = math.pow(((x + x) - (-2.0 - (F * F))), -0.5)
	t_3 = t_2 * F
	tmp = 0
	if F <= -1.4e+61:
		tmp = (-1.0 / math.sin(B)) * (((1.0 + t_0) / F) * F)
	elif F <= -1.1e-171:
		tmp = t_1 + ((t_2 / math.sin(B)) / (1.0 / F))
	elif F <= 7.8e-122:
		tmp = (((-x / math.tan(B)) * (-1.0 * B)) - t_3) / (-1.0 * B)
	elif F <= 4.5e-15:
		tmp = t_1 + ((1.0 / math.sin(B)) * t_3)
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	t_1 = Float64(-Float64(x / B))
	t_2 = Float64(Float64(x + x) - Float64(-2.0 - Float64(F * F))) ^ -0.5
	t_3 = Float64(t_2 * F)
	tmp = 0.0
	if (F <= -1.4e+61)
		tmp = Float64(Float64(-1.0 / sin(B)) * Float64(Float64(Float64(1.0 + t_0) / F) * F));
	elseif (F <= -1.1e-171)
		tmp = Float64(t_1 + Float64(Float64(t_2 / sin(B)) / Float64(1.0 / F)));
	elseif (F <= 7.8e-122)
		tmp = Float64(Float64(Float64(Float64(Float64(-x) / tan(B)) * Float64(-1.0 * B)) - t_3) / Float64(-1.0 * B));
	elseif (F <= 4.5e-15)
		tmp = Float64(t_1 + Float64(Float64(1.0 / sin(B)) * t_3));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = cos(B) * x;
	t_1 = -(x / B);
	t_2 = ((x + x) - (-2.0 - (F * F))) ^ -0.5;
	t_3 = t_2 * F;
	tmp = 0.0;
	if (F <= -1.4e+61)
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	elseif (F <= -1.1e-171)
		tmp = t_1 + ((t_2 / sin(B)) / (1.0 / F));
	elseif (F <= 7.8e-122)
		tmp = (((-x / tan(B)) * (-1.0 * B)) - t_3) / (-1.0 * B);
	elseif (F <= 4.5e-15)
		tmp = t_1 + ((1.0 / sin(B)) * t_3);
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = (-N[(x / B), $MachinePrecision])}, Block[{t$95$2 = N[Power[N[(N[(x + x), $MachinePrecision] - N[(-2 - N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1/2], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * F), $MachinePrecision]}, If[LessEqual[F, -14000000000000000718667586864996145195776999603682877405921280], N[(N[(-1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1 + t$95$0), $MachinePrecision] / F), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1196547670217613/1087770609288739018116276647019455748771006705104961378712461595034426490595025393129804804639189577049885346787832834079429794483512744426310696916513970896780966442670885312576979206144], N[(t$95$1 + N[(N[(t$95$2 / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[(1 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7256757823367339/93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968], N[(N[(N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] * N[(-1 * B), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(-1 * B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 713053462628379/158456325028528675187087900672], N[(t$95$1 + N[(N[(1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(1 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.4000000000000001e61

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   8. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} \cdot \left(\frac{1 + \cos B \cdot x}{F} \cdot F\right)} \]

   if -1.4000000000000001e61 < F < -1.1000000000000001e-171

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{F}{\sin B}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{F}{\sin B}} \]

      4. div-flip-rev N/A

         \[\leadsto \left(-\frac{x}{B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\frac{\sin B}{F}}} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\frac{\sin B}{F}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\color{blue}{\frac{\sin B}{F}}} \]

      8. mult-flip N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\color{blue}{\sin B \cdot \frac{1}{F}}} \]

      9. associate-/r* N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}{\frac{1}{F}}} \]

   6. Applied rewrites 57.3%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B}}{\frac{1}{F}}} \]

   if -1.1000000000000001e-171 < F < 7.7999999999999998e-122

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in B around 0

      \[\leadsto \frac{\frac{-x}{\tan B} \cdot \color{blue}{\left(-1 \cdot B\right)} - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 48.9%

         \[\leadsto \frac{\frac{-x}{\tan B} \cdot \left(-1 \cdot \color{blue}{B}\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B} \]

   6. Applied rewrites 48.9%

      \[\leadsto \frac{\frac{-x}{\tan B} \cdot \color{blue}{\left(-1 \cdot B\right)} - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{\frac{-x}{\tan B} \cdot \left(-1 \cdot B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{\color{blue}{-1 \cdot B}} \]
   8. Step-by-step derivation

      1. lower-*.f64 60.9%

         \[\leadsto \frac{\frac{-x}{\tan B} \cdot \left(-1 \cdot B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-1 \cdot \color{blue}{B}} \]

   9. Applied rewrites 60.9%

      \[\leadsto \frac{\frac{-x}{\tan B} \cdot \left(-1 \cdot B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{\color{blue}{-1 \cdot B}} \]

   if 7.7999999999999998e-122 < F < 4.4999999999999998e-15

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 57.3%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B} \cdot \left({\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F\right)} \]

   if 4.4999999999999998e-15 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 88.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos B \cdot x\\ t_1 := -\frac{x}{B}\\ t_2 := {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}\\ t_3 := t\_2 \cdot F\\ \mathbf{if}\;F \leq -130000000000000003301073569777188864:\\ \;\;\;\;\frac{-1}{\sin B} \cdot \left(\frac{1 + t\_0}{F} \cdot F\right)\\ \mathbf{elif}\;F \leq \frac{-1196547670217613}{1087770609288739018116276647019455748771006705104961378712461595034426490595025393129804804639189577049885346787832834079429794483512744426310696916513970896780966442670885312576979206144}:\\ \;\;\;\;t\_1 + \frac{t\_2}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq \frac{7256757823367339}{93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968}:\\ \;\;\;\;\frac{\frac{-x}{\tan B} \cdot \left(-1 \cdot B\right) - t\_3}{-1 \cdot B}\\ \mathbf{elif}\;F \leq \frac{713053462628379}{158456325028528675187087900672}:\\ \;\;\;\;t\_1 + \frac{1}{\sin B} \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (* (cos B) x))
       (t_1 (- (/ x B)))
       (t_2 (pow (- (+ x x) (- -2 (* F F))) -1/2))
       (t_3 (* t_2 F)))
  (if (<= F -130000000000000003301073569777188864)
    (* (/ -1 (sin B)) (* (/ (+ 1 t_0) F) F))
    (if (<=
         F
         -1196547670217613/1087770609288739018116276647019455748771006705104961378712461595034426490595025393129804804639189577049885346787832834079429794483512744426310696916513970896780966442670885312576979206144)
      (+ t_1 (/ t_2 (/ (sin B) F)))
      (if (<=
           F
           7256757823367339/93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968)
        (/ (- (* (/ (- x) (tan B)) (* -1 B)) t_3) (* -1 B))
        (if (<= F 713053462628379/158456325028528675187087900672)
          (+ t_1 (* (/ 1 (sin B)) t_3))
          (/ (- 1 t_0) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double t_1 = -(x / B);
	double t_2 = pow(((x + x) - (-2.0 - (F * F))), -0.5);
	double t_3 = t_2 * F;
	double tmp;
	if (F <= -1.3e+35) {
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -1.1e-171) {
		tmp = t_1 + (t_2 / (sin(B) / F));
	} else if (F <= 7.8e-122) {
		tmp = (((-x / tan(B)) * (-1.0 * B)) - t_3) / (-1.0 * B);
	} else if (F <= 4.5e-15) {
		tmp = t_1 + ((1.0 / sin(B)) * t_3);
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = cos(b) * x
    t_1 = -(x / b)
    t_2 = ((x + x) - ((-2.0d0) - (f * f))) ** (-0.5d0)
    t_3 = t_2 * f
    if (f <= (-1.3d+35)) then
        tmp = ((-1.0d0) / sin(b)) * (((1.0d0 + t_0) / f) * f)
    else if (f <= (-1.1d-171)) then
        tmp = t_1 + (t_2 / (sin(b) / f))
    else if (f <= 7.8d-122) then
        tmp = (((-x / tan(b)) * ((-1.0d0) * b)) - t_3) / ((-1.0d0) * b)
    else if (f <= 4.5d-15) then
        tmp = t_1 + ((1.0d0 / sin(b)) * t_3)
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.cos(B) * x;
	double t_1 = -(x / B);
	double t_2 = Math.pow(((x + x) - (-2.0 - (F * F))), -0.5);
	double t_3 = t_2 * F;
	double tmp;
	if (F <= -1.3e+35) {
		tmp = (-1.0 / Math.sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -1.1e-171) {
		tmp = t_1 + (t_2 / (Math.sin(B) / F));
	} else if (F <= 7.8e-122) {
		tmp = (((-x / Math.tan(B)) * (-1.0 * B)) - t_3) / (-1.0 * B);
	} else if (F <= 4.5e-15) {
		tmp = t_1 + ((1.0 / Math.sin(B)) * t_3);
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.cos(B) * x
	t_1 = -(x / B)
	t_2 = math.pow(((x + x) - (-2.0 - (F * F))), -0.5)
	t_3 = t_2 * F
	tmp = 0
	if F <= -1.3e+35:
		tmp = (-1.0 / math.sin(B)) * (((1.0 + t_0) / F) * F)
	elif F <= -1.1e-171:
		tmp = t_1 + (t_2 / (math.sin(B) / F))
	elif F <= 7.8e-122:
		tmp = (((-x / math.tan(B)) * (-1.0 * B)) - t_3) / (-1.0 * B)
	elif F <= 4.5e-15:
		tmp = t_1 + ((1.0 / math.sin(B)) * t_3)
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	t_1 = Float64(-Float64(x / B))
	t_2 = Float64(Float64(x + x) - Float64(-2.0 - Float64(F * F))) ^ -0.5
	t_3 = Float64(t_2 * F)
	tmp = 0.0
	if (F <= -1.3e+35)
		tmp = Float64(Float64(-1.0 / sin(B)) * Float64(Float64(Float64(1.0 + t_0) / F) * F));
	elseif (F <= -1.1e-171)
		tmp = Float64(t_1 + Float64(t_2 / Float64(sin(B) / F)));
	elseif (F <= 7.8e-122)
		tmp = Float64(Float64(Float64(Float64(Float64(-x) / tan(B)) * Float64(-1.0 * B)) - t_3) / Float64(-1.0 * B));
	elseif (F <= 4.5e-15)
		tmp = Float64(t_1 + Float64(Float64(1.0 / sin(B)) * t_3));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = cos(B) * x;
	t_1 = -(x / B);
	t_2 = ((x + x) - (-2.0 - (F * F))) ^ -0.5;
	t_3 = t_2 * F;
	tmp = 0.0;
	if (F <= -1.3e+35)
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	elseif (F <= -1.1e-171)
		tmp = t_1 + (t_2 / (sin(B) / F));
	elseif (F <= 7.8e-122)
		tmp = (((-x / tan(B)) * (-1.0 * B)) - t_3) / (-1.0 * B);
	elseif (F <= 4.5e-15)
		tmp = t_1 + ((1.0 / sin(B)) * t_3);
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = (-N[(x / B), $MachinePrecision])}, Block[{t$95$2 = N[Power[N[(N[(x + x), $MachinePrecision] - N[(-2 - N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1/2], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * F), $MachinePrecision]}, If[LessEqual[F, -130000000000000003301073569777188864], N[(N[(-1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1 + t$95$0), $MachinePrecision] / F), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1196547670217613/1087770609288739018116276647019455748771006705104961378712461595034426490595025393129804804639189577049885346787832834079429794483512744426310696916513970896780966442670885312576979206144], N[(t$95$1 + N[(t$95$2 / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7256757823367339/93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968], N[(N[(N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] * N[(-1 * B), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(-1 * B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 713053462628379/158456325028528675187087900672], N[(t$95$1 + N[(N[(1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(1 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.3e35

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   8. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} \cdot \left(\frac{1 + \cos B \cdot x}{F} \cdot F\right)} \]

   if -1.3e35 < F < -1.1000000000000001e-171

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{F}{\sin B}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{F}{\sin B}} \]

      4. div-flip-rev N/A

         \[\leadsto \left(-\frac{x}{B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\frac{\sin B}{F}}} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\frac{\sin B}{F}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}}} \]

      7. lower-/.f64 50.1%

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}}} \]

   6. Applied rewrites 50.1%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}} \]

   if -1.1000000000000001e-171 < F < 7.7999999999999998e-122

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in B around 0

      \[\leadsto \frac{\frac{-x}{\tan B} \cdot \color{blue}{\left(-1 \cdot B\right)} - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 48.9%

         \[\leadsto \frac{\frac{-x}{\tan B} \cdot \left(-1 \cdot \color{blue}{B}\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B} \]

   6. Applied rewrites 48.9%

      \[\leadsto \frac{\frac{-x}{\tan B} \cdot \color{blue}{\left(-1 \cdot B\right)} - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{\frac{-x}{\tan B} \cdot \left(-1 \cdot B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{\color{blue}{-1 \cdot B}} \]
   8. Step-by-step derivation

      1. lower-*.f64 60.9%

         \[\leadsto \frac{\frac{-x}{\tan B} \cdot \left(-1 \cdot B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-1 \cdot \color{blue}{B}} \]

   9. Applied rewrites 60.9%

      \[\leadsto \frac{\frac{-x}{\tan B} \cdot \left(-1 \cdot B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{\color{blue}{-1 \cdot B}} \]

   if 7.7999999999999998e-122 < F < 4.4999999999999998e-15

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 57.3%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B} \cdot \left({\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F\right)} \]

   if 4.4999999999999998e-15 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 10: 87.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos B \cdot x\\ t_1 := -\frac{x}{B}\\ t_2 := {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}\\ \mathbf{if}\;F \leq -130000000000000003301073569777188864:\\ \;\;\;\;\frac{-1}{\sin B} \cdot \left(\frac{1 + t\_0}{F} \cdot F\right)\\ \mathbf{elif}\;F \leq \frac{-3548786815231963}{4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512}:\\ \;\;\;\;t\_1 + \frac{t\_2}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq \frac{7256757823367339}{93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq \frac{713053462628379}{158456325028528675187087900672}:\\ \;\;\;\;t\_1 + \frac{1}{\sin B} \cdot \left(t\_2 \cdot F\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (* (cos B) x))
       (t_1 (- (/ x B)))
       (t_2 (pow (- (+ x x) (- -2 (* F F))) -1/2)))
  (if (<= F -130000000000000003301073569777188864)
    (* (/ -1 (sin B)) (* (/ (+ 1 t_0) F) F))
    (if (<=
         F
         -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512)
      (+ t_1 (/ t_2 (/ (sin B) F)))
      (if (<=
           F
           7256757823367339/93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968)
        (/ (- x) (tan B))
        (if (<= F 713053462628379/158456325028528675187087900672)
          (+ t_1 (* (/ 1 (sin B)) (* t_2 F)))
          (/ (- 1 t_0) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double t_1 = -(x / B);
	double t_2 = pow(((x + x) - (-2.0 - (F * F))), -0.5);
	double tmp;
	if (F <= -1.3e+35) {
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -8.2e-259) {
		tmp = t_1 + (t_2 / (sin(B) / F));
	} else if (F <= 7.8e-122) {
		tmp = -x / tan(B);
	} else if (F <= 4.5e-15) {
		tmp = t_1 + ((1.0 / sin(B)) * (t_2 * F));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(b) * x
    t_1 = -(x / b)
    t_2 = ((x + x) - ((-2.0d0) - (f * f))) ** (-0.5d0)
    if (f <= (-1.3d+35)) then
        tmp = ((-1.0d0) / sin(b)) * (((1.0d0 + t_0) / f) * f)
    else if (f <= (-8.2d-259)) then
        tmp = t_1 + (t_2 / (sin(b) / f))
    else if (f <= 7.8d-122) then
        tmp = -x / tan(b)
    else if (f <= 4.5d-15) then
        tmp = t_1 + ((1.0d0 / sin(b)) * (t_2 * f))
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.cos(B) * x;
	double t_1 = -(x / B);
	double t_2 = Math.pow(((x + x) - (-2.0 - (F * F))), -0.5);
	double tmp;
	if (F <= -1.3e+35) {
		tmp = (-1.0 / Math.sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -8.2e-259) {
		tmp = t_1 + (t_2 / (Math.sin(B) / F));
	} else if (F <= 7.8e-122) {
		tmp = -x / Math.tan(B);
	} else if (F <= 4.5e-15) {
		tmp = t_1 + ((1.0 / Math.sin(B)) * (t_2 * F));
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.cos(B) * x
	t_1 = -(x / B)
	t_2 = math.pow(((x + x) - (-2.0 - (F * F))), -0.5)
	tmp = 0
	if F <= -1.3e+35:
		tmp = (-1.0 / math.sin(B)) * (((1.0 + t_0) / F) * F)
	elif F <= -8.2e-259:
		tmp = t_1 + (t_2 / (math.sin(B) / F))
	elif F <= 7.8e-122:
		tmp = -x / math.tan(B)
	elif F <= 4.5e-15:
		tmp = t_1 + ((1.0 / math.sin(B)) * (t_2 * F))
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	t_1 = Float64(-Float64(x / B))
	t_2 = Float64(Float64(x + x) - Float64(-2.0 - Float64(F * F))) ^ -0.5
	tmp = 0.0
	if (F <= -1.3e+35)
		tmp = Float64(Float64(-1.0 / sin(B)) * Float64(Float64(Float64(1.0 + t_0) / F) * F));
	elseif (F <= -8.2e-259)
		tmp = Float64(t_1 + Float64(t_2 / Float64(sin(B) / F)));
	elseif (F <= 7.8e-122)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 4.5e-15)
		tmp = Float64(t_1 + Float64(Float64(1.0 / sin(B)) * Float64(t_2 * F)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = cos(B) * x;
	t_1 = -(x / B);
	t_2 = ((x + x) - (-2.0 - (F * F))) ^ -0.5;
	tmp = 0.0;
	if (F <= -1.3e+35)
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	elseif (F <= -8.2e-259)
		tmp = t_1 + (t_2 / (sin(B) / F));
	elseif (F <= 7.8e-122)
		tmp = -x / tan(B);
	elseif (F <= 4.5e-15)
		tmp = t_1 + ((1.0 / sin(B)) * (t_2 * F));
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = (-N[(x / B), $MachinePrecision])}, Block[{t$95$2 = N[Power[N[(N[(x + x), $MachinePrecision] - N[(-2 - N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1/2], $MachinePrecision]}, If[LessEqual[F, -130000000000000003301073569777188864], N[(N[(-1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1 + t$95$0), $MachinePrecision] / F), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512], N[(t$95$1 + N[(t$95$2 / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7256757823367339/93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 713053462628379/158456325028528675187087900672], N[(t$95$1 + N[(N[(1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.3e35

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   8. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} \cdot \left(\frac{1 + \cos B \cdot x}{F} \cdot F\right)} \]

   if -1.3e35 < F < -8.1999999999999996e-259

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{F}{\sin B}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{F}{\sin B}} \]

      4. div-flip-rev N/A

         \[\leadsto \left(-\frac{x}{B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\frac{\sin B}{F}}} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\frac{\sin B}{F}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}}} \]

      7. lower-/.f64 50.1%

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}}} \]

   6. Applied rewrites 50.1%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}}} \]

   if -8.1999999999999996e-259 < F < 7.7999999999999998e-122

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.8%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{\cos B}{\sin B}\right)\right) \]

      7. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      10. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(\frac{1}{\tan B} \cdot x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(1 \cdot \frac{1}{\tan B}\right) \cdot x\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\tan B} \cdot x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

   6. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 7.7999999999999998e-122 < F < 4.4999999999999998e-15

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 57.3%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B} \cdot \left({\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F\right)} \]

   if 4.4999999999999998e-15 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 11: 87.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos B \cdot x\\ t_1 := {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}\\ \mathbf{if}\;F \leq -86000000000000002621014052368016515123921215154871214834341745017896304640:\\ \;\;\;\;\frac{-1}{\sin B} \cdot \left(\frac{1 + t\_0}{F} \cdot F\right)\\ \mathbf{elif}\;F \leq \frac{-3548786815231963}{4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512}:\\ \;\;\;\;\frac{t\_1}{\sin B} \cdot F - \frac{x}{B}\\ \mathbf{elif}\;F \leq \frac{7256757823367339}{93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq \frac{713053462628379}{158456325028528675187087900672}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{1}{\sin B} \cdot \left(t\_1 \cdot F\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (* (cos B) x))
       (t_1 (pow (- (+ x x) (- -2 (* F F))) -1/2)))
  (if (<=
       F
       -86000000000000002621014052368016515123921215154871214834341745017896304640)
    (* (/ -1 (sin B)) (* (/ (+ 1 t_0) F) F))
    (if (<=
         F
         -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512)
      (- (* (/ t_1 (sin B)) F) (/ x B))
      (if (<=
           F
           7256757823367339/93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968)
        (/ (- x) (tan B))
        (if (<= F 713053462628379/158456325028528675187087900672)
          (+ (- (/ x B)) (* (/ 1 (sin B)) (* t_1 F)))
          (/ (- 1 t_0) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double t_1 = pow(((x + x) - (-2.0 - (F * F))), -0.5);
	double tmp;
	if (F <= -8.6e+73) {
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -8.2e-259) {
		tmp = ((t_1 / sin(B)) * F) - (x / B);
	} else if (F <= 7.8e-122) {
		tmp = -x / tan(B);
	} else if (F <= 4.5e-15) {
		tmp = -(x / B) + ((1.0 / sin(B)) * (t_1 * F));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(b) * x
    t_1 = ((x + x) - ((-2.0d0) - (f * f))) ** (-0.5d0)
    if (f <= (-8.6d+73)) then
        tmp = ((-1.0d0) / sin(b)) * (((1.0d0 + t_0) / f) * f)
    else if (f <= (-8.2d-259)) then
        tmp = ((t_1 / sin(b)) * f) - (x / b)
    else if (f <= 7.8d-122) then
        tmp = -x / tan(b)
    else if (f <= 4.5d-15) then
        tmp = -(x / b) + ((1.0d0 / sin(b)) * (t_1 * f))
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.cos(B) * x;
	double t_1 = Math.pow(((x + x) - (-2.0 - (F * F))), -0.5);
	double tmp;
	if (F <= -8.6e+73) {
		tmp = (-1.0 / Math.sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -8.2e-259) {
		tmp = ((t_1 / Math.sin(B)) * F) - (x / B);
	} else if (F <= 7.8e-122) {
		tmp = -x / Math.tan(B);
	} else if (F <= 4.5e-15) {
		tmp = -(x / B) + ((1.0 / Math.sin(B)) * (t_1 * F));
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.cos(B) * x
	t_1 = math.pow(((x + x) - (-2.0 - (F * F))), -0.5)
	tmp = 0
	if F <= -8.6e+73:
		tmp = (-1.0 / math.sin(B)) * (((1.0 + t_0) / F) * F)
	elif F <= -8.2e-259:
		tmp = ((t_1 / math.sin(B)) * F) - (x / B)
	elif F <= 7.8e-122:
		tmp = -x / math.tan(B)
	elif F <= 4.5e-15:
		tmp = -(x / B) + ((1.0 / math.sin(B)) * (t_1 * F))
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	t_1 = Float64(Float64(x + x) - Float64(-2.0 - Float64(F * F))) ^ -0.5
	tmp = 0.0
	if (F <= -8.6e+73)
		tmp = Float64(Float64(-1.0 / sin(B)) * Float64(Float64(Float64(1.0 + t_0) / F) * F));
	elseif (F <= -8.2e-259)
		tmp = Float64(Float64(Float64(t_1 / sin(B)) * F) - Float64(x / B));
	elseif (F <= 7.8e-122)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 4.5e-15)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(1.0 / sin(B)) * Float64(t_1 * F)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = cos(B) * x;
	t_1 = ((x + x) - (-2.0 - (F * F))) ^ -0.5;
	tmp = 0.0;
	if (F <= -8.6e+73)
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	elseif (F <= -8.2e-259)
		tmp = ((t_1 / sin(B)) * F) - (x / B);
	elseif (F <= 7.8e-122)
		tmp = -x / tan(B);
	elseif (F <= 4.5e-15)
		tmp = -(x / B) + ((1.0 / sin(B)) * (t_1 * F));
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(x + x), $MachinePrecision] - N[(-2 - N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1/2], $MachinePrecision]}, If[LessEqual[F, -86000000000000002621014052368016515123921215154871214834341745017896304640], N[(N[(-1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1 + t$95$0), $MachinePrecision] / F), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512], N[(N[(N[(t$95$1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7256757823367339/93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 713053462628379/158456325028528675187087900672], N[((-N[(x / B), $MachinePrecision]) + N[(N[(1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8.6000000000000003e73

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   8. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} \cdot \left(\frac{1 + \cos B \cdot x}{F} \cdot F\right)} \]

   if -8.6000000000000003e73 < F < -8.1999999999999996e-259

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-\frac{x}{B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

      5. lower--.f64 49.2%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

   6. Applied rewrites 49.2%

      \[\leadsto \color{blue}{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \color{blue}{\frac{F}{\sin B}} - \frac{x}{B} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{\sin B}} - \frac{x}{B} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \frac{x}{B} \]

      5. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B}} \cdot F - \frac{x}{B} \]

      6. lift-*.f64 57.3%

         \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \frac{x}{B} \]

   8. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F - \frac{x}{B}} \]

   if -8.1999999999999996e-259 < F < 7.7999999999999998e-122

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.8%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{\cos B}{\sin B}\right)\right) \]

      7. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      10. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(\frac{1}{\tan B} \cdot x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(1 \cdot \frac{1}{\tan B}\right) \cdot x\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\tan B} \cdot x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

   6. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 7.7999999999999998e-122 < F < 4.4999999999999998e-15

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 57.3%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B} \cdot \left({\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F\right)} \]

   if 4.4999999999999998e-15 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 12: 86.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos B \cdot x\\ t_1 := {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}\\ \mathbf{if}\;F \leq -86000000000000002621014052368016515123921215154871214834341745017896304640:\\ \;\;\;\;\frac{-1}{\sin B} \cdot \left(\frac{1 + t\_0}{F} \cdot F\right)\\ \mathbf{elif}\;F \leq \frac{-3548786815231963}{4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512}:\\ \;\;\;\;\frac{t\_1}{\sin B} \cdot F - \frac{x}{B}\\ \mathbf{elif}\;F \leq \frac{5470679174164527}{14396524142538228424993723224595141948383030778566133225922417832357880258148761185020930195532450742879746914027266864394266451377581759004827248578768524336431104}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq \frac{713053462628379}{158456325028528675187087900672}:\\ \;\;\;\;t\_1 \cdot \left(\frac{1}{\sin B} \cdot F\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (* (cos B) x))
       (t_1 (pow (- (+ x x) (- -2 (* F F))) -1/2)))
  (if (<=
       F
       -86000000000000002621014052368016515123921215154871214834341745017896304640)
    (* (/ -1 (sin B)) (* (/ (+ 1 t_0) F) F))
    (if (<=
         F
         -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512)
      (- (* (/ t_1 (sin B)) F) (/ x B))
      (if (<=
           F
           5470679174164527/14396524142538228424993723224595141948383030778566133225922417832357880258148761185020930195532450742879746914027266864394266451377581759004827248578768524336431104)
        (/ (- x) (tan B))
        (if (<= F 713053462628379/158456325028528675187087900672)
          (- (* t_1 (* (/ 1 (sin B)) F)) (/ x B))
          (/ (- 1 t_0) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double t_1 = pow(((x + x) - (-2.0 - (F * F))), -0.5);
	double tmp;
	if (F <= -8.6e+73) {
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -8.2e-259) {
		tmp = ((t_1 / sin(B)) * F) - (x / B);
	} else if (F <= 3.8e-148) {
		tmp = -x / tan(B);
	} else if (F <= 4.5e-15) {
		tmp = (t_1 * ((1.0 / sin(B)) * F)) - (x / B);
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(b) * x
    t_1 = ((x + x) - ((-2.0d0) - (f * f))) ** (-0.5d0)
    if (f <= (-8.6d+73)) then
        tmp = ((-1.0d0) / sin(b)) * (((1.0d0 + t_0) / f) * f)
    else if (f <= (-8.2d-259)) then
        tmp = ((t_1 / sin(b)) * f) - (x / b)
    else if (f <= 3.8d-148) then
        tmp = -x / tan(b)
    else if (f <= 4.5d-15) then
        tmp = (t_1 * ((1.0d0 / sin(b)) * f)) - (x / b)
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.cos(B) * x;
	double t_1 = Math.pow(((x + x) - (-2.0 - (F * F))), -0.5);
	double tmp;
	if (F <= -8.6e+73) {
		tmp = (-1.0 / Math.sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -8.2e-259) {
		tmp = ((t_1 / Math.sin(B)) * F) - (x / B);
	} else if (F <= 3.8e-148) {
		tmp = -x / Math.tan(B);
	} else if (F <= 4.5e-15) {
		tmp = (t_1 * ((1.0 / Math.sin(B)) * F)) - (x / B);
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.cos(B) * x
	t_1 = math.pow(((x + x) - (-2.0 - (F * F))), -0.5)
	tmp = 0
	if F <= -8.6e+73:
		tmp = (-1.0 / math.sin(B)) * (((1.0 + t_0) / F) * F)
	elif F <= -8.2e-259:
		tmp = ((t_1 / math.sin(B)) * F) - (x / B)
	elif F <= 3.8e-148:
		tmp = -x / math.tan(B)
	elif F <= 4.5e-15:
		tmp = (t_1 * ((1.0 / math.sin(B)) * F)) - (x / B)
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	t_1 = Float64(Float64(x + x) - Float64(-2.0 - Float64(F * F))) ^ -0.5
	tmp = 0.0
	if (F <= -8.6e+73)
		tmp = Float64(Float64(-1.0 / sin(B)) * Float64(Float64(Float64(1.0 + t_0) / F) * F));
	elseif (F <= -8.2e-259)
		tmp = Float64(Float64(Float64(t_1 / sin(B)) * F) - Float64(x / B));
	elseif (F <= 3.8e-148)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 4.5e-15)
		tmp = Float64(Float64(t_1 * Float64(Float64(1.0 / sin(B)) * F)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = cos(B) * x;
	t_1 = ((x + x) - (-2.0 - (F * F))) ^ -0.5;
	tmp = 0.0;
	if (F <= -8.6e+73)
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	elseif (F <= -8.2e-259)
		tmp = ((t_1 / sin(B)) * F) - (x / B);
	elseif (F <= 3.8e-148)
		tmp = -x / tan(B);
	elseif (F <= 4.5e-15)
		tmp = (t_1 * ((1.0 / sin(B)) * F)) - (x / B);
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(x + x), $MachinePrecision] - N[(-2 - N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1/2], $MachinePrecision]}, If[LessEqual[F, -86000000000000002621014052368016515123921215154871214834341745017896304640], N[(N[(-1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1 + t$95$0), $MachinePrecision] / F), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512], N[(N[(N[(t$95$1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5470679174164527/14396524142538228424993723224595141948383030778566133225922417832357880258148761185020930195532450742879746914027266864394266451377581759004827248578768524336431104], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 713053462628379/158456325028528675187087900672], N[(N[(t$95$1 * N[(N[(1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8.6000000000000003e73

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   8. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} \cdot \left(\frac{1 + \cos B \cdot x}{F} \cdot F\right)} \]

   if -8.6000000000000003e73 < F < -8.1999999999999996e-259

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-\frac{x}{B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

      5. lower--.f64 49.2%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

   6. Applied rewrites 49.2%

      \[\leadsto \color{blue}{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \color{blue}{\frac{F}{\sin B}} - \frac{x}{B} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{\sin B}} - \frac{x}{B} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \frac{x}{B} \]

      5. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B}} \cdot F - \frac{x}{B} \]

      6. lift-*.f64 57.3%

         \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \frac{x}{B} \]

   8. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F - \frac{x}{B}} \]

   if -8.1999999999999996e-259 < F < 3.8000000000000001e-148

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.8%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{\cos B}{\sin B}\right)\right) \]

      7. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      10. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(\frac{1}{\tan B} \cdot x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(1 \cdot \frac{1}{\tan B}\right) \cdot x\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\tan B} \cdot x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

   6. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 3.8000000000000001e-148 < F < 4.4999999999999998e-15

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-\frac{x}{B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

      5. lower--.f64 49.2%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

   6. Applied rewrites 49.2%

      \[\leadsto \color{blue}{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \color{blue}{\frac{F}{\sin B}} - \frac{x}{B} \]

      2. mult-flip N/A

         \[\leadsto {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} - \frac{x}{B} \]

      3. *-commutative N/A

         \[\leadsto {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} - \frac{x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} - \frac{x}{B} \]

      5. lower-/.f64 49.2%

         \[\leadsto {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \left(\color{blue}{\frac{1}{\sin B}} \cdot F\right) - \frac{x}{B} \]

   8. Applied rewrites 49.2%

      \[\leadsto {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} - \frac{x}{B} \]

   if 4.4999999999999998e-15 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 13: 86.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -86000000000000002621014052368016515123921215154871214834341745017896304640:\\ \;\;\;\;\frac{-1}{\sin B} \cdot \left(\frac{1 + t\_0}{F} \cdot F\right)\\ \mathbf{elif}\;F \leq \frac{-3548786815231963}{4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512}:\\ \;\;\;\;\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F - \frac{x}{B}\\ \mathbf{elif}\;F \leq \frac{7256757823367339}{93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq \frac{713053462628379}{158456325028528675187087900672}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (* (cos B) x)))
  (if (<=
       F
       -86000000000000002621014052368016515123921215154871214834341745017896304640)
    (* (/ -1 (sin B)) (* (/ (+ 1 t_0) F) F))
    (if (<=
         F
         -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512)
      (-
       (* (/ (pow (- (+ x x) (- -2 (* F F))) -1/2) (sin B)) F)
       (/ x B))
      (if (<=
           F
           7256757823367339/93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968)
        (/ (- x) (tan B))
        (if (<= F 713053462628379/158456325028528675187087900672)
          (+ (- (/ x B)) (/ (* F (pow (+ 2 (* 2 x)) -1/2)) (sin B)))
          (/ (- 1 t_0) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -8.6e+73) {
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -8.2e-259) {
		tmp = ((pow(((x + x) - (-2.0 - (F * F))), -0.5) / sin(B)) * F) - (x / B);
	} else if (F <= 7.8e-122) {
		tmp = -x / tan(B);
	} else if (F <= 4.5e-15) {
		tmp = -(x / B) + ((F * pow((2.0 + (2.0 * x)), -0.5)) / sin(B));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(b) * x
    if (f <= (-8.6d+73)) then
        tmp = ((-1.0d0) / sin(b)) * (((1.0d0 + t_0) / f) * f)
    else if (f <= (-8.2d-259)) then
        tmp = (((((x + x) - ((-2.0d0) - (f * f))) ** (-0.5d0)) / sin(b)) * f) - (x / b)
    else if (f <= 7.8d-122) then
        tmp = -x / tan(b)
    else if (f <= 4.5d-15) then
        tmp = -(x / b) + ((f * ((2.0d0 + (2.0d0 * x)) ** (-0.5d0))) / sin(b))
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.cos(B) * x;
	double tmp;
	if (F <= -8.6e+73) {
		tmp = (-1.0 / Math.sin(B)) * (((1.0 + t_0) / F) * F);
	} else if (F <= -8.2e-259) {
		tmp = ((Math.pow(((x + x) - (-2.0 - (F * F))), -0.5) / Math.sin(B)) * F) - (x / B);
	} else if (F <= 7.8e-122) {
		tmp = -x / Math.tan(B);
	} else if (F <= 4.5e-15) {
		tmp = -(x / B) + ((F * Math.pow((2.0 + (2.0 * x)), -0.5)) / Math.sin(B));
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.cos(B) * x
	tmp = 0
	if F <= -8.6e+73:
		tmp = (-1.0 / math.sin(B)) * (((1.0 + t_0) / F) * F)
	elif F <= -8.2e-259:
		tmp = ((math.pow(((x + x) - (-2.0 - (F * F))), -0.5) / math.sin(B)) * F) - (x / B)
	elif F <= 7.8e-122:
		tmp = -x / math.tan(B)
	elif F <= 4.5e-15:
		tmp = -(x / B) + ((F * math.pow((2.0 + (2.0 * x)), -0.5)) / math.sin(B))
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -8.6e+73)
		tmp = Float64(Float64(-1.0 / sin(B)) * Float64(Float64(Float64(1.0 + t_0) / F) * F));
	elseif (F <= -8.2e-259)
		tmp = Float64(Float64(Float64((Float64(Float64(x + x) - Float64(-2.0 - Float64(F * F))) ^ -0.5) / sin(B)) * F) - Float64(x / B));
	elseif (F <= 7.8e-122)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 4.5e-15)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)) / sin(B)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = cos(B) * x;
	tmp = 0.0;
	if (F <= -8.6e+73)
		tmp = (-1.0 / sin(B)) * (((1.0 + t_0) / F) * F);
	elseif (F <= -8.2e-259)
		tmp = (((((x + x) - (-2.0 - (F * F))) ^ -0.5) / sin(B)) * F) - (x / B);
	elseif (F <= 7.8e-122)
		tmp = -x / tan(B);
	elseif (F <= 4.5e-15)
		tmp = -(x / B) + ((F * ((2.0 + (2.0 * x)) ^ -0.5)) / sin(B));
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -86000000000000002621014052368016515123921215154871214834341745017896304640], N[(N[(-1 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1 + t$95$0), $MachinePrecision] / F), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512], N[(N[(N[(N[Power[N[(N[(x + x), $MachinePrecision] - N[(-2 - N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1/2], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7256757823367339/93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 713053462628379/158456325028528675187087900672], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F * N[Power[N[(2 + N[(2 * x), $MachinePrecision]), $MachinePrecision], -1/2], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8.6000000000000003e73

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   8. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} \cdot \left(\frac{1 + \cos B \cdot x}{F} \cdot F\right)} \]

   if -8.6000000000000003e73 < F < -8.1999999999999996e-259

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-\frac{x}{B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

      5. lower--.f64 49.2%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

   6. Applied rewrites 49.2%

      \[\leadsto \color{blue}{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \color{blue}{\frac{F}{\sin B}} - \frac{x}{B} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{\sin B}} - \frac{x}{B} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \frac{x}{B} \]

      5. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B}} \cdot F - \frac{x}{B} \]

      6. lift-*.f64 57.3%

         \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F} - \frac{x}{B} \]

   8. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}}}{\sin B} \cdot F - \frac{x}{B}} \]

   if -8.1999999999999996e-259 < F < 7.7999999999999998e-122

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.8%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{\cos B}{\sin B}\right)\right) \]

      7. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      10. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(\frac{1}{\tan B} \cdot x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(1 \cdot \frac{1}{\tan B}\right) \cdot x\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\tan B} \cdot x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

   6. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 7.7999999999999998e-122 < F < 4.4999999999999998e-15

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around 0

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      2. metadata-eval N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\sin B}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin \color{blue}{B}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      8. metadata-eval N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      9. metadata-eval N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B} \]

      10. lower-sin.f64 35.7%

          \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B} \]

   7. Applied rewrites 35.7%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} \]

   if 4.4999999999999998e-15 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 14: 86.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}\\ \mathbf{if}\;F \leq \frac{-6284457040522883}{1461501637330902918203684832716283019655932542976}:\\ \;\;\;\;\frac{1 + x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq \frac{-3548786815231963}{4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq \frac{7256757823367339}{93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq \frac{713053462628379}{158456325028528675187087900672}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0
        (+ (- (/ x B)) (/ (* F (pow (+ 2 (* 2 x)) -1/2)) (sin B)))))
  (if (<=
       F
       -6284457040522883/1461501637330902918203684832716283019655932542976)
    (/ (+ 1 (* x (cos B))) (- (sin B)))
    (if (<=
         F
         -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512)
      t_0
      (if (<=
           F
           7256757823367339/93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968)
        (/ (- x) (tan B))
        (if (<= F 713053462628379/158456325028528675187087900672)
          t_0
          (/ (- 1 (* (cos B) x)) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B) + ((F * pow((2.0 + (2.0 * x)), -0.5)) / sin(B));
	double tmp;
	if (F <= -4.3e-33) {
		tmp = (1.0 + (x * cos(B))) / -sin(B);
	} else if (F <= -8.2e-259) {
		tmp = t_0;
	} else if (F <= 7.8e-122) {
		tmp = -x / tan(B);
	} else if (F <= 4.5e-15) {
		tmp = t_0;
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x / b) + ((f * ((2.0d0 + (2.0d0 * x)) ** (-0.5d0))) / sin(b))
    if (f <= (-4.3d-33)) then
        tmp = (1.0d0 + (x * cos(b))) / -sin(b)
    else if (f <= (-8.2d-259)) then
        tmp = t_0
    else if (f <= 7.8d-122) then
        tmp = -x / tan(b)
    else if (f <= 4.5d-15) then
        tmp = t_0
    else
        tmp = (1.0d0 - (cos(b) * x)) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x / B) + ((F * Math.pow((2.0 + (2.0 * x)), -0.5)) / Math.sin(B));
	double tmp;
	if (F <= -4.3e-33) {
		tmp = (1.0 + (x * Math.cos(B))) / -Math.sin(B);
	} else if (F <= -8.2e-259) {
		tmp = t_0;
	} else if (F <= 7.8e-122) {
		tmp = -x / Math.tan(B);
	} else if (F <= 4.5e-15) {
		tmp = t_0;
	} else {
		tmp = (1.0 - (Math.cos(B) * x)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x / B) + ((F * math.pow((2.0 + (2.0 * x)), -0.5)) / math.sin(B))
	tmp = 0
	if F <= -4.3e-33:
		tmp = (1.0 + (x * math.cos(B))) / -math.sin(B)
	elif F <= -8.2e-259:
		tmp = t_0
	elif F <= 7.8e-122:
		tmp = -x / math.tan(B)
	elif F <= 4.5e-15:
		tmp = t_0
	else:
		tmp = (1.0 - (math.cos(B) * x)) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-Float64(x / B)) + Float64(Float64(F * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)) / sin(B)))
	tmp = 0.0
	if (F <= -4.3e-33)
		tmp = Float64(Float64(1.0 + Float64(x * cos(B))) / Float64(-sin(B)));
	elseif (F <= -8.2e-259)
		tmp = t_0;
	elseif (F <= 7.8e-122)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 4.5e-15)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x / B) + ((F * ((2.0 + (2.0 * x)) ^ -0.5)) / sin(B));
	tmp = 0.0;
	if (F <= -4.3e-33)
		tmp = (1.0 + (x * cos(B))) / -sin(B);
	elseif (F <= -8.2e-259)
		tmp = t_0;
	elseif (F <= 7.8e-122)
		tmp = -x / tan(B);
	elseif (F <= 4.5e-15)
		tmp = t_0;
	else
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x / B), $MachinePrecision]) + N[(N[(F * N[Power[N[(2 + N[(2 * x), $MachinePrecision]), $MachinePrecision], -1/2], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6284457040522883/1461501637330902918203684832716283019655932542976], N[(N[(1 + N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512], t$95$0, If[LessEqual[F, 7256757823367339/93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 713053462628379/158456325028528675187087900672], t$95$0, N[(N[(1 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.3000000000000003e-33

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{1 + \color{blue}{x \cdot \cos B}}{-\sin B} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1 + x \cdot \color{blue}{\cos B}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1 + x \cdot \cos B}{-\sin B} \]

      3. lower-cos.f64 55.8%

         \[\leadsto \frac{1 + x \cdot \cos B}{-\sin B} \]

   9. Applied rewrites 55.8%

      \[\leadsto \frac{1 + \color{blue}{x \cdot \cos B}}{-\sin B} \]

   if -4.3000000000000003e-33 < F < -8.1999999999999996e-259 or 7.7999999999999998e-122 < F < 4.4999999999999998e-15

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around 0

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      2. metadata-eval N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\sin B}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin \color{blue}{B}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      8. metadata-eval N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      9. metadata-eval N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B} \]

      10. lower-sin.f64 35.7%

          \[\leadsto \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B} \]

   7. Applied rewrites 35.7%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} \]

   if -8.1999999999999996e-259 < F < 7.7999999999999998e-122

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.8%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{\cos B}{\sin B}\right)\right) \]

      7. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      10. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(\frac{1}{\tan B} \cdot x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(1 \cdot \frac{1}{\tan B}\right) \cdot x\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\tan B} \cdot x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

   6. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 4.4999999999999998e-15 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 86.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left(x + x\right) - -2\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq \frac{-6284457040522883}{1461501637330902918203684832716283019655932542976}:\\ \;\;\;\;\frac{1 + x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq \frac{-3548786815231963}{4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq \frac{5470679174164527}{14396524142538228424993723224595141948383030778566133225922417832357880258148761185020930195532450742879746914027266864394266451377581759004827248578768524336431104}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq \frac{713053462628379}{158456325028528675187087900672}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (- (* (pow (- (+ x x) -2) -1/2) (/ F (sin B))) (/ x B))))
  (if (<=
       F
       -6284457040522883/1461501637330902918203684832716283019655932542976)
    (/ (+ 1 (* x (cos B))) (- (sin B)))
    (if (<=
         F
         -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512)
      t_0
      (if (<=
           F
           5470679174164527/14396524142538228424993723224595141948383030778566133225922417832357880258148761185020930195532450742879746914027266864394266451377581759004827248578768524336431104)
        (/ (- x) (tan B))
        (if (<= F 713053462628379/158456325028528675187087900672)
          t_0
          (/ (- 1 (* (cos B) x)) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (pow(((x + x) - -2.0), -0.5) * (F / sin(B))) - (x / B);
	double tmp;
	if (F <= -4.3e-33) {
		tmp = (1.0 + (x * cos(B))) / -sin(B);
	} else if (F <= -8.2e-259) {
		tmp = t_0;
	} else if (F <= 3.8e-148) {
		tmp = -x / tan(B);
	} else if (F <= 4.5e-15) {
		tmp = t_0;
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((x + x) - (-2.0d0)) ** (-0.5d0)) * (f / sin(b))) - (x / b)
    if (f <= (-4.3d-33)) then
        tmp = (1.0d0 + (x * cos(b))) / -sin(b)
    else if (f <= (-8.2d-259)) then
        tmp = t_0
    else if (f <= 3.8d-148) then
        tmp = -x / tan(b)
    else if (f <= 4.5d-15) then
        tmp = t_0
    else
        tmp = (1.0d0 - (cos(b) * x)) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (Math.pow(((x + x) - -2.0), -0.5) * (F / Math.sin(B))) - (x / B);
	double tmp;
	if (F <= -4.3e-33) {
		tmp = (1.0 + (x * Math.cos(B))) / -Math.sin(B);
	} else if (F <= -8.2e-259) {
		tmp = t_0;
	} else if (F <= 3.8e-148) {
		tmp = -x / Math.tan(B);
	} else if (F <= 4.5e-15) {
		tmp = t_0;
	} else {
		tmp = (1.0 - (Math.cos(B) * x)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (math.pow(((x + x) - -2.0), -0.5) * (F / math.sin(B))) - (x / B)
	tmp = 0
	if F <= -4.3e-33:
		tmp = (1.0 + (x * math.cos(B))) / -math.sin(B)
	elif F <= -8.2e-259:
		tmp = t_0
	elif F <= 3.8e-148:
		tmp = -x / math.tan(B)
	elif F <= 4.5e-15:
		tmp = t_0
	else:
		tmp = (1.0 - (math.cos(B) * x)) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64((Float64(Float64(x + x) - -2.0) ^ -0.5) * Float64(F / sin(B))) - Float64(x / B))
	tmp = 0.0
	if (F <= -4.3e-33)
		tmp = Float64(Float64(1.0 + Float64(x * cos(B))) / Float64(-sin(B)));
	elseif (F <= -8.2e-259)
		tmp = t_0;
	elseif (F <= 3.8e-148)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 4.5e-15)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((((x + x) - -2.0) ^ -0.5) * (F / sin(B))) - (x / B);
	tmp = 0.0;
	if (F <= -4.3e-33)
		tmp = (1.0 + (x * cos(B))) / -sin(B);
	elseif (F <= -8.2e-259)
		tmp = t_0;
	elseif (F <= 3.8e-148)
		tmp = -x / tan(B);
	elseif (F <= 4.5e-15)
		tmp = t_0;
	else
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[Power[N[(N[(x + x), $MachinePrecision] - -2), $MachinePrecision], -1/2], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6284457040522883/1461501637330902918203684832716283019655932542976], N[(N[(1 + N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512], t$95$0, If[LessEqual[F, 5470679174164527/14396524142538228424993723224595141948383030778566133225922417832357880258148761185020930195532450742879746914027266864394266451377581759004827248578768524336431104], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 713053462628379/158456325028528675187087900672], t$95$0, N[(N[(1 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.3000000000000003e-33

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{1 + \color{blue}{x \cdot \cos B}}{-\sin B} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1 + x \cdot \color{blue}{\cos B}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1 + x \cdot \cos B}{-\sin B} \]

      3. lower-cos.f64 55.8%

         \[\leadsto \frac{1 + x \cdot \cos B}{-\sin B} \]

   9. Applied rewrites 55.8%

      \[\leadsto \frac{1 + \color{blue}{x \cdot \cos B}}{-\sin B} \]

   if -4.3000000000000003e-33 < F < -8.1999999999999996e-259 or 3.8000000000000001e-148 < F < 4.4999999999999998e-15

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-\frac{x}{B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

      5. lower--.f64 49.2%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

   6. Applied rewrites 49.2%

      \[\leadsto \color{blue}{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto {\left(\left(x + x\right) - \color{blue}{-2}\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.1%

      \[\leadsto {\left(\left(x + x\right) - \color{blue}{-2}\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B} - \frac{x}{B} \]

   if -8.1999999999999996e-259 < F < 3.8000000000000001e-148

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.8%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{\cos B}{\sin B}\right)\right) \]

      7. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      10. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(\frac{1}{\tan B} \cdot x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(1 \cdot \frac{1}{\tan B}\right) \cdot x\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\tan B} \cdot x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

   6. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 4.4999999999999998e-15 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 16: 84.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq \frac{-1010998000018149}{3369993333393829974333376885877453834204643052817571560137951281152}:\\ \;\;\;\;\frac{1 + x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq \frac{5057235284857433}{3064991081731777716716694054300618367237478244367204352}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<=
     F
     -1010998000018149/3369993333393829974333376885877453834204643052817571560137951281152)
  (/ (+ 1 (* x (cos B))) (- (sin B)))
  (if (<=
       F
       5057235284857433/3064991081731777716716694054300618367237478244367204352)
    (/ (- x) (tan B))
    (/ (- 1 (* (cos B) x)) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-52) {
		tmp = (1.0 + (x * cos(B))) / -sin(B);
	} else if (F <= 1.65e-39) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3d-52)) then
        tmp = (1.0d0 + (x * cos(b))) / -sin(b)
    else if (f <= 1.65d-39) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 - (cos(b) * x)) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-52) {
		tmp = (1.0 + (x * Math.cos(B))) / -Math.sin(B);
	} else if (F <= 1.65e-39) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 - (Math.cos(B) * x)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3e-52:
		tmp = (1.0 + (x * math.cos(B))) / -math.sin(B)
	elif F <= 1.65e-39:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 - (math.cos(B) * x)) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e-52)
		tmp = Float64(Float64(1.0 + Float64(x * cos(B))) / Float64(-sin(B)));
	elseif (F <= 1.65e-39)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3e-52)
		tmp = (1.0 + (x * cos(B))) / -sin(B);
	elseif (F <= 1.65e-39)
		tmp = -x / tan(B);
	else
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1010998000018149/3369993333393829974333376885877453834204643052817571560137951281152], N[(N[(1 + N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 5057235284857433/3064991081731777716716694054300618367237478244367204352], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3e-52

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{1 + \color{blue}{x \cdot \cos B}}{-\sin B} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1 + x \cdot \color{blue}{\cos B}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1 + x \cdot \cos B}{-\sin B} \]

      3. lower-cos.f64 55.8%

         \[\leadsto \frac{1 + x \cdot \cos B}{-\sin B} \]

   9. Applied rewrites 55.8%

      \[\leadsto \frac{1 + \color{blue}{x \cdot \cos B}}{-\sin B} \]

   if -3e-52 < F < 1.6499999999999999e-39

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.8%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{\cos B}{\sin B}\right)\right) \]

      7. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      10. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(\frac{1}{\tan B} \cdot x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(1 \cdot \frac{1}{\tan B}\right) \cdot x\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\tan B} \cdot x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

   6. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 1.6499999999999999e-39 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 76.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -800000000000:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x}{F} - \frac{1}{F}\right)\right)}{-\sin B}\\ \mathbf{elif}\;F \leq \frac{-3548786815231963}{4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{elif}\;F \leq \frac{5057235284857433}{3064991081731777716716694054300618367237478244367204352}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -800000000000)
  (/ (* -1 (* F (- (* -1 (/ x F)) (/ 1 F)))) (- (sin B)))
  (if (<=
       F
       -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512)
    (+
     (- (/ x B))
     (* (/ F B) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2)))))
    (if (<=
         F
         5057235284857433/3064991081731777716716694054300618367237478244367204352)
      (/ (- x) (tan B))
      (/ (- 1 (* (cos B) x)) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -800000000000.0) {
		tmp = (-1.0 * (F * ((-1.0 * (x / F)) - (1.0 / F)))) / -sin(B);
	} else if (F <= -8.2e-259) {
		tmp = -(x / B) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else if (F <= 1.65e-39) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-800000000000.0d0)) then
        tmp = ((-1.0d0) * (f * (((-1.0d0) * (x / f)) - (1.0d0 / f)))) / -sin(b)
    else if (f <= (-8.2d-259)) then
        tmp = -(x / b) + ((f / b) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    else if (f <= 1.65d-39) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 - (cos(b) * x)) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -800000000000.0) {
		tmp = (-1.0 * (F * ((-1.0 * (x / F)) - (1.0 / F)))) / -Math.sin(B);
	} else if (F <= -8.2e-259) {
		tmp = -(x / B) + ((F / B) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else if (F <= 1.65e-39) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 - (Math.cos(B) * x)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -800000000000.0:
		tmp = (-1.0 * (F * ((-1.0 * (x / F)) - (1.0 / F)))) / -math.sin(B)
	elif F <= -8.2e-259:
		tmp = -(x / B) + ((F / B) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	elif F <= 1.65e-39:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 - (math.cos(B) * x)) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -800000000000.0)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(-1.0 * Float64(x / F)) - Float64(1.0 / F)))) / Float64(-sin(B)));
	elseif (F <= -8.2e-259)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	elseif (F <= 1.65e-39)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -800000000000.0)
		tmp = (-1.0 * (F * ((-1.0 * (x / F)) - (1.0 / F)))) / -sin(B);
	elseif (F <= -8.2e-259)
		tmp = -(x / B) + ((F / B) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	elseif (F <= 1.65e-39)
		tmp = -x / tan(B);
	else
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -800000000000], N[(N[(-1 * N[(F * N[(N[(-1 * N[(x / F), $MachinePrecision]), $MachinePrecision] - N[(1 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2), $MachinePrecision] + N[(2 * x), $MachinePrecision]), $MachinePrecision], (-N[(1 / 2), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5057235284857433/3064991081731777716716694054300618367237478244367204352], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8e11

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.9%

      \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

   if -8e11 < F < -8.1999999999999996e-259

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 35.5%

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   7. Applied rewrites 35.5%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if -8.1999999999999996e-259 < F < 1.6499999999999999e-39

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.8%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{\cos B}{\sin B}\right)\right) \]

      7. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      10. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(\frac{1}{\tan B} \cdot x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(1 \cdot \frac{1}{\tan B}\right) \cdot x\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\tan B} \cdot x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

   6. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 1.6499999999999999e-39 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{{\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F - \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \frac{\color{blue}{1} - \cos B \cdot x}{\sin B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 18: 71.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -800000000000:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x}{F} - \frac{1}{F}\right)\right)}{-\sin B}\\ \mathbf{elif}\;F \leq \frac{-3548786815231963}{4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{elif}\;F \leq 1850000000000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (- (/ x B))))
  (if (<= F -800000000000)
    (/ (* -1 (* F (- (* -1 (/ x F)) (/ 1 F)))) (- (sin B)))
    (if (<=
         F
         -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512)
      (+ t_0 (* (/ F B) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2)))))
      (if (<= F 1850000000000)
        (/ (- x) (tan B))
        (+ t_0 (/ 1 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -800000000000.0) {
		tmp = (-1.0 * (F * ((-1.0 * (x / F)) - (1.0 / F)))) / -sin(B);
	} else if (F <= -8.2e-259) {
		tmp = t_0 + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else if (F <= 1850000000000.0) {
		tmp = -x / tan(B);
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x / b)
    if (f <= (-800000000000.0d0)) then
        tmp = ((-1.0d0) * (f * (((-1.0d0) * (x / f)) - (1.0d0 / f)))) / -sin(b)
    else if (f <= (-8.2d-259)) then
        tmp = t_0 + ((f / b) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    else if (f <= 1850000000000.0d0) then
        tmp = -x / tan(b)
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -800000000000.0) {
		tmp = (-1.0 * (F * ((-1.0 * (x / F)) - (1.0 / F)))) / -Math.sin(B);
	} else if (F <= -8.2e-259) {
		tmp = t_0 + ((F / B) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else if (F <= 1850000000000.0) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x / B)
	tmp = 0
	if F <= -800000000000.0:
		tmp = (-1.0 * (F * ((-1.0 * (x / F)) - (1.0 / F)))) / -math.sin(B)
	elif F <= -8.2e-259:
		tmp = t_0 + ((F / B) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	elif F <= 1850000000000.0:
		tmp = -x / math.tan(B)
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -800000000000.0)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(-1.0 * Float64(x / F)) - Float64(1.0 / F)))) / Float64(-sin(B)));
	elseif (F <= -8.2e-259)
		tmp = Float64(t_0 + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	elseif (F <= 1850000000000.0)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x / B);
	tmp = 0.0;
	if (F <= -800000000000.0)
		tmp = (-1.0 * (F * ((-1.0 * (x / F)) - (1.0 / F)))) / -sin(B);
	elseif (F <= -8.2e-259)
		tmp = t_0 + ((F / B) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	elseif (F <= 1850000000000.0)
		tmp = -x / tan(B);
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -800000000000], N[(N[(-1 * N[(F * N[(N[(-1 * N[(x / F), $MachinePrecision]), $MachinePrecision] - N[(1 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, -3548786815231963/4327788799063369698118367518036104040602397294887907509272254128346512609744690048814426160231687683233172643784762398137404191207445999921156415311568401014033503715319849649510248592805285405106374515984066055406780647774220793764564147394699562815402300054714269682368512], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2), $MachinePrecision] + N[(2 * x), $MachinePrecision]), $MachinePrecision], (-N[(1 / 2), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1850000000000], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8e11

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.9%

      \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

   if -8e11 < F < -8.1999999999999996e-259

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 35.5%

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   7. Applied rewrites 35.5%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if -8.1999999999999996e-259 < F < 1.85e12

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.8%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{\cos B}{\sin B}\right)\right) \]

      7. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      10. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(\frac{1}{\tan B} \cdot x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(1 \cdot \frac{1}{\tan B}\right) \cdot x\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\tan B} \cdot x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

   6. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 1.85e12 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 36.4%

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{1}{\sin B} \]

   7. Applied rewrites 36.4%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 19: 71.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq \frac{-631383297997835}{332306998946228968225951765070086144}:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x}{F} - \frac{1}{F}\right)\right)}{-\sin B}\\ \mathbf{elif}\;F \leq 1850000000000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -631383297997835/332306998946228968225951765070086144)
  (/ (* -1 (* F (- (* -1 (/ x F)) (/ 1 F)))) (- (sin B)))
  (if (<= F 1850000000000)
    (/ (- x) (tan B))
    (+ (- (/ x B)) (/ 1 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9e-21) {
		tmp = (-1.0 * (F * ((-1.0 * (x / F)) - (1.0 / F)))) / -sin(B);
	} else if (F <= 1850000000000.0) {
		tmp = -x / tan(B);
	} else {
		tmp = -(x / B) + (1.0 / sin(B));
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.9d-21)) then
        tmp = ((-1.0d0) * (f * (((-1.0d0) * (x / f)) - (1.0d0 / f)))) / -sin(b)
    else if (f <= 1850000000000.0d0) then
        tmp = -x / tan(b)
    else
        tmp = -(x / b) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9e-21) {
		tmp = (-1.0 * (F * ((-1.0 * (x / F)) - (1.0 / F)))) / -Math.sin(B);
	} else if (F <= 1850000000000.0) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = -(x / B) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.9e-21:
		tmp = (-1.0 * (F * ((-1.0 * (x / F)) - (1.0 / F)))) / -math.sin(B)
	elif F <= 1850000000000.0:
		tmp = -x / math.tan(B)
	else:
		tmp = -(x / B) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.9e-21)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(-1.0 * Float64(x / F)) - Float64(1.0 / F)))) / Float64(-sin(B)));
	elseif (F <= 1850000000000.0)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.9e-21)
		tmp = (-1.0 * (F * ((-1.0 * (x / F)) - (1.0 / F)))) / -sin(B);
	elseif (F <= 1850000000000.0)
		tmp = -x / tan(B);
	else
		tmp = -(x / B) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -631383297997835/332306998946228968225951765070086144], N[(N[(-1 * N[(F * N[(N[(-1 * N[(x / F), $MachinePrecision]), $MachinePrecision] - N[(1 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 1850000000000], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(1 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.8999999999999999e-21

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)}\right)}{-\sin B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \color{blue}{\frac{1}{F}}\right)\right)}{-\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{\color{blue}{1}}{F}\right)\right)}{-\sin B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

      8. lower-/.f64 51.4%

         \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{\color{blue}{F}}\right)\right)}{-\sin B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F} - \frac{1}{F}\right)\right)}}{-\sin B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.9%

      \[\leadsto \frac{-1 \cdot \left(F \cdot \left(-1 \cdot \frac{x}{F} - \frac{1}{F}\right)\right)}{-\sin B} \]

   if -1.8999999999999999e-21 < F < 1.85e12

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.8%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{\cos B}{\sin B}\right)\right) \]

      7. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      10. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(\frac{1}{\tan B} \cdot x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(1 \cdot \frac{1}{\tan B}\right) \cdot x\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\tan B} \cdot x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

   6. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 1.85e12 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 36.4%

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{1}{\sin B} \]

   7. Applied rewrites 36.4%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 69.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq \frac{-631383297997835}{332306998946228968225951765070086144}:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1850000000000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (- (/ x B))))
  (if (<= F -631383297997835/332306998946228968225951765070086144)
    (+ t_0 (/ -1 (sin B)))
    (if (<= F 1850000000000)
      (/ (- x) (tan B))
      (+ t_0 (/ 1 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -1.9e-21) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 1850000000000.0) {
		tmp = -x / tan(B);
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x / b)
    if (f <= (-1.9d-21)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 1850000000000.0d0) then
        tmp = -x / tan(b)
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -1.9e-21) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 1850000000000.0) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x / B)
	tmp = 0
	if F <= -1.9e-21:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 1850000000000.0:
		tmp = -x / math.tan(B)
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -1.9e-21)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 1850000000000.0)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x / B);
	tmp = 0.0;
	if (F <= -1.9e-21)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 1850000000000.0)
		tmp = -x / tan(B);
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -631383297997835/332306998946228968225951765070086144], N[(t$95$0 + N[(-1 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1850000000000], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.8999999999999999e-21

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 35.7%

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{\sin B} \]

   7. Applied rewrites 35.7%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   if -1.8999999999999999e-21 < F < 1.85e12

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.8%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{\cos B}{\sin B}\right)\right) \]

      7. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      10. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(\frac{1}{\tan B} \cdot x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(1 \cdot \frac{1}{\tan B}\right) \cdot x\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\tan B} \cdot x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

   6. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 1.85e12 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 36.4%

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{1}{\sin B} \]

   7. Applied rewrites 36.4%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 63.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq \frac{-631383297997835}{332306998946228968225951765070086144}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2800000000000000113078950170696485178350383232651583988151595665389099233453737098594544983240697983719967744055948864694408373044400218446508962682114742997152086973947001523074882120184614624849536236760622355503693081243909389889239515136:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -631383297997835/332306998946228968225951765070086144)
  (+ (- (/ x B)) (/ -1 (sin B)))
  (if (<=
       F
       2800000000000000113078950170696485178350383232651583988151595665389099233453737098594544983240697983719967744055948864694408373044400218446508962682114742997152086973947001523074882120184614624849536236760622355503693081243909389889239515136)
    (/ (- x) (tan B))
    (/ -1 (- (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9e-21) {
		tmp = -(x / B) + (-1.0 / sin(B));
	} else if (F <= 2.8e+240) {
		tmp = -x / tan(B);
	} else {
		tmp = -1.0 / -sin(B);
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.9d-21)) then
        tmp = -(x / b) + ((-1.0d0) / sin(b))
    else if (f <= 2.8d+240) then
        tmp = -x / tan(b)
    else
        tmp = (-1.0d0) / -sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9e-21) {
		tmp = -(x / B) + (-1.0 / Math.sin(B));
	} else if (F <= 2.8e+240) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = -1.0 / -Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.9e-21:
		tmp = -(x / B) + (-1.0 / math.sin(B))
	elif F <= 2.8e+240:
		tmp = -x / math.tan(B)
	else:
		tmp = -1.0 / -math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.9e-21)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 2.8e+240)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(-1.0 / Float64(-sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.9e-21)
		tmp = -(x / B) + (-1.0 / sin(B));
	elseif (F <= 2.8e+240)
		tmp = -x / tan(B);
	else
		tmp = -1.0 / -sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -631383297997835/332306998946228968225951765070086144], N[((-N[(x / B), $MachinePrecision]) + N[(-1 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2800000000000000113078950170696485178350383232651583988151595665389099233453737098594544983240697983719967744055948864694408373044400218446508962682114742997152086973947001523074882120184614624849536236760622355503693081243909389889239515136], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(-1 / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.8999999999999999e-21

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.2%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 35.7%

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{\sin B} \]

   7. Applied rewrites 35.7%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   if -1.8999999999999999e-21 < F < 2.8000000000000001e240

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.8%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{\cos B}{\sin B}\right)\right) \]

      7. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      10. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(\frac{1}{\tan B} \cdot x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(1 \cdot \frac{1}{\tan B}\right) \cdot x\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\tan B} \cdot x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

   6. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 2.8000000000000001e240 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{-1}}{-\sin B} \]
   5. Step-by-step derivation

   6. Applied rewrites 17.1%

      \[\leadsto \frac{\color{blue}{-1}}{-\sin B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 56.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq 2800000000000000113078950170696485178350383232651583988151595665389099233453737098594544983240697983719967744055948864694408373044400218446508962682114742997152086973947001523074882120184614624849536236760622355503693081243909389889239515136:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<=
     F
     2800000000000000113078950170696485178350383232651583988151595665389099233453737098594544983240697983719967744055948864694408373044400218446508962682114742997152086973947001523074882120184614624849536236760622355503693081243909389889239515136)
  (/ (- x) (tan B))
  (/ -1 (- (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 2.8e+240) {
		tmp = -x / tan(B);
	} else {
		tmp = -1.0 / -sin(B);
	}
	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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 2.8d+240) then
        tmp = -x / tan(b)
    else
        tmp = (-1.0d0) / -sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 2.8e+240) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = -1.0 / -Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 2.8e+240:
		tmp = -x / math.tan(B)
	else:
		tmp = -1.0 / -math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 2.8e+240)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(-1.0 / Float64(-sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 2.8e+240)
		tmp = -x / tan(B);
	else
		tmp = -1.0 / -sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 2800000000000000113078950170696485178350383232651583988151595665389099233453737098594544983240697983719967744055948864694408373044400218446508962682114742997152086973947001523074882120184614624849536236760622355503693081243909389889239515136], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(-1 / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.8000000000000001e240

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.8%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{\cos B}{\sin B}\right)\right) \]

      7. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

      10. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(1 \cdot \left(\frac{1}{\tan B} \cdot x\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(1 \cdot \frac{1}{\tan B}\right) \cdot x\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\tan B} \cdot x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

   6. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 2.8000000000000001e240 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}}\right)\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      7. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(\mathsf{neg}\left(\sin B\right)\right) - F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\mathsf{neg}\left(\sin B\right)}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{-x}{\tan B} \cdot \left(-\sin B\right) - {\left(\left(x + x\right) - \left(-2 - F \cdot F\right)\right)}^{\frac{-1}{2}} \cdot F}{-\sin B}} \]

   4. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{-1}}{-\sin B} \]
   5. Step-by-step derivation

   6. Applied rewrites 17.1%

      \[\leadsto \frac{\color{blue}{-1}}{-\sin B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 55.9% accurate, 3.2× speedup

Math

\[\frac{-x}{\tan B} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (/ (- x) (tan B)))

C

double code(double F, double B, double x) {
	return -x / tan(B);
}

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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -x / tan(b)
end function

Java

public static double code(double F, double B, double x) {
	return -x / Math.tan(B);
}

Python

def code(F, B, x):
	return -x / math.tan(B)

Julia

function code(F, B, x)
	return Float64(Float64(-x) / tan(B))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / tan(B);
end

Wolfram

code[F_, B_, x_] := N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

   2. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

   3. lower-*.f64 N/A

      \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

   4. lower-cos.f64 N/A

      \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   5. lower-sin.f64 55.8%

      \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

4. Applied rewrites 55.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

   2. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

   3. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(1 \cdot \frac{x \cdot \cos B}{\sin B}\right) \]

   6. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{\cos B}{\sin B}\right)\right) \]

   7. div-flip-rev N/A

      \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

   8. lift-sin.f64 N/A

      \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

   9. lift-cos.f64 N/A

      \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right)\right) \]

   10. tan-quot N/A

       \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

   11. lift-tan.f64 N/A

       \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

   12. lift-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(1 \cdot \left(x \cdot \frac{1}{\tan B}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(1 \cdot \left(\frac{1}{\tan B} \cdot x\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{neg}\left(\left(1 \cdot \frac{1}{\tan B}\right) \cdot x\right) \]

   15. *-lft-identity N/A

       \[\leadsto \mathsf{neg}\left(\frac{1}{\tan B} \cdot x\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

6. Applied rewrites 55.9%

   \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]
7. Add Preprocessing

Alternative 24: 21.3% accurate, 8.4× speedup

Math

\[\left(-\frac{x}{B}\right) + \frac{F}{B} \cdot \frac{-1}{F} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (+ (- (/ x B)) (* (/ F B) (/ -1 F))))

C

double code(double F, double B, double x) {
	return -(x / B) + ((F / B) * (-1.0 / F));
}

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(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x / b) + ((f / b) * ((-1.0d0) / f))
end function

Java

public static double code(double F, double B, double x) {
	return -(x / B) + ((F / B) * (-1.0 / F));
}

Python

def code(F, B, x):
	return -(x / B) + ((F / B) * (-1.0 / F))

Julia

function code(F, B, x)
	return Float64(Float64(-Float64(x / B)) + Float64(Float64(F / B) * Float64(-1.0 / F)))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -(x / B) + ((F / B) * (-1.0 / F));
end

Wolfram

code[F_, B_, x_] := N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[(-1 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in B around 0

   \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
3. Step-by-step derivation

   1. lower-/.f64 49.2%

      \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

4. Applied rewrites 49.2%

   \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

5. Taylor expanded in F around -inf

   \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{-1}{F}} \]
6. Step-by-step derivation

   1. lower-/.f64 27.9%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{-1}{\color{blue}{F}} \]

7. Applied rewrites 27.9%

   \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{-1}{F}} \]

8. Taylor expanded in B around 0

   \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{B}} \cdot \frac{-1}{F} \]
9. Step-by-step derivation

   1. lower-/.f64 21.3%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\color{blue}{B}} \cdot \frac{-1}{F} \]

10. Applied rewrites 21.3%

    \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{B}} \cdot \frac{-1}{F} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))