VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.0% → 99.6%

Time: 6.9s

Alternatives: 22

Speedup: 1.4×

Specification

Math

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

FPCore

(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (tan B))))
  (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))

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[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 77.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (tan B))))
  (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))

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[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+30}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 110000000:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e+30)
   (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 (sin B)))
   (if (<= F 110000000.0)
     (+
      (- (/ (* x 1.0) (tan B)))
      (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
     (/ (- 1.0 (* (cos B) x)) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e+30) {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / sin(B));
	} else if (F <= 110000000.0) {
		tmp = -((x * 1.0) / tan(B)) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} 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) :: tmp
    if (f <= (-3d+30)) then
        tmp = -(x * (1.0d0 / tan(b))) + ((-1.0d0) / sin(b))
    else if (f <= 110000000.0d0) then
        tmp = -((x * 1.0d0) / tan(b)) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    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+30) {
		tmp = -(x * (1.0 / Math.tan(B))) + (-1.0 / Math.sin(B));
	} else if (F <= 110000000.0) {
		tmp = -((x * 1.0) / Math.tan(B)) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = (1.0 - (Math.cos(B) * x)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3e+30:
		tmp = -(x * (1.0 / math.tan(B))) + (-1.0 / math.sin(B))
	elif F <= 110000000.0:
		tmp = -((x * 1.0) / math.tan(B)) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	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+30)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / sin(B)));
	elseif (F <= 110000000.0)
		tmp = Float64(Float64(-Float64(Float64(x * 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)))));
	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+30)
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / sin(B));
	elseif (F <= 110000000.0)
		tmp = -((x * 1.0) / tan(B)) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	else
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e+30], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 110000000.0], N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.99999999999999978e30

   1. Initial program 58.0%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -2.99999999999999978e30 < F < 1.1e8

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.5

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

   3. Applied rewrites 99.5%

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

   if 1.1e8 < F

   1. Initial program 56.7%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -7.4 \cdot 10^{+25}:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 110000000:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<= F -7.4e+25)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 110000000.0)
       (+ t_0 (* (/ F (sin B)) (sqrt (/ 1.0 (fma F F 2.0)))))
       (/ (- 1.0 (* (cos B) x)) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B)));
	double tmp;
	if (F <= -7.4e+25) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 110000000.0) {
		tmp = t_0 + ((F / sin(B)) * sqrt((1.0 / fma(F, F, 2.0))));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -7.4e+25)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 110000000.0)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / fma(F, F, 2.0)))));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -7.4e+25], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 110000000.0], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.3999999999999998e25

   1. Initial program 58.4%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -7.3999999999999998e25 < F < 1.1e8

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 99.3

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \]

   4. Applied rewrites 99.3%

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

   if 1.1e8 < F

   1. Initial program 56.7%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 3: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+17}:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<= F -2e+17)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 1.4)
       (+ t_0 (* (/ F (sin B)) (sqrt 0.5)))
       (/ (- 1.0 (* (cos B) x)) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B)));
	double tmp;
	if (F <= -2e+17) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 1.4) {
		tmp = t_0 + ((F / sin(B)) * sqrt(0.5));
	} 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 * (1.0d0 / tan(b)))
    if (f <= (-2d+17)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 1.4d0) then
        tmp = t_0 + ((f / sin(b)) * sqrt(0.5d0))
    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 * (1.0 / Math.tan(B)));
	double tmp;
	if (F <= -2e+17) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 1.4) {
		tmp = t_0 + ((F / Math.sin(B)) * Math.sqrt(0.5));
	} else {
		tmp = (1.0 - (Math.cos(B) * x)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x * (1.0 / math.tan(B)))
	tmp = 0
	if F <= -2e+17:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 1.4:
		tmp = t_0 + ((F / math.sin(B)) * math.sqrt(0.5))
	else:
		tmp = (1.0 - (math.cos(B) * x)) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -2e+17)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 1.4)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * sqrt(0.5)));
	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 * (1.0 / tan(B)));
	tmp = 0.0;
	if (F <= -2e+17)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 1.4)
		tmp = t_0 + ((F / sin(B)) * sqrt(0.5));
	else
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -2e+17], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e17

   1. Initial program 59.4%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -2e17 < F < 1.3999999999999999

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 99.3

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \]

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in F around 0

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

   7. Applied rewrites 96.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \sqrt{0.5} \]

   if 1.3999999999999999 < F

   1. Initial program 57.3%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 99.4

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

   4. Applied rewrites 99.4%

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

Alternative 4: 92.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\\ \mathbf{if}\;F \leq -2 \cdot 10^{+17}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-67}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{t\_0}}\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F \cdot {t\_0}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (fma 2.0 x (fma F F 2.0))))
   (if (<= F -2e+17)
     (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 (sin B)))
     (if (<= F 4e-67)
       (+ (/ (- x) (tan B)) (* (/ F B) (sqrt (/ 1.0 t_0))))
       (if (<= F 0.0024)
         (+ (- (/ x B)) (/ (* F (pow t_0 -0.5)) (sin B)))
         (/ (- 1.0 (* (cos B) x)) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = fma(2.0, x, fma(F, F, 2.0));
	double tmp;
	if (F <= -2e+17) {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / sin(B));
	} else if (F <= 4e-67) {
		tmp = (-x / tan(B)) + ((F / B) * sqrt((1.0 / t_0)));
	} else if (F <= 0.0024) {
		tmp = -(x / B) + ((F * pow(t_0, -0.5)) / sin(B));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(2.0, x, fma(F, F, 2.0))
	tmp = 0.0
	if (F <= -2e+17)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / sin(B)));
	elseif (F <= 4e-67)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(Float64(F / B) * sqrt(Float64(1.0 / t_0))));
	elseif (F <= 0.0024)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F * (t_0 ^ -0.5)) / sin(B)));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+17], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4e-67], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0024], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F * N[Power[t$95$0, -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2e17

   1. Initial program 59.4%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -2e17 < F < 3.99999999999999977e-67

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 83.7

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 83.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

      7. *-rgt-identity N/A

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

      8. lift-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-tan.f64 83.8

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

      11. lift-+.f64 N/A

          \[\leadsto \frac{-x}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   6. Applied rewrites 83.8%

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

   if 3.99999999999999977e-67 < F < 0.00239999999999999979

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} \]

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 79.5

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   6. Applied rewrites 79.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   if 0.00239999999999999979 < F

   1. Initial program 57.9%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 98.5

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

   4. Applied rewrites 98.5%

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

Alternative 5: 92.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ t_1 := \mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\\ \mathbf{if}\;F \leq -2 \cdot 10^{+17}:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-67}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{t\_1}}\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F \cdot {t\_1}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)) (t_1 (fma 2.0 x (fma F F 2.0))))
   (if (<= F -2e+17)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F 4e-67)
       (+ (/ (- x) (tan B)) (* (/ F B) (sqrt (/ 1.0 t_1))))
       (if (<= F 0.0024)
         (+ (- (/ x B)) (/ (* F (pow t_1 -0.5)) (sin B)))
         (/ (- 1.0 t_0) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double t_1 = fma(2.0, x, fma(F, F, 2.0));
	double tmp;
	if (F <= -2e+17) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= 4e-67) {
		tmp = (-x / tan(B)) + ((F / B) * sqrt((1.0 / t_1)));
	} else if (F <= 0.0024) {
		tmp = -(x / B) + ((F * pow(t_1, -0.5)) / sin(B));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	t_1 = fma(2.0, x, fma(F, F, 2.0))
	tmp = 0.0
	if (F <= -2e+17)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= 4e-67)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(Float64(F / B) * sqrt(Float64(1.0 / t_1))));
	elseif (F <= 0.0024)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F * (t_1 ^ -0.5)) / sin(B)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+17], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 4e-67], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0024], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F * N[Power[t$95$1, -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2e17

   1. Initial program 59.4%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -2e17 < F < 3.99999999999999977e-67

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 83.7

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 83.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

      7. *-rgt-identity N/A

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

      8. lift-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-tan.f64 83.8

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

      11. lift-+.f64 N/A

          \[\leadsto \frac{-x}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   6. Applied rewrites 83.8%

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

   if 3.99999999999999977e-67 < F < 0.00239999999999999979

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} \]

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 79.5

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   6. Applied rewrites 79.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   if 0.00239999999999999979 < F

   1. Initial program 57.9%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 98.5

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

   4. Applied rewrites 98.5%

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

Alternative 6: 81.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\\ t_1 := \left(-\frac{x}{B}\right) + \frac{F \cdot {t\_0}^{-0.5}}{\sin B}\\ \mathbf{if}\;F \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -5 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-67}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{t\_0}}\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (fma 2.0 x (fma F F 2.0)))
        (t_1 (+ (- (/ x B)) (/ (* F (pow t_0 -0.5)) (sin B)))))
   (if (<= F -1.35e+154)
     (/ (- -1.0 x) B)
     (if (<= F -5e-24)
       t_1
       (if (<= F 4e-67)
         (+ (/ (- x) (tan B)) (* (/ F B) (sqrt (/ 1.0 t_0))))
         (if (<= F 0.0024) t_1 (/ (- 1.0 (* (cos B) x)) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = fma(2.0, x, fma(F, F, 2.0));
	double t_1 = -(x / B) + ((F * pow(t_0, -0.5)) / sin(B));
	double tmp;
	if (F <= -1.35e+154) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -5e-24) {
		tmp = t_1;
	} else if (F <= 4e-67) {
		tmp = (-x / tan(B)) + ((F / B) * sqrt((1.0 / t_0)));
	} else if (F <= 0.0024) {
		tmp = t_1;
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(2.0, x, fma(F, F, 2.0))
	t_1 = Float64(Float64(-Float64(x / B)) + Float64(Float64(F * (t_0 ^ -0.5)) / sin(B)))
	tmp = 0.0
	if (F <= -1.35e+154)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -5e-24)
		tmp = t_1;
	elseif (F <= 4e-67)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(Float64(F / B) * sqrt(Float64(1.0 / t_0))));
	elseif (F <= 0.0024)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[(x / B), $MachinePrecision]) + N[(N[(F * N[Power[t$95$0, -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.35e+154], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -5e-24], t$95$1, If[LessEqual[F, 4e-67], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0024], t$95$1, N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.35000000000000003e154

   1. Initial program 34.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 27.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.0%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.35000000000000003e154 < F < -4.9999999999999998e-24 or 3.99999999999999977e-67 < F < 0.00239999999999999979

   1. Initial program 93.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} \]

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 78.0

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   6. Applied rewrites 78.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   if -4.9999999999999998e-24 < F < 3.99999999999999977e-67

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 84.8

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 84.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

      7. *-rgt-identity N/A

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

      8. lift-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-tan.f64 84.9

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

      11. lift-+.f64 N/A

          \[\leadsto \frac{-x}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   6. Applied rewrites 84.9%

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

   if 0.00239999999999999979 < F

   1. Initial program 57.9%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 98.5

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

   4. Applied rewrites 98.5%

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

Alternative 7: 77.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ t_1 := \frac{-1 - x}{B}\\ t_2 := \frac{F}{\sin B}\\ t_3 := \left(-x \cdot \frac{1}{\tan B}\right) + t\_2 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -60:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (+
          (/ (- x) (tan B))
          (* (/ F B) (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))))))
        (t_1 (/ (- -1.0 x) B))
        (t_2 (/ F (sin B)))
        (t_3
         (+
          (- (* x (/ 1.0 (tan B))))
          (* t_2 (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))))
   (if (<= t_3 (- INFINITY))
     t_1
     (if (<= t_3 -60.0)
       t_0
       (if (<= t_3 5.0)
         (* (sqrt (/ 1.0 (fma F F 2.0))) t_2)
         (if (<= t_3 INFINITY) t_0 t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = (-x / tan(B)) + ((F / B) * sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))));
	double t_1 = (-1.0 - x) / B;
	double t_2 = F / sin(B);
	double t_3 = -(x * (1.0 / tan(B))) + (t_2 * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_3 <= -60.0) {
		tmp = t_0;
	} else if (t_3 <= 5.0) {
		tmp = sqrt((1.0 / fma(F, F, 2.0))) * t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(-x) / tan(B)) + Float64(Float64(F / B) * sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0))))))
	t_1 = Float64(Float64(-1.0 - x) / B)
	t_2 = Float64(F / sin(B))
	t_3 = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(t_2 * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_3 <= -60.0)
		tmp = t_0;
	elseif (t_3 <= 5.0)
		tmp = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * t_2);
	elseif (t_3 <= Inf)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$2 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(t$95$2 * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$1, If[LessEqual[t$95$3, -60.0], t$95$0, If[LessEqual[t$95$3, 5.0], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -inf.0 or +inf.0 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 20.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.8%

      \[\leadsto \frac{-1 - x}{B} \]

   if -inf.0 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -60 or 5 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < +inf.0

   1. Initial program 96.5%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 94.8

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 94.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

      7. *-rgt-identity N/A

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

      8. lift-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-tan.f64 95.0

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

      11. lift-+.f64 N/A

          \[\leadsto \frac{-x}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   6. Applied rewrites 95.0%

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

   if -60 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 5

   1. Initial program 75.3%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      6. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{\sin B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B} \]

      8. lift-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B} \]

      9. lift-/.f64 47.5

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\color{blue}{\sin B}} \]

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 77.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}\\ t_1 := -x \cdot \frac{1}{\tan B}\\ t_2 := t\_1 + \frac{F}{B} \cdot t\_0\\ t_3 := \frac{-1 - x}{B}\\ t_4 := \frac{F}{\sin B}\\ t_5 := t\_1 + t\_4 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq -60:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 5:\\ \;\;\;\;t\_0 \cdot t\_4\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (fma F F 2.0))))
        (t_1 (- (* x (/ 1.0 (tan B)))))
        (t_2 (+ t_1 (* (/ F B) t_0)))
        (t_3 (/ (- -1.0 x) B))
        (t_4 (/ F (sin B)))
        (t_5
         (+ t_1 (* t_4 (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))))
   (if (<= t_5 (- INFINITY))
     t_3
     (if (<= t_5 -60.0)
       t_2
       (if (<= t_5 5.0) (* t_0 t_4) (if (<= t_5 INFINITY) t_2 t_3))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / fma(F, F, 2.0)));
	double t_1 = -(x * (1.0 / tan(B)));
	double t_2 = t_1 + ((F / B) * t_0);
	double t_3 = (-1.0 - x) / B;
	double t_4 = F / sin(B);
	double t_5 = t_1 + (t_4 * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_5 <= -60.0) {
		tmp = t_2;
	} else if (t_5 <= 5.0) {
		tmp = t_0 * t_4;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / fma(F, F, 2.0)))
	t_1 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	t_2 = Float64(t_1 + Float64(Float64(F / B) * t_0))
	t_3 = Float64(Float64(-1.0 - x) / B)
	t_4 = Float64(F / sin(B))
	t_5 = Float64(t_1 + Float64(t_4 * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_5 <= -60.0)
		tmp = t_2;
	elseif (t_5 <= 5.0)
		tmp = Float64(t_0 * t_4);
	elseif (t_5 <= Inf)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$2 = N[(t$95$1 + N[(N[(F / B), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$4 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 + N[(t$95$4 * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], t$95$3, If[LessEqual[t$95$5, -60.0], t$95$2, If[LessEqual[t$95$5, 5.0], N[(t$95$0 * t$95$4), $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$2, t$95$3]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -inf.0 or +inf.0 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 20.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.8%

      \[\leadsto \frac{-1 - x}{B} \]

   if -inf.0 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -60 or 5 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < +inf.0

   1. Initial program 96.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 96.4

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \]

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{B}} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 94.8%

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

   if -60 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 5

   1. Initial program 75.3%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      6. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{\sin B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B} \]

      8. lift-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B} \]

      9. lift-/.f64 47.5

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\color{blue}{\sin B}} \]

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-11}:\\ \;\;\;\;-\cos B \cdot \frac{x}{\sin B}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-24}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -5.4e-11)
   (- (* (cos B) (/ x (sin B))))
   (if (<= x 1.1e-24)
     (+ (- (/ x B)) (/ (* F (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (sin B)))
     (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -5.4e-11) {
		tmp = -(cos(B) * (x / sin(B)));
	} else if (x <= 1.1e-24) {
		tmp = -(x / B) + ((F * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)) / sin(B));
	} else {
		tmp = -(x * (1.0 / tan(B))) + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -5.4e-11)
		tmp = Float64(-Float64(cos(B) * Float64(x / sin(B))));
	elseif (x <= 1.1e-24)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)) / sin(B)));
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -5.4e-11], (-N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[x, 1.1e-24], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.40000000000000009e-11

   1. Initial program 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 73.4

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

   3. Applied rewrites 73.4%

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

   4. Taylor expanded in F around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sin.f64 93.1

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

   6. Applied rewrites 93.1%

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

   if -5.40000000000000009e-11 < x < 1.10000000000000001e-24

   1. Initial program 72.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} \]

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 64.1

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   6. Applied rewrites 64.1%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   if 1.10000000000000001e-24 < x

   1. Initial program 86.6%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 85.4

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 94.4

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

   7. Applied rewrites 94.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{B}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-24}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 B))))
   (if (<= x -1.75e-10)
     t_0
     (if (<= x 1.1e-24)
       (+ (- (/ x B)) (/ (* F (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (sin B)))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B))) + (1.0 / B);
	double tmp;
	if (x <= -1.75e-10) {
		tmp = t_0;
	} else if (x <= 1.1e-24) {
		tmp = -(x / B) + ((F * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)) / sin(B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(1.0 / B))
	tmp = 0.0
	if (x <= -1.75e-10)
		tmp = t_0;
	elseif (x <= 1.1e-24)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)) / sin(B)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.75e-10], t$95$0, If[LessEqual[x, 1.1e-24], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7499999999999999e-10 or 1.10000000000000001e-24 < x

   1. Initial program 82.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 81.5

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 93.9

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

   7. Applied rewrites 93.9%

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

   if -1.7499999999999999e-10 < x < 1.10000000000000001e-24

   1. Initial program 72.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} \]

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 64.1

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   6. Applied rewrites 64.1%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 71.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B}\\ t_1 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -6.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -2 \cdot 10^{+17}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + t\_0 \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-33}:\\ \;\;\;\;t\_1 + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}\\ \mathbf{elif}\;F \leq 108000000:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sin B))) (t_1 (- (* x (/ 1.0 (tan B))))))
   (if (<= F -6.8e+69)
     (/ (- -1.0 x) B)
     (if (<= F -2e+17)
       (+ (- (/ x B)) (* t_0 (/ -1.0 F)))
       (if (<= F 3.1e-33)
         (+ t_1 (* (/ F B) (sqrt (/ 1.0 (fma 2.0 x 2.0)))))
         (if (<= F 108000000.0)
           (* (sqrt (/ 1.0 (fma F F 2.0))) t_0)
           (+ t_1 (/ 1.0 B))))))))

C

double code(double F, double B, double x) {
	double t_0 = F / sin(B);
	double t_1 = -(x * (1.0 / tan(B)));
	double tmp;
	if (F <= -6.8e+69) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -2e+17) {
		tmp = -(x / B) + (t_0 * (-1.0 / F));
	} else if (F <= 3.1e-33) {
		tmp = t_1 + ((F / B) * sqrt((1.0 / fma(2.0, x, 2.0))));
	} else if (F <= 108000000.0) {
		tmp = sqrt((1.0 / fma(F, F, 2.0))) * t_0;
	} else {
		tmp = t_1 + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F / sin(B))
	t_1 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -6.8e+69)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -2e+17)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(t_0 * Float64(-1.0 / F)));
	elseif (F <= 3.1e-33)
		tmp = Float64(t_1 + Float64(Float64(F / B) * sqrt(Float64(1.0 / fma(2.0, x, 2.0)))));
	elseif (F <= 108000000.0)
		tmp = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * t_0);
	else
		tmp = Float64(t_1 + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -6.8e+69], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -2e+17], N[((-N[(x / B), $MachinePrecision]) + N[(t$95$0 * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.1e-33], N[(t$95$1 + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 108000000.0], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$1 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -6.79999999999999973e69

   1. Initial program 52.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 34.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.4%

      \[\leadsto \frac{-1 - x}{B} \]

   if -6.79999999999999973e69 < F < -2e17

   1. Initial program 96.2%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 77.8

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

   7. Applied rewrites 77.8%

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

   if -2e17 < F < 3.09999999999999997e-33

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 83.0

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 81.7

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \]

   7. Applied rewrites 81.7%

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

   if 3.09999999999999997e-33 < F < 1.08e8

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      6. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{\sin B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B} \]

      8. lift-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B} \]

      9. lift-/.f64 62.3

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\color{blue}{\sin B}} \]

   4. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}} \]

   if 1.08e8 < F

   1. Initial program 56.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 44.2

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 73.5

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

   7. Applied rewrites 73.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\color{blue}{B}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 12: 69.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{B}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-50}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 B))))
   (if (<= x -2e-50)
     t_0
     (if (<= x 1.25e-33) (* (sqrt (/ 1.0 (fma F F 2.0))) (/ F (sin B))) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B))) + (1.0 / B);
	double tmp;
	if (x <= -2e-50) {
		tmp = t_0;
	} else if (x <= 1.25e-33) {
		tmp = sqrt((1.0 / fma(F, F, 2.0))) * (F / sin(B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(1.0 / B))
	tmp = 0.0
	if (x <= -2e-50)
		tmp = t_0;
	elseif (x <= 1.25e-33)
		tmp = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * Float64(F / sin(B)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e-50], t$95$0, If[LessEqual[x, 1.25e-33], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000002e-50 or 1.25000000000000007e-33 < x

   1. Initial program 81.6%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 78.9

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 88.1

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

   7. Applied rewrites 88.1%

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

   if -2.00000000000000002e-50 < x < 1.25000000000000007e-33

   1. Initial program 72.7%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      6. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{\sin B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B} \]

      8. lift-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B} \]

      9. lift-/.f64 51.6

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\color{blue}{\sin B}} \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 50.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 4.8:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 4.8)
   (/ (- (* (/ 1.0 (sqrt (fma F F (fma 2.0 x 2.0)))) F) x) B)
   (* (sqrt (/ 1.0 (fma F F 2.0))) (/ F (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 4.8) {
		tmp = (((1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B;
	} else {
		tmp = sqrt((1.0 / fma(F, F, 2.0))) * (F / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 4.8)
		tmp = Float64(Float64(Float64(Float64(1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * Float64(F / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 4.8], N[(N[(N[(N[(1.0 / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.79999999999999982

   1. Initial program 74.5%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 57.0%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      6. sqrt-div N/A

         \[\leadsto \frac{\frac{\sqrt{1}}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      10. pow2 N/A

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

      11. count-2-rev N/A

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

      12. associate-+l+ N/A

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

      13. pow2 N/A

          \[\leadsto \frac{\frac{1}{\sqrt{F \cdot F + \left(2 \cdot x + 2\right)}} \cdot F - x}{B} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2 \cdot x + 2\right)}} \cdot F - x}{B} \]

      15. lower-fma.f64 57.0

          \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B} \]

   6. Applied rewrites 57.0%

      \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B} \]

   if 4.79999999999999982 < B

   1. Initial program 84.5%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      6. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{\sin B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B} \]

      8. lift-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B} \]

      9. lift-/.f64 30.3

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\color{blue}{\sin B}} \]

   4. Applied rewrites 30.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 50.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+29}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+29)
   (/ (- -1.0 x) B)
   (if (<= F 7e+117)
     (/ (- (* (sqrt (/ 1.0 (+ (fma F F (+ x x)) 2.0))) F) x) B)
     (+
      (- (* x (/ 1.0 (* (fma 0.3333333333333333 (* B B) 1.0) B))))
      (/ 1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+29) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 7e+117) {
		tmp = ((sqrt((1.0 / (fma(F, F, (x + x)) + 2.0))) * F) - x) / B;
	} else {
		tmp = -(x * (1.0 / (fma(0.3333333333333333, (B * B), 1.0) * B))) + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+29)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 7e+117)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / Float64(fma(F, F, Float64(x + x)) + 2.0))) * F) - x) / B);
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / Float64(fma(0.3333333333333333, Float64(B * B), 1.0) * B)))) + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+29], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7e+117], N[(N[(N[(N[Sqrt[N[(1.0 / N[(N[(F * F + N[(x + x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-N[(x * N[(1.0 / N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.99999999999999983e29

   1. Initial program 58.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 36.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.99999999999999983e29 < F < 6.99999999999999965e117

   1. Initial program 98.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   if 6.99999999999999965e117 < F

   1. Initial program 39.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 34.7

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 34.7%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot \color{blue}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot \color{blue}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      3. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\left(\frac{1}{3} \cdot {B}^{2} + 1\right) \cdot B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, 1\right) \cdot B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      5. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      6. lift-*.f64 11.3

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   7. Applied rewrites 11.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   8. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}\right) + \frac{1}{\color{blue}{B}} \]
   9. Step-by-step derivation

      1. lift-/.f64 51.3

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) + \frac{1}{B} \]

   10. Applied rewrites 51.3%

       \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) + \frac{1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 50.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e+154)
   (/ (- -1.0 x) B)
   (if (<= F 5e+117)
     (/ (- (* (/ 1.0 (sqrt (fma F F (fma 2.0 x 2.0)))) F) x) B)
     (+
      (- (* x (/ 1.0 (* (fma 0.3333333333333333 (* B B) 1.0) B))))
      (/ 1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e+154) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5e+117) {
		tmp = (((1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B;
	} else {
		tmp = -(x * (1.0 / (fma(0.3333333333333333, (B * B), 1.0) * B))) + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e+154)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 5e+117)
		tmp = Float64(Float64(Float64(Float64(1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / Float64(fma(0.3333333333333333, Float64(B * B), 1.0) * B)))) + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e+154], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5e+117], N[(N[(N[(N[(1.0 / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-N[(x * N[(1.0 / N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.00000000000000004e154

   1. Initial program 34.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 27.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.0%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.00000000000000004e154 < F < 4.99999999999999983e117

   1. Initial program 96.6%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      6. sqrt-div N/A

         \[\leadsto \frac{\frac{\sqrt{1}}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      10. pow2 N/A

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

      11. count-2-rev N/A

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

      12. associate-+l+ N/A

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

      13. pow2 N/A

          \[\leadsto \frac{\frac{1}{\sqrt{F \cdot F + \left(2 \cdot x + 2\right)}} \cdot F - x}{B} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2 \cdot x + 2\right)}} \cdot F - x}{B} \]

      15. lower-fma.f64 51.1

          \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B} \]

   6. Applied rewrites 51.1%

      \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B} \]

   if 4.99999999999999983e117 < F

   1. Initial program 39.5%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 34.8

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 34.8%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot \color{blue}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot \color{blue}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      3. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\left(\frac{1}{3} \cdot {B}^{2} + 1\right) \cdot B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, 1\right) \cdot B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      5. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      6. lift-*.f64 11.3

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   7. Applied rewrites 11.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   8. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}\right) + \frac{1}{\color{blue}{B}} \]
   9. Step-by-step derivation

      1. lift-/.f64 51.3

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) + \frac{1}{B} \]

   10. Applied rewrites 51.3%

       \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) + \frac{1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 50.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e+154)
   (/ (- -1.0 x) B)
   (if (<= F 0.0024)
     (/ (- (* (/ 1.0 (sqrt (fma F F (fma 2.0 x 2.0)))) F) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e+154) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 0.0024) {
		tmp = (((1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e+154)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 0.0024)
		tmp = Float64(Float64(Float64(Float64(1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e+154], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.0024], N[(N[(N[(N[(1.0 / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.00000000000000004e154

   1. Initial program 34.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 27.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.0%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.00000000000000004e154 < F < 0.00239999999999999979

   1. Initial program 97.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.4%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      6. sqrt-div N/A

         \[\leadsto \frac{\frac{\sqrt{1}}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      10. pow2 N/A

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

      11. count-2-rev N/A

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

      12. associate-+l+ N/A

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

      13. pow2 N/A

          \[\leadsto \frac{\frac{1}{\sqrt{F \cdot F + \left(2 \cdot x + 2\right)}} \cdot F - x}{B} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2 \cdot x + 2\right)}} \cdot F - x}{B} \]

      15. lower-fma.f64 51.4

          \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B} \]

   6. Applied rewrites 51.4%

      \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B} \]

   if 0.00239999999999999979 < F

   1. Initial program 57.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 36.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.3%

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

Alternative 17: 49.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 102000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{F}{B}, \sqrt{0.5}, \frac{-x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.4e+25)
   (/ (- -1.0 x) B)
   (if (<= F 102000000.0)
     (fma (/ F B) (sqrt 0.5) (/ (- x) B))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.4e+25) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 102000000.0) {
		tmp = fma((F / B), sqrt(0.5), (-x / B));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.4e+25)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 102000000.0)
		tmp = fma(Float64(F / B), sqrt(0.5), Float64(Float64(-x) / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.4e+25], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 102000000.0], N[(N[(F / B), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + N[((-x) / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.3999999999999998e25

   1. Initial program 58.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.0%

      \[\leadsto \frac{-1 - x}{B} \]

   if -7.3999999999999998e25 < F < 1.02e8

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 33.5

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 33.5%

      \[\leadsto \frac{-x}{B} \]

   8. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \frac{x}{B} + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} + -1 \cdot \color{blue}{\frac{x}{B}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2 + 2 \cdot x}}, -1 \cdot \frac{x}{B}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2 + 2 \cdot x}}, -1 \cdot \frac{x}{B}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2 + 2 \cdot x}}, -1 \cdot \frac{x}{B}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2 + 2 \cdot x}}, -1 \cdot \frac{x}{B}\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2 \cdot x + 2}}, -1 \cdot \frac{x}{B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, -1 \cdot \frac{x}{B}\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{-1 \cdot x}{B}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{\mathsf{neg}\left(x\right)}{B}\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{-x}{B}\right) \]

      11. lower-/.f64 49.8

          \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{-x}{B}\right) \]

   10. Applied rewrites 49.8%

       \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}}, \frac{-x}{B}\right) \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2}}, \frac{-x}{B}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 49.8%

       \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{0.5}, \frac{-x}{B}\right) \]

   if 1.02e8 < F

   1. Initial program 56.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 35.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.9%

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

Alternative 18: 49.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 102000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{F}{B}, \sqrt{0.5}, \frac{-x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.4e+25)
   (/ (- -1.0 x) B)
   (if (<= F 102000000.0)
     (fma (/ F B) (sqrt 0.5) (/ (- x) B))
     (/ (- (fma (/ (fma 2.0 x 2.0) (* F F)) -0.5 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.4e+25) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 102000000.0) {
		tmp = fma((F / B), sqrt(0.5), (-x / B));
	} else {
		tmp = (fma((fma(2.0, x, 2.0) / (F * F)), -0.5, 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.4e+25)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 102000000.0)
		tmp = fma(Float64(F / B), sqrt(0.5), Float64(Float64(-x) / B));
	else
		tmp = Float64(Float64(fma(Float64(fma(2.0, x, 2.0) / Float64(F * F)), -0.5, 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.4e+25], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 102000000.0], N[(N[(F / B), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + N[((-x) / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.3999999999999998e25

   1. Initial program 58.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.0%

      \[\leadsto \frac{-1 - x}{B} \]

   if -7.3999999999999998e25 < F < 1.02e8

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 33.5

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 33.5%

      \[\leadsto \frac{-x}{B} \]

   8. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \frac{x}{B} + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} + -1 \cdot \color{blue}{\frac{x}{B}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2 + 2 \cdot x}}, -1 \cdot \frac{x}{B}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2 + 2 \cdot x}}, -1 \cdot \frac{x}{B}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2 + 2 \cdot x}}, -1 \cdot \frac{x}{B}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2 + 2 \cdot x}}, -1 \cdot \frac{x}{B}\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2 \cdot x + 2}}, -1 \cdot \frac{x}{B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, -1 \cdot \frac{x}{B}\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{-1 \cdot x}{B}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{\mathsf{neg}\left(x\right)}{B}\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{-x}{B}\right) \]

      11. lower-/.f64 49.8

          \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{-x}{B}\right) \]

   10. Applied rewrites 49.8%

       \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}}, \frac{-x}{B}\right) \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2}}, \frac{-x}{B}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 49.8%

       \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{0.5}, \frac{-x}{B}\right) \]

   if 1.02e8 < F

   1. Initial program 56.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 35.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + 2}} \cdot F - x}{B} \]
   6. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(x + x\right) + 2}} \cdot F - x}{B} \]

      2. lift-+.f64 12.3

         \[\leadsto \frac{\sqrt{\frac{1}{\left(x + x\right) + 2}} \cdot F - x}{B} \]

   7. Applied rewrites 12.3%

      \[\leadsto \frac{\sqrt{\frac{1}{\left(x + x\right) + 2}} \cdot F - x}{B} \]

   8. Taylor expanded in F around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. div-add N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

   10. Applied rewrites 50.9%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 43.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.00115:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -1.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.00115)
   (/ (- -1.0 x) B)
   (if (<= F -1.2e-146)
     (/ (* F (sqrt 0.5)) B)
     (if (<= F 5.2e-72) (/ (- x) B) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.00115) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -1.2e-146) {
		tmp = (F * sqrt(0.5)) / B;
	} else if (F <= 5.2e-72) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / 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 <= (-0.00115d0)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= (-1.2d-146)) then
        tmp = (f * sqrt(0.5d0)) / b
    else if (f <= 5.2d-72) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.00115) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -1.2e-146) {
		tmp = (F * Math.sqrt(0.5)) / B;
	} else if (F <= 5.2e-72) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.00115:
		tmp = (-1.0 - x) / B
	elif F <= -1.2e-146:
		tmp = (F * math.sqrt(0.5)) / B
	elif F <= 5.2e-72:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.00115)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -1.2e-146)
		tmp = Float64(Float64(F * sqrt(0.5)) / B);
	elseif (F <= 5.2e-72)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.00115)
		tmp = (-1.0 - x) / B;
	elseif (F <= -1.2e-146)
		tmp = (F * sqrt(0.5)) / B;
	elseif (F <= 5.2e-72)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.00115], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -1.2e-146], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5.2e-72], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.00115

   1. Initial program 61.5%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 38.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.8%

      \[\leadsto \frac{-1 - x}{B} \]

   if -0.00115 < F < -1.2000000000000001e-146

   1. Initial program 99.5%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.8

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 26.8%

      \[\leadsto \frac{-x}{B} \]

   8. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \frac{x}{B} + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} + -1 \cdot \color{blue}{\frac{x}{B}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2 + 2 \cdot x}}, -1 \cdot \frac{x}{B}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2 + 2 \cdot x}}, -1 \cdot \frac{x}{B}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2 + 2 \cdot x}}, -1 \cdot \frac{x}{B}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2 + 2 \cdot x}}, -1 \cdot \frac{x}{B}\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{2 \cdot x + 2}}, -1 \cdot \frac{x}{B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, -1 \cdot \frac{x}{B}\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{-1 \cdot x}{B}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{\mathsf{neg}\left(x\right)}{B}\right) \]

      10. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{-x}{B}\right) \]

      11. lower-/.f64 50.4

          \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{-x}{B}\right) \]

   10. Applied rewrites 50.4%

       \[\leadsto \mathsf{fma}\left(\frac{F}{B}, \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}}, \frac{-x}{B}\right) \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{B} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{B} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{B} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{B} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{B} \]

       5. metadata-eval 26.3

          \[\leadsto \frac{F \cdot \sqrt{0.5}}{B} \]

   13. Applied rewrites 26.3%

       \[\leadsto \frac{F \cdot \sqrt{0.5}}{B} \]

   if -1.2000000000000001e-146 < F < 5.19999999999999992e-72

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.8

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 39.8%

      \[\leadsto \frac{-x}{B} \]

   if 5.19999999999999992e-72 < F

   1. Initial program 64.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 38.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.8%

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

Alternative 20: 42.9% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.6e-34)
   (/ (- -1.0 x) B)
   (if (<= F 5.2e-72) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e-34) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.2e-72) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / 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 <= (-1.6d-34)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 5.2d-72) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e-34) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.2e-72) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.6e-34:
		tmp = (-1.0 - x) / B
	elif F <= 5.2e-72:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.6e-34)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 5.2e-72)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.6e-34)
		tmp = (-1.0 - x) / B;
	elseif (F <= 5.2e-72)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.6e-34], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5.2e-72], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.60000000000000001e-34

   1. Initial program 64.0%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 39.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.1%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.60000000000000001e-34 < F < 5.19999999999999992e-72

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 37.1

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 37.1%

      \[\leadsto \frac{-x}{B} \]

   if 5.19999999999999992e-72 < F

   1. Initial program 64.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 38.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.8%

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

Alternative 21: 35.9% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.6e-34) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e-34) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / 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 <= (-1.6d-34)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e-34) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.6e-34:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.6e-34)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.6e-34)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.6e-34], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.60000000000000001e-34

   1. Initial program 64.0%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 39.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.1%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.60000000000000001e-34 < F

   1. Initial program 82.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 45.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 30.9

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 30.9%

      \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 28.5% accurate, 21.7× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / 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 / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

code[F_, B_, x_] := N[((-x) / B), $MachinePrecision]

Derivation

1. Initial program 77.0%

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

2. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 43.6%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

5. Taylor expanded in F around 0

   \[\leadsto \frac{-1 \cdot x}{B} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f64 28.5

      \[\leadsto \frac{-x}{B} \]

7. Applied rewrites 28.5%

   \[\leadsto \frac{-x}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025114 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))